American-Type Options. Volume 1 American-Type Options: Stochastic Approximation Methods, Volume 1 9783110329827, 9783110329674

Table of contents :
Preface
1 Multivariate modulated Markov log-price processes (LPP)
1.1 Markov LPP
1.2 LPP represented by random walks
1.3 Autoregressive LPP
1.4 Autoregressive stochastic volatility LPP
2 American-type options
2.1 American-type options
2.2 Pay-off functions
2.3 Reward and log-reward functions
2.4 Optimal stopping times
2.5 American-type knockout options
3 Backward recurrence reward algorithms
3.1 Binomial tree reward algorithms
3.2 Trinomial tree reward algorithms
3.3 Random walk reward algorithms
3.4 Markov chain reward algorithms
4 Upper bounds for option rewards
4.1 Markov LPP with bounded characteristics
4.2 LPP represented by random walks
4.3 Markov LPP with unbounded characteristics
4.4 Univariate Markov Gaussian LPP
4.5 Multivariate modulated Markov Gaussian LPP
5 Convergence of option rewards – I
5.1 Asymptotically uniform upper bounds for rewards – I
5.2 Modulated Markov LPP with bounded characteristics
5.3 LPP represented by modulated random walks
6 Convergence of option rewards – II
6.1 Asymptotically uniform upper bounds for rewards – II
6.2 Univariate modulated LPP with unbounded characteristics
6.3 Asymptotically uniform upper bounds for rewards – III
6.4 Multivariate modulated LPP with unbounded characteristics
6.5 Conditions of convergence for Markov price processes
7 Space-skeleton reward approximations
7.1 Atomic approximation models
7.2 Univariate Markov LPP with bounded characteristics
7.3 MultivariateMarkov LPP with bounded characteristics
7.4 LPP represented by multivariate modulated random walks
7.5 MultivariateMarkov LPP with unbounded characteristics
8 Convergence of rewards for Markov Gaussian LPP
8.1 Univariate Markov Gaussian LPP
8.2 Multivariate modulated Markov Gaussian LPP
8.3 Markov Gaussian LPP with estimated characteristics
8.4 Skeleton reward approximations for Markov Gaussian LPP
8.5 LPP represented by Gaussian random walks
9 Tree-type approximations for Markov Gaussian LPP
9.1 Univariate binomial tree approximations
9.2 Multivariate binomial tree approximations
9.3 Multivariate trinomial tree approximations
9.4 Inhomogeneous in space binomial approximations
9.5 Inhomogeneous in time and space trinomial approximations
10 Convergence of tree-type reward approximations
10.1 Univariate binomial tree approximation models
10.2 Multivariate homogeneous in space tree models
10.3 Univariate inhomogeneous in space tree models
10.4 Multivariate inhomogeneous in space tree models
Bibliographical Remarks
Bibliography
Index

Citation preview

Dmitrii S. Silvestrov American-Type Options

De Gruyter Studies in Mathematics

| Edited by Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany

Volume 56

Dmitrii S. Silvestrov

American-Type Options | Stochastic Approximation Methods Volume 1

Mathematics Subject Classification 2010 Primary: 91G20, 91G60, 60J05, 60G40, 60J22; Secondary: 60J10, 60G15, 60G50, 65C40, 62L15 Author Prof. Dr. Dmitrii S. Silvestrov Stockholm University Department of Mathematics SE-106 91 Stockholm Sweden [email protected]

ISBN 978-3-11-032967-4 e-ISBN 978-3-11-032982-7 Set-ISBN 978-3-11-032983-4 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ♾Printed on acid-free paper Printed in Germany www.degruyter.com

Preface American-type options are one of the very important financial instruments and at the same time one of the most interesting and popular objects for research in financial mathematics. The main mathematical problems connected with such options relate to finding of the optimal expected option rewards, in particular, fair prices of options, as well as finding the optimal strategies for buyers of options that are optimal stopping times for execution of options. In this way, the theory of American-type options is connected with optimal stop­ ping problems for stochastic processes, which play an important role in the theory of stochastic processes and its applications in sequential analysis, control theory and many other applied areas. As is well known, analytical solutions for American-type options are available only in some special cases and, even in such cases, the corresponding formulas are not easily computable. These difficulties dramatically increase in the case of multivariate log-price processes and nonstandard pay-off functions. Approximation methods are a reasonable alternative that can be used in cases, where analytical solutions are not available. The main classes of approximation methods are: stochastic approximation methods based on approximation of the cor­ responding stochastic log-price processes by simpler processes, for which optimal expected rewards can be effectively computed; integrodifferential approximation methods based on approximation of integrodifferential equations that can be derived for optimal expected rewards by their difference analogs; and Monte Carlo methods based on simulation of the corresponding log-price processes. Stochastic approximation methods have important advantages in comparison with other two methods. They let one usually impose weaker smoothness conditions on transition probabilities and pay-off functions in comparison with integrodiffer­ ential approximation methods and they are also computationally more effective in comparison with Monte Carlo-based methods. This book is devoted to systematical presentation of stochastic approximation methods for models of American-type options with general pay-off functions for dis­ crete time Markov log-price processes. The principal novelty of our studies is that we systematically consider discrete time multivariate modulated Markov log-price processes and general pay-off func­ tions, which can depend not only on price but also an additional stochastic index com­ ponent with a general phase space. We also impose minimal conditions of smoothness on transition probabilities and pay-off functions as well as minimal moment compact­ ness conditions on log-price processes and restrictions on the rate of growth for pay-off function in price arguments. The book contains ten chapters.

vi | Preface In Chapter 1, we introduce models of multivariate modulated Markov log-price processes and consider main examples of such processes, including log-price pro­ cesses represented by multivariate modulated random walks and various autoregres­ sive and autoregressive moving average-type log-price processes as well as autoregres­ sive and autoregressive and moving average stochastic volatility models. In Chapter 2, we define American-type options and present basic examples such as call and put options, exchange of assets options, digital options, and knockout op­ tions, and their portfolios. We also introduce basic objects connected with Americantype options, namely reward functions, optimal expected rewards, optimal stopping times and present related basic results, namely, backward recurrence relations for re­ ward functions and results concerning hitting structure of optimal stopping times for American-type options. In Chapter 3, we investigate the structure of backward recurrence relations for logprice processes represented by atomic Markov chains, which transition probabilities are concentrated on finite sets. Such log-price processes play the role of approximat­ ing processes in the corresponding stochastic approximation reward algorithms. The so-called recombining conditions, which guarantee that the number of time–space nodes for trajectories of such atomic Markov chains has not more than polynomial rate of growth as a function of number of time steps, play here the key role. We inves­ tigate in details these conditions, for univariate and multivariate binomial and trino­ mial models both homogeneous and inhomogeneous in time and space, as well as for general discrete random walks and atomic Markov chains. In Chapter 4, we give upper bounds for reward functions and moments of random rewards for American-type options for multivariate modulated Markov log-price pro­ cesses with bounded and unbounded characteristics and with general pay-off func­ tions, which have not more than polynomial rate of growth in price arguments. We also specify these results for log-price processes represented by modulated random walks and multivariate Markov Gaussian processes with bounded and unbounded drift and volatility coefficients. In Chapters 5 and 6, we present our main convergence results for rewards of Amer­ ican-type options with general perturbed pay-off functions for perturbed multivariate modulated Markov log-price processes, respectively, with bounded and unbounded characteristics. It is important that we impose minimal conditions of smoothness on the limiting transition probabilities and pay-off functions. For the basic case, where the transition probabilities have densities with respect to some pivotal Lebesgue-type measure, it is usually required that the sets of weak discontinuity for the limiting tran­ sition probabilities, as functions of initial points, and the sets of discontinuity for pay-off functions are zero sets with respect to the above pivotal measure. In fact, such assumptions make it possible for the transition probabilities and pay-off functions to be very irregular. In Chapter 7, we preset general results about so-called space-skeleton approxima­ tions for rewards of American-type options with general pay-off functions for multi­

Preface | vii

variate modulated Markov log-price processes. In this model, the space discretization of underlying Markov log-price processes is used. The approximating atomic Markov chains have finite numbers of nodes located on some regular finite greeds for each moment of time. The corresponding backward recurrence relations let one effectively compute the approximating reward functions and optimal expected rewards. Their convergence to the corresponding limiting quantities is derived using the general con­ vergence theorems given in Chapters 5 and 6. In Chapter 8, we present results about convergence of option rewards and spaceskeleton reward approximations for multivariate modulated Markov Gaussian logprice processes with bounded and unbounded drift and volatility coefficients. We also give results of almost sure and weak consistency of estimates for rewards of Americantype options for Markov Gaussian log-price processes with estimated coefficients. In Chapters 9 and 10, we present results about fitting of coefficients and conver­ gence for reward functions of American-type options for binomial and trinomial tree approximations for univariate and multivariate Markov Gaussian log-price processes homogeneous and inhomogeneous in time and space. The bibliography, which contains more than 500 references, is also preceded by brief remarks. It should be noted that models of log-price processes with discrete time have their own very important value. However, they can also be imbedded in models of log-price processes with continuous time. In such framework, American-type options for dis­ crete time model should be considered as Bermudian type options for the correspond­ ing continuous time models. The discrete time models are also used as a main tool in approximation reward algorithms for continuous time models. This book is the first volume of a comprehensive two volumes monograph. Let us shortly present the content of the second volume. It will include: results about spaceskeleton approximations for rewards of American-type options for various autoregres­ sive and autoregressive moving average-type log-price processes and autoregressive and autoregressive moving average stochastic volatility log-price processes; Monte Carlo-based approximations for rewards of American-type options; results about mul­ tithreshold structure of stopping domains for optimal stopping times; time skeleton re­ ward approximations connecting discrete and continuous time models; convergence and approximation results for option rewards for continuous time multivariate modu­ lated Markov log-price processes, log-price processes represented by modulated Lévy processes, and multivariate modulated diffusion log-price processes; applications to European type options, American-type options connected with reselling of European options, and knockout options; as well as results of experimental studies. The presentation of material in this book is organized in a way that will hopefully be appreciated by readers. Each chapter has a preamble in which the main results are outlined and the content of the chapter is thereafter presented in sections. Each section is broken up into titled subsections.

viii | Preface I would also like to comment on the notation system used in the book. Throughout the text I make use of several basic classes of conditions. Conditions that belong to a specific class are denoted by the same letter. For example, the letter B is used for conditions restricting the rate of growth of pay-off functions in price arguments, the letter C for moment compactness conditions imposed on log-price processes, and so forth. Conditions belonging to a specific class have subscripts numbering conditions in the class. A list of all conditions is given in the Index. Chapters and sections have, respectively, a singular and a double numeration. Formulas also have a double numeration. For example, label (1.2) refers to formula 2 in Chapter 1. Subsections, theorems, lemmas, definitions and remarks have a triple numeration. For example, Theorem 1.2.3 means Theorem 3 in Section 1.2. I hope that the publication of this new book and the comprehensive bibliography of works related to problems of stochastic approximation methods, optimal stopping, and convergence for American-type options will be a useful contribution to the con­ tinuing intensive studies in this actual area of financial mathematics. In addition to its employment for research and reference purposes, the book can be used in special courses related to stochastic approximation methods, optimal stopping problems and option pricing and as a complementary reading in general courses on stochastic pro­ cesses. In this respect, it may be useful for specialists as well as doctoral and advanced undergraduate students. I am much indebted to Dr. Evelina Silvestrova for her continuous encouragement and support of various aspects of my work on this book. I would like to thank my collaborators for their fruitful contribution at different stages of my research studies in the area; Professor Alexander Kukush, Professor Raimondo Manca, Dr. Henrik Jönson, Dr. Robin Lundgren, Dr. Anatoliy Malyarenko, Dr. Evelina Silvestrova, and Dr. Fredrik Srenberg for the fruitful cooperation. I would also like to thank all my colleagues at the Department of Mathematics, Stockholm University, Stockholm, Sweden, and the School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden, for creating the inspiring research environment and friendly atmosphere which so stimulated my work. I also would like to thank the Riksbankens Jubileumsfond for the very important and stimulating support of my research in the area and work on this book. Stockholm, March 2013

Dmitrii Silvestrov

Contents Preface | v 1 1.1 1.2 1.3 1.4

Multivariate modulated Markov log-price processes (LPP) | 1 Markov LPP | 1 LPP represented by random walks | 8 Autoregressive LPP | 18 Autoregressive stochastic volatility LPP | 28

2 2.1 2.2 2.3 2.4 2.5

American-type options | 44 American-type options | 44 Pay-off functions | 47 Reward and log-reward functions | 53 Optimal stopping times | 63 American-type knockout options | 72

3 3.1 3.2 3.3 3.4

Backward recurrence reward algorithms | 76 Binomial tree reward algorithms | 76 Trinomial tree reward algorithms | 88 Random walk reward algorithms | 100 Markov chain reward algorithms | 106

4 4.1 4.2 4.3 4.4 4.5

Upper bounds for option rewards | 115 Markov LPP with bounded characteristics | 115 LPP represented by random walks | 127 Markov LPP with unbounded characteristics | 133 Univariate Markov Gaussian LPP | 154 Multivariate modulated Markov Gaussian LPP | 159

5 5.1 5.2 5.3

Convergence of option rewards – I | 167 Asymptotically uniform upper bounds for rewards – I | 168 Modulated Markov LPP with bounded characteristics | 180 LPP represented by modulated random walks | 194

6 6.1 6.2 6.3 6.4 6.5

Convergence of option rewards – II | 203 Asymptotically uniform upper bounds for rewards – II | 204 Univariate modulated LPP with unbounded characteristics | 214 Asymptotically uniform upper bounds for rewards – III | 220 Multivariate modulated LPP with unbounded characteristics | 231 Conditions of convergence for Markov price processes | 238

x | Contents 7 7.1 7.2 7.3 7.4 7.5

Space-skeleton reward approximations | 241 Atomic approximation models | 242 Univariate Markov LPP with bounded characteristics | 251 Multivariate Markov LPP with bounded characteristics | 262 LPP represented by multivariate modulated random walks | 275 Multivariate Markov LPP with unbounded characteristics | 294

8 8.1 8.2 8.3 8.4 8.5

Convergence of rewards for Markov Gaussian LPP | 303 Univariate Markov Gaussian LPP | 303 Multivariate modulated Markov Gaussian LPP | 312 Markov Gaussian LPP with estimated characteristics | 321 Skeleton reward approximations for Markov Gaussian LPP | 335 LPP represented by Gaussian random walks | 347

9 9.1 9.2 9.3 9.4 9.5

Tree-type approximations for Markov Gaussian LPP | 357 Univariate binomial tree approximations | 358 Multivariate binomial tree approximations | 367 Multivariate trinomial tree approximations | 379 Inhomogeneous in space binomial approximations | 394 Inhomogeneous in time and space trinomial approximations | 398

10 10.1 10.2 10.3 10.4

Convergence of tree-type reward approximations | 413 Univariate binomial tree approximation models | 413 Multivariate homogeneous in space tree models | 424 Univariate inhomogeneous in space tree models | 441 Multivariate inhomogeneous in space tree models | 456

Bibliographical Remarks | 465 Bibliography | 475 Index | 501

1 Multivariate modulated Markov log-price processes (LPP) In this chapter, we introduce models of multivariate modulated Markov log-price and price processes. In Section 1.1, we introduce models of multivariate modulated Markov log-price and price processes. In particular, we discuss and comment the sense of introducing a stochastic modulating index component and consider different variants of mod­ ulation. We also discuss variants of definition for multivariate modulated Markov log-price and price processes in a stochastic dynamic form. In Section 1.2, we present different models of multivariate modulated Markov log-price and price processes represented, respectively, by usual and exponential ran­ dom walks. We show in which way a multivariate modulated Markov log-price and price processes inhomogeneous in time can be interpreted as multivariate modulated Markov random walks inhomogeneous in time and space. Then, we consider different particular variants of such models. These are mod­ els of log-price and price processes represented, respectively, by usual or exponential multivariate random walks with independent jumps, multivariate modulated Markov random walks with inhomogeneous in time and space scale-location parameters, and multivariate modulated random walks with Markov and semi-Markov modulation. In Sections 1.3 and 1.4, we present various models of modulated autoregressive and autoregressive moving average-type log-price and price processes and autore­ gressive and autoregressive and moving average stochastic volatility-type log-price and price processes. These are nonmodulated and modulated autoregressive (AR), autoregressive moving average (ARMA), Cox, Ingersoll, Ross (CIR), centered autore­ gressive conditional heteroskedastic (ARCH), autoregressive/centered autoregressive conditional heteroskedastic (AR/ARCH), and generalized autoregressive conditional heteroskedastic (GARCH) models. We show in which way the above autoregressive models of log-price and pice processes can be imbedded in the class of multivariate modulated Markov log-price and price processes.

1.1 Markov LPP In this section, we introduce models of discrete time multivariate modulated Markov-­ type log-price and price processes that are studied in this book.

2 | 1 Multivariate modulated Markov log-price processes (LPP)

1.1.1 Discrete time multivariate modulated log-price and price processes Let Rk be a k-dimensional Euclidean space and Bk is the corresponding Borel σ-algebra of subsets of Rk . Also let X be a measurable space with a σ-algebra of measurable subsets BX . A typical example is, where X is a complete, separable, metric space (Polish space) with a metric dX (x , x ) and BX be the corresponding Borel σ-algebra of subsets of X (the minimal σ-algebra containing all balls R d (x) = {x ∈ X : dX (x, x ) ≤ d} in the space X). Let us consider the space Z = Rk × X with the σ-algebra of measurable subsets BZ = Bk × BX , which is the minimal σ-algebra containing all the sets B × C, where B ∈ Bk , C ∈ BX . In the case, where X is a Polish metric space with a metric dX (x , x ), the space Z z ,  z  ) =  = Rk × X is also a Polish metric space with the natural metric dZ (       | y − y  |2 + dX (x , x )2 ,  z = ( y , x ),  z = ( y , x ) ∈ Z. Let BZ be the corre­ sponding Borel σ-algebra of subsets of Z. We also use the space R+ s = (s1 , . . . , s k ) : s1 , . . . , s k > 0} that is the k = { + k-dimensional product of the interval R+ 1 = (0, ∞), and the space V = Rk × X,. The  v , v  ) = space V is also a Polish metric space with the metric dV (          2   2 | s − s | + dX (x , x ) ,  v = ( s , x ),  v = ( s , x ) ∈ V. The corresponding Borel σ-algebras for the spaces R+ and V are denoted, respectively, as B+ k and BV . n = Let us consider a random sequence (discrete time stochastic process) Y (Y n,1 , . . . , Y n,k ), n = 0, 1, . . . with the phase space Rk , a random sequence X n ,  n , X n ), n = 0, 1, 2, . . . with the phase space X, and the random sequence  Z n = (Y n = 0, 1, 2, . . . with the phase space Z.  n and X n and, thus, Z n as well are We always assume that the random sequences Y defined on the same probability space  Ω, F , P. S n = (S n,1 , . . . , S n,k ), n = 0, 1, . . . con­ We also consider the random sequence  n = (Y n,1 , . . . , Y n,k ), n = 0, 1, . . . , by the follow­ nected with the random sequence Y ing relations:   n = ln  Sn = e Yn , Y S n , n = 0, 1, . . . . (1.1) Here and henceforth, the notations, ey = (e y1 , . . . , e y k ),  y = ( y1 , . . . , y k ) ∈ Rk , and ln  s = (ln s1 , . . . , ln s k ),  s = (s1 , . . . , s k ) ∈ R+ are used. k  n = ( S n , X n ), n = 0, 1 . . . . Let us also consider the random sequence V n , X n ,   n have, respectively, the By the definition, the processes Y Zn,  S n , and V + phase spaces, Rk , X, Z, Rk , and V. n as a multivariate log-price process with discrete time, X n as a We interpret Y n , and  stochastic index modulating the log-price process Y Z n as a multivariate mod­ ulated log-price process (the extended log-price process supplemented by the addi­ tional index component X n ).

1.1 Markov LPP

| 3

  n as a multivariate modu­ S n is interpreted as a multivariate price process and V lated price process (the extended price process supplemented by the additional index component X n ). We do prefer to use log-price processes as initial objects instead of price processes. This is because of log-price processes usually have an additive structure of increments, while price processes usually have a multiplicative structure of increments. The addi­ tive structure of increments is, as a rule, simpler and more convenient for analysis than the multiplicative one. The character of increments mentioned above also explains using of exponential transformation which connects log-price processes with the corresponding price pro­ cesses.

1.1.2 Information filtrations generated by log-price and price processes 0 , . . . , Y  n ], n = 0, 1, . . . , be a natural filtration generated by the log-price Let Fn = σ [Y  process Y n that is the family of σ-algebras of random events generated by the random 0 , . . . , Y n , for n = 0, 1, . . . . Also let Fn = σ [ variables Y Z0 , . . . ,  Z n ], n = 0, 1, . . . , be a natural filtration generated by the process  Zn. It is useful to note that Fn and Fn coincide with the natural filtrations generated, n . respectively, by the processes  S n and V  By the definition, Fn ⊆ Fn , n = 0, 1, . . . , i.e. filtration Fn is an extension of filtration Fn . Thus, the component X n represents an additional market information that be­ n come available at the moment n additionally to the value of the log-price process Y  or the corresponding price process S n . This information can be supplied by some additional log-price or price process n or X n , which values are observable but not included in the vector log-price process Y  the price process S n . Another variant is where X n represents stochastic dynamics of some parameters  n , for example, its stochastic volatility. for the log-price process Y The third variant is where X n is, indeed, a market index, for example, a global price index “controlling” market prices, or a process representing some market regime index, for example, indicating growing, declining, or stable market situation.

1.1.3 Markov log-price and price processes In what follows, we always assume that  Z n , n = 0, 1, . . . , is an inhomogeneous in time Markov chain with a phase space Z, initial distribution P0 (A) = P{ Z0 ∈ A} defined for A ∈ BZ and the one-step transition probabilities defined for  z ∈ Z, A ∈ BZ ,

4 | 1 Multivariate modulated Markov log-price processes (LPP) n = 1, 2, . . . ,

  n ∈ A/ P n ( z , A) = P Z Z n −1 =  z .

(1.2)

 n is also an inhomogeneous Markov chain with the phase space V. Obviously, V The initial distribution P˙ 0 (B), and the transition probabilities P˙ n ( v , B) of the  Markov chain V n are connected with the initial distribution and the transition proba­ bilities of the Markov chain  Z n by the following relations, for  v = ( s , x) ∈ V, B ∈ BV , n = 1, 2, . . . , v , B) = P n ( zv , A B ) , (1.3) P˙ 0 (B) = P(A B ) , P˙ n (

zv = (ln  s , x) ∈ Z, A B = { zv :  v ∈ B} ∈ BZ . where  n and the corresponding price pro­ It is worth noting that the log-price process Y cess  S n themselves may not be Markov processes. The component X n represents in­  n or the formation which addition to information supplied by the log-price process Y    price process S n makes the corresponding extended processes Z n = (S n , X n ) and  n = ( V S n , X n ) Markov processes.  n are homogeneous in Z n and V An important case is where the Markov chains  time. This is the case, where the transition probabilities P n ( z , A) = P( z , A) and P˙ n ( v , A) = P˙ ( v , A) do not depend on time n = 0, 1, . . . .

1.1.4 Dynamic representations for Markov log-price and price processes n , X n ) can be given in a stochastic dynamic form, i.e. be The Markov process  Z n = (Y defined by the following stochastic transition dynamic relation:  Z n = A n ( Z n −1 , U n ) ,

n = 1, 2, . . . ,

(1.4)

Z0 is a random variable taking values in the space Z, (b) U n , n = 1, 2, . . . , is a where (a)  sequence of “noise” independent random variables taking values in some measurable space U with a σ-algebra of measurable subsets BU , (c) the random variable  Z0 and z , u ), the sequence of random variables U n , n = 1, 2, . . . , are independent, and (d) A n ( n = 1, 2, . . . , are the measurable functions acting from the space Z × U to the space Z. Usually, the initial state can also be represented in the form  Z0 = A0 (U0 ), where (e) U0 is a random variable taking values in the space U, (f) the random variable U0 and the sequence of random variables U n , n = 1, 2, . . . , are independent, and (g) A0 (u ), n = 1, 2, . . . , is a measurable function acting from the space U to the space Z. Z n has a Polish phase space Z and is given in the standard form If the Markov chain  by its initial distribution and transition probabilities, it is always possible to construct on some probability space a Markov chain  Z n , which is stochastically equivalent to the  Markov chain Z n in the distributional sense and is given in the above dynamic form. As is shown, for example, in Gikhman and Skorokhod (1977), in this case, it is possible to construct the measurable functions A n ( z , u ), n = 1, 2, . . . , acting from the

1.1 Markov LPP

| 5

space Z × [0, 1] to the space Z and a function A0 (u ) acting from the interval [0, 1] to the space Z such that, for any sequence of the i.i.d. random variables U n , n = 0, 1, . . . , uniformly distributed in the interval [0, 1], the random sequence  Z n = A n (Z n−1 , U n ),   n = 1, 2, . . . Z0 = A0 (U0 ) is a Markov chain with the space space Z, the initial distribution P0 (A), and the transition probabilities P n ( z , A ). The above remarks, let us, in most of the cases, define a Markov log-price process  Z n in the dynamic form without any loss of generality. The transition dynamic functions A n ( z , u ) can be represented in the form A n ( z , u) y , x, u ), C n ( y , x, u )),  z = ( y , x) ∈ Z, u ∈ U, n = 1, 2, . . . , where B n ( y , x, u ) = (B n ( and C n ( y , x, u ) are the components of functions A n ( z , u ) acting from the space space Rk × X × U, respectively, to the space Rk and X. Analogously, the function A0 (u ) can be represented in the form A0 (u ) = (B0 (u ), C0 (u )), u ∈ U, where B0 (u ) and C0 (u ) are the components of the functions A0 (u ) acting from the space space U, respectively, to the space Rk and X. The stochastic transition dynamic relation (1.4) can be rewritten in the following form: ⎧  n = B n (Y  n −1 , X n −1 , U n ) , ⎪ Y ⎪ ⎪ ⎨  n −1 , X n −1 , U n ) , (1.5) X n = C n (Y ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 , X0 ) is a random variable taking values in the space Z, (b) U n , Z0 = (Y where (a)  n = 1, 2, . . . , is a sequence of “noise” independent random variables taking values in some measurable space U with a σ-algebra of measurable subsets BU , (c) the random variable  Z0 and the sequence of random variables U n , n = 1, 2, . . . , are independent, y , x, u ), n = 1, 2, . . . , are the measurable functions acting from the space Rk × (d) B n ( y , x, u ), n = 1, 2, . . . , are the measurable functions X × U to the space Rk , and (e) C n ( acting from the space Rk × X × U to the space X. The stochastic transition dynamic relation (1.5) can also be rewritten in terms of n , the extended price process V ⎧   ⎪ S = e B n (ln S n−1 ,X n−1 ,U n ) , ⎪ ⎪ ⎨ n (1.6) X n = C n (ln  S n −1 , X n −1 , U n ) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

Note that the transition probabilities P n ( z , A) of the Markov chain  Z n are con­ z , u ) by the following relation, for nected with the dynamic transition functions A n (  z ∈ Z, A ∈ BZ , n = 1, 2, . . . : P n ( z , A ) = P{  Z n ∈ A/ Z n −1 =  z } = P{A n ( z , U n ) ∈ A}.

(1.7)

6 | 1 Multivariate modulated Markov log-price processes (LPP)

1.1.5 Modulated Markov log-price and price processes The index component X n can also be interpreted as a modulator of the corresponding  n or the price process  Sn . log-price process Y To see this, let us first consider the case, where the phase space X = {1, 2, . . . , m} is a finite set. In this case, as usual, BX is the σ-field of all subsets of X. Let us introduce conditional marginal transition probabilities for the index X n ,    n −1 =  y , X n −1 = x P n,y (x, x ) = P X n = x /Y (1.8) y , x), Rk × {x }) = P n (( n and conditional marginal transition probabilities for the log-price process Y    n −1 =  n ∈ B/Y P n,xx ( y , B) = P Y y , X n−1 = x, X n = x = P n (( y , x), B × {x })/P n,y (x, x ) .

(1.9)

y , B) in the cases, where Relation (1.9) defines the conditional probabilities P n,xx ( the probabilities P n,y (x, x ) > 0. Otherwise, an arbitrary probability measure (as a function of B) can play the role of this conditional transition probability. y , x) ∈ Z, A ∈ BZ are uniquely deter­ The transition probabilities P n (z, A), z = ( mined by their values on cylindric sets A = B × C, where B ∈ Bk , C ∈ BX . For such sets, these probabilities can be represented via the corresponding marginal transition probabilities introduced above,

P n (( y , x), B × C) = P n,xx ( y , B)P n,y (x, x ) . (1.10) x ∈C

In this representation let us consider the index X n as a modulator of the log-price n . In fact, relation (1.10) shows that some kind of mutual cross modula­ component Y  n and X n takes place. tion for processes Y There is, however, the important case where the index X n modulates the log-price  n but not vice versa. This is the case, where the marginal transition proba­ process Y bilities for the index do not depend on values of the log-price process, i.e. P n,y (x, x ) = P n (x, x ) .

(1.11)

In what follows, we always assume that the corresponding relation holds for all admissible values of arguments if this is not specified in this relation. It is easy to show that, in this case, the index X n is a Markov chain with the phase space X and the one-step transition probabilities P n (x, x ). If the model is defined in the dynamic form given by relation (1.5) then the follow­ ing relation replaces (1.11): C n ((y, x), U n ) = C n (x, U n ) .

(1.12)

1.1 Markov LPP

| 7

Let us now consider the case, where X is a general space with the σ-algebra of measurable subsets BX . Let us introduce conditional marginal transition probabilities for the stochastic index X n    n −1 =  P n,y (x, C) = P X n ∈ C/Y y , X n −1 = x (1.13)  = P n ( y , x) , Rk × C . The transition probabilities P n ( z , A ),  z = ( y , x) ∈ Z, A ∈ BZ are uniquely deter­ mined by their values on cylindric sets A = B × C, where B ∈ Bk , C ∈ BX . By the definition, P n (( y , x), B × C) ≤ P n (( y , x), Rk × C), C ∈ BX , for every ( y , x) ∈ Z, B ∈ Bk . Thus, by the Radon–Nikodym theorem, P n (( y , x), B × C), can be represented in the following form:

y , x), B × C) = P n,x,x( y , B)P n,y (x, dx ) , (1.14) P n (( C

y , B) is the Radon–Nikodym derivative of measure P n (( y , x), B × C), C ∈ where P n,x,x ( BX with respect to measure P n (( y , x), Rk × C), C ∈ BX . Under some minor conditions, for example, if X is a Polish space and BX is the corresponding Borel σ-algebra of subsets of X, there exists the regular variant of the y , B), which is a measurable function in argument (x, x ,  y) ∈ X × derivative P n,x,x ( y) ∈ X × Rk , for every B ∈ Bk and a probability measure in B ∈ Bk for every (x, x ,  X × X × Rk . This derivative is a regular conditional distribution    n −1 =  n ∈ B/Y y , B) = P Y y , X n−1 = x, X n = x . (1.15) P n,xx (  n . In One can consider the index X n as a modulator of the log-price component Y n and X n , in some sense, cross-modulate fact, relation (1.15) shows that processes Y each other. There is, however, the important case, where the index X n modulates the log-price n but not vice versa. This is the case, where the marginal transition proba­ process Y bilities for the index X n do not depend on the values of the log-price process, i.e.

P n,y (x, C) = P n (x, C) .

(1.16)

It is possible to show that, in this case, the index X n is a Markov chain with the phase space X and the one-step transition probabilities P n (x, C). The stochastic transition dynamic relation (1.5) takes, in this case, the following simpler form: ⎧  = B n (Y ⎪ n−1 , X n−1 , X n , U n ) , ⎪Y ⎪ ⎨ n X n = C n (X n−1 , U n ) , (1.17) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 , X0 ) is a random variable taking values in the space Z, (b) U n , U n , where (a)  Z0 = (Y n = 1, 2, . . . is a family of independent random variables taking values in some

8 | 1 Multivariate modulated Markov log-price processes (LPP) measurable space U with a σ-algebra of measurable subsets BU , (c) the random vari­ able  Z0 and the family of random variables U n , U n , n = 1, 2, . . . , are independent, (d) B n ( y , x, x , u ), n = 1, 2, . . . , are the measurable functions acting from the space Rk × X × X × U to Rk , and (e) C n (x, u ), n = 1, 2, . . . , are the measurable functions acting from the space X × U to the space X. Relation (1.17) takes the following exponential form: ⎧   ⎪ S n = e B n (ln S n−1 ,X n−1 ,X n ,U n ) , ⎪ ⎪ ⎨ (1.18) X n = C n (X n−1 , U n ) , ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . ,

1.2 LPP represented by random walks In this section, we consider a number of typical examples of discrete time modulated Markov log-price and price processes represented by multivariate modulated random walks.

1.2.1 Log-price and price processes represented by Markov random walks inhomogeneous in space and time n , X n ) can always be represented in the form of the A modulated Markov chain  Z n = (Y modulated Markov random walk, given by the following stochastic transition dynamic relation, ⎧ ⎨( Y 0 , X 0 ) if n = 0 , n , X n ) =  Z n = (Y (1.19) ⎩( Y  n −1 + W  n , X n ) if n = 1, 2, . . . ,

where n − Y  n −1 , n = Y W

n = 1, 2, . . .

(1.20)

In this case, it is convenient to operate with the transition jump probabilities ˜ n (  n , X n ) ∈ A/Y  n −1 =  P y , x, A) = P{(W y, X n−1 = x}, which are connected with the transition probabilities of the Markov chain  Z n by the following relations, for  z = ( y , x) ∈ Z, A ∈ BZ , n = 0, 1, . . . :   ˜ n (  n , X n ) ∈ A/Y  n −1 =  P y , x, A) = P (W y , X n −1 = x    n −1 =  n −  = P (Y y , X n ) ∈ A/Y y , X n−1 = x = P n ( z , A[y] ) , (1.21) where A[y] = { z  = ( y  , x): ( y −  y , x) ∈ A}.

1.2 LPP represented by random walks |

9

A dynamic analog of this representation can be obtained by simple transforma­ tion of the stochastic transition dynamic relation (1.5). Indeed, by introducing a new ˜ n ( dynamic transition function B y , x, u ) = B n ( y , x, u ) −  y, one can rewrite relation (1.5) in the following form: ⎧ ˜ n (Y  n −1 + B  n −1 , X n −1 , U n ) , n = Y ⎪ Y ⎪ ⎪ ⎨  n −1 , X n −1 , U n ) , (1.22) X n = C n (Y ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 , X0 ) is a random variable taking values in the space Z = Rk × X, where (a)  Z0 = (Y (b) U n , n = 1, 2, . . . , is a sequence of independent random variables taking value in some measurable space U with a σ-algebra of measurable subsets BU , (c) the Z0 and the random sequence U n , n = 1, 2, . . . , are independent, random variable  ˜ n ( (d) B y , x, u ), n = 1, 2, . . . , are the vector functions in which components are the measurable functions acting from the space Rk × X × U to Rk , (e) C n ( y , x, u ), n = 1, 2, . . . , are the measurable functions acting from the space Rk × X × U to the space X.  n is concerned, it will be given in this case As far as the modulated price process V by the following multiplicative analog of stochastic transition dynamic relation: (1.22): ⎧ ˜   ⎪ S = S n−1 · e B n (ln S n−1 ,X n−1 ,U n ) , ⎪ ⎪ ⎨ n (1.23) X n = C n (ln  S n −1 , X n −1 , U n ) , ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

c = a· b should be understood as the Here and henceforth, the vector product  vector with the components c i = a i b i , i = 1, . . . , k. In the following subsections, we consider typical examples of log-price and price processes represented, respectively, by random walks and exponential random walks.

1.2.2 Log-price and price processes represented by standard random walks  n is a standard ran­ The simplest example is where a multivariate Markov log-price Y dom walk  n −1 + W  n , n = 1, 2, . . . , n = Y Y (1.24) 0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking values in the space Rk , where (a) Y  n = (W n,1 , . . . , W n,k ), n = 1, 2, . . . , is a sequence of independent random vec­ (b) W 0 and the random sequence W  n, tors taking values in Rk , and (c) the random vector Y

n = 1, 2, . . . , are independent. In this case, the model without index component is considered. In fact, one can always define and include in the model a “virtual” index component X n , which takes a constant value x0 ∈ X, for every n = 0, 1, . . . .

10 | 1 Multivariate modulated Markov log-price processes (LPP) n in the dynamic form. Also note that in this Relation (1.24) defines the process Y  n , for case, the role of “noise” random variable U n is played by the random vector W n = 1, 2, . . . . Thus, the space U = Rk . S n is concerned, it will be given in this case by the As far as the price process  following multiplicative analog of relation (1.24):   Sn =  S n −1 · e W n ,

n = 1, 2, . . . .

(1.25)

The process  S n is called a standard exponential random walk.  n and  It is useful to note that the random walks Y S n are homogeneous in time if  n , n = 1, 2, . . . , are identically distributed. the random vectors W There are several the most frequently used models. The first and the simplest one is the binomial model, where components W n,1 , . . . ,  n , n = 1, 2, . . . , are binary random variables, i.e. random W n,k of random vectors W variables taking just two different values. Usually, one of these values is positive while n and  another is negative. In this case, Y S n are, respectively, a multivariate binomial random walk and a multivariate exponential binomial random walk. Slightly more complex is the trinomial model, where the components W n,1 , . . . ,  n , n = 1, 2, . . . , are trinary random variables, i.e. random W n,k of the random vectors W variables taking just three different values. Usually one of these values is positive, one n and  S n are, respectively, a multivariate is zero and one is negative. In this case, Y trinomial random walk and a multivariate exponential trinomial random walk.  n , n = 1, 2, . . . , Also, variants of the model with more general random vectors W with a discrete sets of possible values are considered. Discrete time models are often appear as a result of some time discretization proce­ dures applied to the continuous time models of log-price processes with independent increments. In such cases, the following three variants of log-price processes repre­ sented by random walks appear in a natural way. For example, application of the time discretization procedure mentioned above to a continuous time log-price process represented by a multivariate Brownian motion, n , n = 1, 2, . . . , repre­ results with the model of discrete time log-price processes Y  sented by a standard random walk with jumps W n , n = 1, 2, . . . , which have multi­  n and  variate normal distributions. In this case, Y S n are discrete time analogs of, re­ spectively, a multivariate Brownian motion and a multivariate exponential Brownian motion. In the case, where the corresponding continuous time model is represented by a so-called jump-diffusion log-price process, application of the time discretization pro­ cedure mentioned above results with the model of a discrete time log-price processes  n + W  n , n , n = 1, 2, . . . , represented by a standard random walk with jumps W n = W Y  n and W  n , are indepen­ n = 1, 2, . . . . Here, the additive components of the jumps, W   dent, the random vectors W n have a multivariate normal distributions while the ran­  n have vector compound Poisson distributions (i.e., W  n can be repre­ dom vectors W sented in the form of random sum, where summands are i.i.d. random vectors inde­

1.2 LPP represented by random walks

| 11

pendent of a random index, which counts number of summands and has a Poisson distribution). Finally, if the corresponding continuous time model is represented by the Lévy process, application of the time discretization procedure mentioned above results with  n , n = 1, 2, . . . , with jumps W  n, n = the model of a discrete time log-price processes Y 1, 2, . . . , possessing infinitely divisible distributions.

1.2.3 Log-price and price processes represented by inhomogeneous in time and space random walks This model is also convenient to introduce using the stochastic transition dynamic relation ˜ n (Y  n −1 + B n−1 , U n ) , n = 0, 1, . . . , n = Y Y (1.26) 0 is a random vector taking values in the space Rk , (b) U n , n = 1, 2, . . . , is a where (a) Y sequence of independent random variables taking value in some measurable space U 0 and the random with a σ-algebra of measurable subsets BU , (c) the random vector Y ˜ n ( sequence U n , n = 1, 2, . . . , are independent, (d) B y , u ) are the measurable functions acting from the space Rk × U to Rk . As given in the previous subsection, the model without index component is con­ sidered. In fact, one can always to define a “virtual” index component X n which takes a constant value x0 , for every n = 0, 1, . . . . S n is concerned, it is given in this case by the following As far as the price process  formula: ˜   Sn =  S n−1 · e B n (ln S n−1 ,U n ) , n = 1, 2, . . . . (1.27)

One of the important variants is the model, where the log-price process is a ran­ dom walk with drift and volatility functional coefficients inhomogeneous in space and time,  n −1 +   n −1 ) + Σ n ( Y  n −1 ) W  n , n = 1, 2, . . . , n = Y Y μ n (Y (1.28) 0 = (Y 0,1 , . . . , Y 0,k ) is a k-dimensional random vector, (b) W  n = (W n,1 , . . . , where (a) Y W n,k ), n = 1, 2, . . . , is a sequence of k-dimensional independent random vectors, 0 and the random sequence W  n , n = 1, 2, . . . , are indepen­ (c) the random vector Y dent, (d)  μ n ( y ) = (μ n,1( y), . . . , μ n,k ( y)), n = 1, 2, . . . , are the vector functions in which components are the measurable functions acting from Rk to R1 , and (e) Σ n ( y) = σ n,ij( y) , n = 1, 2, . . . , are the k × k matrix functions with elements σ n,ij( y ) which are the measurable functions acting from Rk to R1 . It is useful to note that here and henceforth we interpret vectors  b = (b1 , . . . , b k ) as column vectors and use an ordinary definition for a product of a k × k matrix A =  b as a column vector A  b= c = (c1 , . . . , c k ), where c i = kj=1 a i,j b j , a i,j by a vector  i = 1, . . . , k.

12 | 1 Multivariate modulated Markov log-price processes (LPP) As far as the price process  S n is concerned, it will be given in this case by the following multiplicative analog of relation (1.28):     Sn =  S n−1 · eμ n (ln S n−1 )+Σ n (ln S n−1 )W n ,

n = 1, 2, . . . .

(1.29)

 n and the price process  The log-price process Y S n described above have no index component. These processes are discrete time Markov chains.  n, n = These Markov chains are homogeneous in time if the random vectors W ¯ ( 1, 2, . . . , are identically distributed and functions μ¯ n ( y) = μ y) and Σ n ( y ) = Σ( y ) do not dependent on n. Relation (1.28) can be rewritten in the form of stochastic difference equation  n −1 =   n −1 ) + Σ n ( Y  n −1 ) W n , n − Y Y μ n (Y

n = 1, 2, . . . .

(1.30)

Equation (1.30) can be considered as a discrete time analog of a stochastic differ­ ential equation.  n , n = 1, 2, . . . , have normal If it is additionally assumed that random vectors W n , n = 0, 1, . . . , is the discrete time analog of a diffusion distributions, the process Y process, while  S n , n = 0, 1, . . . , is the discrete time analog of an exponential diffusion process. The following two examples of the above model appear in many applications. In the one-dimensional case, this is a discrete time analog of an Ornstein–Uhlen­ beck-type process given by the following stochastic difference equation:  Y n − Y n−1 = λ n ( Y n −1 − λ n ) + σ n W n ,

n = 1, 2, . . . ,

(1.31)

where (a) Y 0 is the real-valued random variable, (b) W n , n = 1, 2, . . . , is a sequence of real-valued independent random variables, (c) the random variable Y0 and the ran­ dom sequence W n , n = 1, 2, . . . , are independent, and (d) λn , λ n , σ n , n = 1, 2, . . . , are the real-valued constants. It is useful to note that the process Y n possesses so-called mean-reverse property if the constants λ n ≤ 0, n = 1, 2, . . . . In the multivariate case, a discrete time Ornstein–Uhlenbeck-type process is given by the following vector stochastic difference equation:  n −1 = Λ n ( Y  n −1 −  n − Y n , Y λn ) + Σn W

n = 1, 2, . . . ,

(1.32)

0 = (Y 0,1 , . . . , Y 0,k ) is a k-dimensional random vector with real-valued where (a) Y  n = (W n,1 , . . . , W n,k ), n = 1, 2, . . . , is a sequence of indepen­ components, (b) W 0 and dent random vectors taking values in the space Rk , (c) the random vector Y   the random sequence W n , n = 1, 2, . . . , are independent, (d) λ n = (λ n,1 , . . . , λ n,k ), n = 1, 2, . . . , are the k-dimensional nonrandom vectors with real-valued components, and (e) Λ n = λ n,ij , Σ n = σ n,ij , n = 1, 2, . . . , are the k × k matrices with real-valued elements. n is a homogeneous in time Markov chain, The Ornstein–Uhlenbeck-type process Y  n , n = 1, 2, . . . , are identically distributed and the parameters if the random vectors W  λn =  λ, Λ n = Λ, Σ n = Σ, n = 1, 2, . . . , do not depend on n.

1.2 LPP represented by random walks |

13

1.2.4 Log-price and price processes represented by multivariate modulated random walks This model can be introduced by the following stochastic transition dynamic relation: which is a particular case of relation (1.22): ⎧ ˜   ⎪ ⎪ ⎪ Y n = Y n −1 + B n ( X n −1 , U n ) , ⎨ X n = C n ( X n −1 , U n ) , (1.33) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 , X0 ) = ((Y 1 , . . . , Y k ), X0 ) is a random vector taking values in space where (a) (Y Z, (b) U n , n = 1, 2, . . . , is a sequence of independent random variables taking value in some measurable space U with a σ-algebra of measurable subsets BU , (c) the ran­ 0 , X0 ) and the random sequence U n , n = 1, 2, . . . , are independent, dom vector (Y ˜ ˜ n,1(x, u ), . . . , B ˜ n,k (x, u )), n = 1, 2, . . . , are the measurable functions (d) B n (x, u ) = (B acting from X × U to Rk , and (e) C n (x, u ), n = 1, 2, . . . , are the measurable functions acting from X × U to X.  n = ( As far as the price process V S n , X n ) is concerned, it will be given in this case by the following multiplicative analog of formula (1.33): ⎧ ˜ ⎪  S = S n−1 · e B n (X n−1 ,U n ) , ⎪ ⎪ ⎨ n (1.34) X n = C n ( X n −1 , U n ) , ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

˜ n ( Note that, in this case, the transition probabilities P y , x, A) take the following form, for  z = ( y , x) ∈ Z, A ∈ BZ , n = 1, 2, . . . ,   ˜ n (  n , X n ) ∈ A/Y  n −1 =  y , x, A) = P (W y , X n −1 = x P   ˜ n (X n−1 , U n ), C n (X n−1 , U n )) ∈ A = Pz ,n (B   ˜ n (x, U n ), C n (x, U n )) ∈ A = P ˜ n (x, A). = P (B (1.35) ˜ n (x, u ) and C n (x, u ) as well as the In this case, the transition dynamic functions B ˜ ˜ transition probabilities P n ( y , x, A) = P n (x, A) do not depend on the log-price argu­ y. ment  Due to this, the index component X n itself is a Markov chain modulating the n . log-price component Y The transition probabilities of the Markov chain X n take the following form, for x ∈ X, C ∈ BX , n = 1, 2, . . . : P n (x, C) = P {X n ∈ C/X n−1 = x} ˜ n (x, Rk × C) . = P {C n (x, U n ) ∈ C} = P

(1.36)

14 | 1 Multivariate modulated Markov log-price processes (LPP) n , X n ) and V  n = ( The random sequences  Z n = (Y S n , X n ) are Markov chains. These Markov chains are homogeneous in time if the random variables U n , n = ˜ (x, u ) and C n (x, u ) ˜ n (x, u ) = B 1, 2, . . . , are identically distributed and functions B = C(x, u ) do not depend on n. ˜ n (x, U n ) and C n (x, U n ) are dependent. In general, random variables B However, there is a case, where these random variables are independent. Let us assume that the space U = U ×U is the product of two spaces and that the σ-algebras of measurable subsets BU is the minimal σ-algebra that contains all sets A × B, A ∈ BU , B ∈ BU , where BU and BU , are σ-algebras of measurable subsets, respectively, for spaces U and U . In this case, the random variables U n = (U n , U n ), n = 1, 2, . . . , can be repre­ sented in the form of random vectors with components U n and U n , which are random variables taking values, respectively, in the spaces U and U . ˜ n (x, u ) = B ˜ n (x, u  ) Let us also assume that the transition dynamic functions B and C n (x, u ) = Cn (x, u  ) for u = (u  , u  ) ∈ U, u  ∈ U , u  ∈ U , i.e. these functions depend only of the corresponding components of the vector argument u = (u  , u  ). In this case, the stochastic transition dynamic relation (1.33) takes the following form: ⎧  ˜  ⎪  ⎪ ⎪ Y n = Y n −1 + B n ( X n −1 , U n ) , ⎨ X n = Cn (X n−1 , U n ) , (1.37) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

If the random variables U n and U n are independent for every n = 1, 2, . . . , then ˜ n (x, U n ) and Cn (x, U n ) will be also independent random variables random variables B ˜ n (x, B × C) take for every n = 0, 1, . . . , and, therefore, the transition probabilities P the following form, for x ∈ X, B ∈ BRk , C ∈ BX , n = 1, 2, . . . , ˜ n (x, U n ) ∈ B}P{Cn (x, U n ) ∈ C} . ˜ n (x, B × C) = P{B P

(1.38)

 n is a random An important variant is the model, where the log-price process Y walk with modulated scale-location parameters. n is given by the following In this model, the modulated log-price process Y stochastic transition dynamic relation: ⎧   n , ⎪ μ n ( X n −1 ) + Σ n ( X n −1 ) W ⎪ ⎪ Y n = Y n −1 +  ⎨ X n = C n ( X n −1 , W n ) , (1.39) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking values in space Rk , (b) W n = where (a) Y (W n,1 , . . . , W n,k ), n = 1, 2, . . . , is a sequence independent random vectors taking values in space Rk , (c) X0 is a random variable taking values in space X, (d) W n , n = 1, 2, . . . , is a sequence of independent random variables taking value in some measurable space W with a σ-algebra of measurable subsets BW , (e) the random vec­ 0 , X0 ) and the random sequences (W  n , W n ), n = 1, 2, . . . , are independent, tor (Y

1.2 LPP represented by random walks |

15

μ n (x) = (μ n,1(x), . . . , μ n,k (x)), n = 1, 2, . . . , are the vector functions which com­ (f)  ponents are the measurable functions acting from the space X to R1 , (g) Σ n (x) = σ n,ij(x) , n = 1, 2, . . . , are k × k matrix functions with elements σ n,ij (x) which are the measurable functions acting from the space X to R1 , and (h) C n (x, w), n = 1, 2, . . . , are the measurable functions acting from the space X × W to the space X. In this case, the role of random vectors U n = (U n , U n ) is played by the random  n , W n ) taking values in the space U = Rk × W. vectors (W As far as the price process  S n is concerned, it will be given in this case by the following multiplicative analog of the first formula in (1.39):   Sn =  S n−1 · eμ n (X n−1 )+Σ n (X n−1 )W n ,

n = 1, 2, . . . .

(1.40)

n , X n ) and V  n = ( The random sequences  Z n = (Y S n , X n ) are Markov chains.  n , W n ), These Markov chains are homogeneous in time if the random vectors (W μ n (x) =  μ ( x ), n = 1, 2, . . . , are identically distributed and the transition functions  Σ n (x) = Σ(x) and C n (x, w) = C(x, w) do not depend on n.

1.2.5 Log-price and price processes represented by random walks modulated by Markov chains For simplicity, let us restrict consideration by the case when the phase space X = n is given by the following {1, 2, . . . , } is a discrete set, and the log-price process Y stochastic transition dynamic relation:  n −1 + W  n ( X n −1 ) , n = Y Y

n = 1, 2, . . . ,

(1.41)

0 = (Y 0,1 , . . . , Y 0,k ) is a k-dimensional random vector, (b) W  n (x) = where (a) Y (W n,1 (x), . . . , W n,k (x)), n = 1, 2, . . . , x ∈ X is a family of k-dimensional indepen­ dent random vectors, (c) X n , n = 0, 1, . . . , is a Markov chain with the phase space X, an initial distribution P(C) = P{X0 ∈ C} and the one-step transition probabilities 0 , P n (x, y) = P{X n = y/X n−1 = x}, x, y ∈ X, n = 1, 2, . . . , and (d) the random vector Y  n (x), n = 1, 2, . . . , x ∈ X, and the Markov chain X n , the family of random variables W n = 0, 1, . . . , are independent. As far as the price process  S n is concerned, it will be given in this case by the following multiplicative analog of relation (1.41),   Sn =  S n − 1 · e W n ( X n −1 ) ,

n = 1, 2, . . . .

(1.42)

1.2.6 Log-price and price processes represented by random walks modulated by semi-Markov indices n is a random walk Let us consider another example, where the log-price process Y modulated by a semi-Markov stochastic index.

16 | 1 Multivariate modulated Markov log-price processes (LPP) Let us first introduce a discrete time semi-Markov process X n , n = 0, 1, . . . . We consider a two-dimensional Markov chain (J n , T n ), n = 0, 1, . . . , with a phase space X × N, where N = {0, 1, . . . }, an initial distribution Q(C) = P{J 0 ∈ C} = P{J 0 ∈ C, T0 = 0} and the one-step transition probabilities Q n (x, C, r) = P{J n ∈ C, T n ≤ r/J n−1 = x, T n−1 = l}, n = 1, 2, . . . , which do not depend on the current state of the second component. Such Markov chain can be referred to as a discrete time Markov renewal process. Then a discrete time semi-Markov process X n can be defined in the following way: X n = J N (n) ,

n = 0, 1, . . . ,

(1.43)

where N n = max(r : T1 + · · · + T r ≤ n) ,

n = 0, 1, . . . .

(1.44)

The random variables J n are sequential states of the semi-Markov process X n at moments of jumps, while the random variables T n are the corresponding interjump times. An usual assumption Q n (x, X, 0) = 0, x ∈ X, n = 1, 2, . . . , is also made in order to exclude instant jumps. In this case, N n ≤ n, n = 0, 1, . . . , with probability 1. Note that we introduced above an inhomogeneous in time Markov renewal pro­ cess (J n , T n ). Consequently, the corresponding semi-Markov process X n defined above is also inhomogeneous in time. The semi-Markov process X n is homogeneous in time if the Markov renewal pro­ cess (J n , T n ) is homogeneous in time Markov chain. This is where its transition prob­ abilities Q n (x, C, r) = Q(x, C, r), x ∈ X, C ∈ BX , n, r = 1, 2, . . . , do not depend on n. As is well known, the semi-Markov process X n is not in general case a Markov chain. However, one can add additional components to the random process X n which will make the corresponding process Markov. Let us define the process that counts time from the moment of last jump before moment n and n, M n = n − T N n , n = 0, 1, . . . . (1.45) ¯ n = (X n , M n , N n ), n = 0, 1, . . . , is a homo­ In this case, the discrete time process X ¯ = X × N × N, the initial distribution geneous Markov chain with the phase space X P(C) = P{X0 ∈ C} = P{X0 ∈ C, M0 = 0, N0 = 0} and the one-step transition proba­ bilities P(x, m, r, C, m , r )   = P X n+1 ∈ C, M n+1 = m , N n+1 = r /X n = x, M n = m, N n = r ⎧ 1− Q r+1 ( x, X ,m )   ⎪ ⎪ ⎪I (x ∈ C) 1−Q r+1 (x, X,m−1) , if m = m + 1, r = r , ⎪ ⎨ = Q r+1 (x,C,m)−Q r+1(x,C,m−1) , if m = 0, r = r + 1 , ⎪ 1− Q r+1 ( x, X ,m −1) ⎪ ⎪ ⎪ ⎩0 , otherwise .

(1.46)

1.2 LPP represented by random walks

| 17

If the semi-Markov chain X n is homogeneous in time, the component N n may be ¯ n = (X n , M n ), n = 0, 1, . . . , is a ho­ omitted. In this case, the discrete time process X ¯ = X × N, the initial distribution mogeneous Markov chain with the phase space X Q(C) = P{X0 ∈ C} = P{X0 ∈ C, M0 = 0} and the one-step transition probabilities P(x, m, C, m )   = P X n+1 ∈ C, M n+1 = m /X n = x, M n = m ⎧ ⎪ ⎪ I (x ∈ C) 1−1−QQ(x,(x,XX,m,m−)1) , if m = m + 1, ⎪ ⎨ )− Q(x,C,m−1) = Q(x,C,m , if m = 0, 1− Q( x, X ,m −1) ⎪ ⎪ ⎪ ⎩0 , otherwise .

(1.47)

A random walk modulated by a semi-Markov stochastic index can be introduced in the way analogous to those used in stochastic transition relation (1.39) defining a random walk modulated by a Markov stochastic index,  n −1 +  n , n = Y Y μ n ( X n −1 ) + Σ n ( X n −1 ) W

n = 1, 2, . . . ,

(1.48)

0 is a k-dimensional random vector, (b) W  n = (W n,1 , . . . , W n,k ), n = where (a) Y 1, 2, . . . , is a sequence of k-dimensional independent random vectors with real-val­ ued components, (c) X n , n = 0, 1, . . . , is a discrete time semi-Markov process defined in (1.43) and (1.44), (d) the semi-Markov process X n , n = 0, 1, . . . , the random vec­ 0 , and the random sequence W  n , n = 1, 2, . . . , are independent, (e)  μ n (x) = tor Y (μ n,1(x), . . . , μ n,k(x)), n = 1, 2, . . . , are the vector functions which components are the measurable functions acting from the space X to R1 , and (f) Σ n (x) = σ n,ij(x) , n = 1, 2, . . . , are the measurable matrix functions with elements σ n,ij(x) which are the measurable functions acting from the space X to R1 . As far as the price process  S n is concerned, it will be given in this case by the following multiplicative analog of formula (1.48):   Sn =  S n−1 · eμ n (X n−1 )+Σ n (X n−1 )W n ,

n = 1, 2, . . . .

(1.49)

n , X n ) is not a Markov chain. In order to have this Note that the process  Z n = (Y ¯ n as the stochastic index modulating the property, one should consider the process X ¯ n ) and V n , X n = corresponding log-price or price process. The processes  Z n = (Y ¯ n ) are Markov chains. Sn , X (  n, n = These Markov chains are homogeneous in time if the random vectors W 1, 2, . . . , are identically distributed and the transition functions  μ n (x) =  μ(x) and  n is homogeneous in Σ n (x) = Σ(x) do not depend on n. Note that, the Markov chain X time due to involvement of additional counting components.

18 | 1 Multivariate modulated Markov log-price processes (LPP)

1.3 Autoregressive LPP In this section, we consider a number of typical examples of discrete time modulated Markov log-price and price processes of the autoregressive type.

1.3.1 Autoregressive-type log-price and price processes Let us consider a model known as an autoregressive AR(p) model. For simplicity, we consider a univariate log-price process defined by the following stochastic transition dynamic relation: Y n = a0 + a1 Y n−1 + · · · + a p Y n−p + σW n ,

n = 1, 2, . . . ,

(1.50)

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y components, (b) W1 , W2 , . . . , is a sequence of real-valued i.i.d. random variables, 0 and the random sequence W1 , W2 , . . . , are independent, (c) the random vector Y (d) p is a positive integer number, and (e) a0 , a1 , . . . , a p , σ are the real-valued con­ stants. The corresponding price process, which can be referred to as an exponential au­ toregressive process, is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.51)

The standard case is where the random variables W1 , W2 , . . . , have the standard normal distribution with the parameters 0 and 1. It is useful to note that the above autoregressive model can be defined in the equiv­ alent form of stochastic difference equation Y n − Y n−1 = a0 + (a1 − 1)Y n−1 + · · · + a p Y n−p + σW n ,

n = 1, 2, . . . .

(1.52)

Also note that the above stochastic difference equation can be represented in the equivalent alternative form Y n − Y n−1 = a0 + a1 (Y n−1 − Y n−2 ) + · · · + ap−1 (Y n−p+1 − Y n−p ) + σ  W n ,

n = 1, 2, . . . .

(1.53)

Indeed, the stochastic difference equation (1.53) can be rewritten in the form of equation (1.52) with the coefficients a0 = a0 , a1 = a1 + 1, a2 = a2 − a1 , . . . , a p−1 = ap−1 − ap−2 , a p = −ap−1 , σ = σ  . The log-price process Y n is not a Markov chain. However, this process can be con­ sidered as the first component of some p-dimensional Markov chain.

1.3 Autoregressive LPP | 19

Let us introduce the p-dimensional vector process n = (Y n,1 , . . . , Y n,p ) = (Y n , . . . , Y n−p+1 ) , Y

n = 1, 2, . . . ,

(1.54)

n = 1, 2, . . . .

(1.55)

and the sequence of p-dimensional i.i.d. random vectors  n = (W n,1 , . . . , W n,p ) = (W n , 0, . . . , 0) , W

Using the above formula and relations (1.50), (1.54), and (1.55), we get the follow­ ing vector stochastic transition dynamic relation (written as a system of stochastic transition dynamic relations): ⎧ ⎪ Y n,1 = a0 + a1 Y n−1,1 + · · · + a p Y n−1,p + σW n,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y = Y n−1,1 + W n,2 , ⎪ ⎪ ⎨ n,2 ... ... (1.56) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p = Y n−1,p−1 + W n,p , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . . This dynamic relation can be rewritten in the equivalent form of vector autore­ gressive stochastic difference equation (written as a system of stochastic difference equations), ⎧ ⎪ Y n,1 − Y n−1,1 = a0 + (a1 − 1)Y n−1,1 + · · · + a p Y n−1,p + σW n,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y − Y n−1,2 = Y n−1,1 − Y n−1,2 + W n,2 , ⎪ ⎪ ⎨ n,2 ... ... (1.57) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p − Y n−1,p = Y n−1,p−1 − Y n−1,p + W n,p , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . . The vector stochastic difference equation (1.57) is a particular case of the vector stochastic difference equation (1.32).  n , n = 0, 1, . . . , is a discrete time homogeneous multivariate Thus, the process Y 0 and vector and Ornstein–Uhlenbeck-type process, with the initial random value Y λ and Λ, Σ, determined by relations (1.50) and (1.54)–(1.57). matrix parameters 

1.3.2 Nonlinear autoregressive-type log-price and price processes Let us consider a model that can be interpreted as a nonlinear autoregressive-type pro­ cess. For simplicity, we consider a univariate log-price process defined by the following stochastic transition dynamic relation: Y n = A n ( Y n −1 , . . . , Y n − p , W n ) ,

n = 1, 2, . . . ,

(1.58)

20 | 1 Multivariate modulated Markov log-price processes (LPP) 0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y components, (b) W1 , W2 , . . . , is a sequence of real-valued independent random vari­ 0 and the random sequence W1 , W2 , . . . , are indepen­ ables, (c) the random vector Y dent, (d) p is a positive integer number, and (e) A n (y1 , . . . , y p , w), n = 1, 2, . . . , are the measurable functions acting from the space Rp+1 to R1 . The corresponding price process, which can be referred to as a nonlinear expo­ nential autoregressive-type process, is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.59)

The log-process Y n is not a Markov chain. Let us show in which way this model can be imbedded in a model of multivariate Markov log-price process. As given in Subsection 1.3.1, let us introduce the p-dimensional vector process  n = (Y n,1 , . . . , Y n,p ) = (Y n , . . . , Y n−p+1) , Y

n = 1, 2, . . . ,

(1.60)

and the sequence of p-dimensional independent random vectors  n = (W n,1 , . . . , W n,p ) = (W n , 0, . . . , 0) , W

n = 1, 2, . . . .

(1.61)

Using relations (1.58), (1.60), and (1.61), we get the following vector stochastic transition dynamic relation (written as a system of stochastic transition dynamic re­ lations): ⎧ ⎪ Y n,1 = A n (Y n−1,1, . . . , Y n−1,p , W n,1) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y = Y n−1,1 + W n,2 , ⎪ ⎪ ⎨ n,2 ... ... (1.62) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p = Y n−1,p−1 + W n,p , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . . The vector transition dynamic relation (1.62) is a particular case of the vector tran­ sition dynamic relation (1.4).  n , n = 0, 1, . . . , is an inhomogeneous multivariate Thus, the vector process Y 0 and the corresponding vector tran­ Markov chain with the initial random value Y sition dynamic functions determined by relations (1.58) and (1.62). A particular case of the above model is an inhomogeneous in time autoregres­ sive-type model, where the log-price process Y n is given by the following transition dynamic relation: Y n = a n,0 + a n,1 Y n−1 + · · · + a n,p Y n−p + σ n W n ,

n = 1, 2, . . . ,

(1.63)

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y components, (b) W1 , W2 , . . . , is a sequence of independent real-valued random vari­ 0 and the random sequence W1 , W2 , . . . , are indepen­ ables, (c) the random vector Y dent, (d) p is a positive integer number, and (e) a n,0 , a n,1 , . . . , a n,p , σ n , n = 1, 2, . . . , are the real-valued constants.

1.3 Autoregressive LPP

| 21

1.3.3 Modulated autoregressive log-price and price processes Let us consider a process that can be interpreted as a modulated nonlinear autoregres­ sive-type process. For simplicity, we consider a log-price process Z n = (Y n , X n ) with a univariate log-price component Y n . Here and henceforth X(r) = X × · · · × X is r-times product of the space X, where X is a measurable space with a σ-algebra of measurable subsets BX . The log-price process Z n is defined by the following stochastic transition dynamic relation: ⎧  ⎪ Y = A n Y n −1 , . . . , Y n − p , X n −1 , . . . , X n − r , W n , ⎪ ⎪ ⎨ n X n = C n ( X n −1 , . . . , X n − r , U n ) , (1.64) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y  0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector with com­ components, (b) X ponents taking values in a measurable space X(r) , (c) (W1 , U1 ), (W2 , U2 ) . . . , is a se­ quence of independent random vectors taking values in the space R1 × U, where U is a measurable space with a σ-algebra of measurable subsets BU , (d) the random 0 , X  0 ) and the random sequence (W1 , U1 ), (W2 , U2 ) . . . , are indepen­ vector  Z0 = (Y dent, (e) p and r are the positive integer numbers, (f) A n (y1 , . . . , y p , x1 , . . . , x r , w), n = 1, 2, . . . , are the measurable functions acting from the space Rp × X(r) × R1 to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X. The corresponding modulated price process, which can be referred to as a mod­ ulated nonlinear exponential autoregressive type process, is defined by the following relation: V n = (S n , X n ) = (e Y n , X n ) , n = 0, 1, . . . . (1.65)

The log-process Y n is not a Markov chain. Let us show in which way this model can be imbedded in a model of the multivariate Markov log-price process. As given in Subsection 1.3.2, let us introduce the p-dimensional vector pro­  n = (Y n,1 , . . . , Y n,p ) = (Y n , . . . , Y n−p+1) , n = 1, 2, . . . , and the sequence cess, Y  n = (W n,1 , . . . , W n,p ) = (W n , 0, of p-dimensional independent random vectors W . . . , 0) , n = 1, 2, . . . , respectively, according to relations (1.60) and (1.61). Also, let us introduce the r-dimensional vector process  n = (X n,1 , . . . , X n,r ) = (X n , . . . , X n−r+1 ) , X

n = 1, 2, . . . .

(1.66)

Using relations (1.60), (1.61), (1.64), and (1.66), we get the following vector stochas­ tic transition dynamic relation (written as a system of stochastic transition dynamic

22 | 1 Multivariate modulated Markov log-price processes (LPP) relations): ⎧ ⎪ Y n,1 = A n (Y n−1,1 , . . . , Y n−1,p , X n−1,1 , . . . , X n−1,r , W n,1) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y = Y n−1,p−1 + W n,p , ⎪ ⎪ ⎨ n,p X n,1 = C n (X n−1,1 , . . . , X n−1,r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎪ X n,2 = X n−1,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,r = X n−1,r−1 , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

(1.67)

The vector stochastic transition dynamic relation (1.67) is a particular case of the vector transition dynamic relation (1.4). n , X  n ), n = 0, 1, . . . , is inhomogeneous multi­ Thus, the vector process  Z n = (Y variate Markov chain with the initial random value  Z0 and the corresponding vector transition dynamic functions determined by relations (1.64) and (1.67). Note that in this case, the role of the index component is played by the process  n , which has the phase space X(r) . X A particular case of the above model is an inhomogeneous in time autoregres­ sive-type model, where the modulated log-price process Z n = (Y n , X n ) is given by the following stochastic transition dynamic relation: ⎧ ⎪ Y n = a n,0 (X n−1 , . . . , X n−r ) + a n,1(X n−1 , . . . , X n−r )Y n−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + · · · + a n,p (X n−1 , . . . , X n−r )Y n−p ⎪ ⎪ ⎨ + σ n ( X n −1 , . . . , X n − r ) W n , (1.68) ⎪ ⎪ ⎪ ⎪ ⎪ X n = C n ( X n −1 , . . . , X n − r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-val­ where (a) Y 0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector with ued components (b) X components taking values in the space X(r) , (c) (W1 , U1 ), (W2 , U2 ) . . . , is a sequence of independent random vectors taking values in the space R1 × U, (d) the random 0 , X 0 ) and the random sequence (W1 , U1 ), (W2 , U2 ) . . . , are indepen­ vector  Z0 = (Y dent, (e) p and r are positive integer numbers, (f) a n,i (x1 , . . . , x r ), σ n (x1 , . . . , x r ), i = 1, . . . , p, n = 1, 2, . . . , are the measurable functions acting from the space X(r) to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X.

1.3 Autoregressive LPP

| 23

1.3.4 Autoregressive moving average-type log-price and price processes Let us consider a model known as an autoregressive moving average model, ARMA(p, q). For simplicity, we consider a univariate log-price process defined by the following stochastic transition dynamic relation: Y n = a 0 + a 1 Y n −1 + · · · + a p Y n − p + b 1 W n−1 + · · · + b q W n−q + σW n ,

n = 1, 2, . . . ,

(1.69)

0 = (Y 0 , . . . , Y −p+1 , W0 , . . . , W−q+1 ) is a (p + q)-dimensional ran­ where (a) Y dom vector with real-valued components, (b) W1 , W2 , . . . , is a sequence of real0 and the random sequence valued i.i.d. random variables, (c) the random vector Y W1 , W2 , . . . , are independent, (d) p and q are the positive integer numbers, and (e) a0 , a1 , . . . , a p , b 1 , . . . , b q , σ are the real-valued constants. The standard case is where the random variables W1 , W2 , . . . , have the standard normal distribution with the parameters 0 and 1. The corresponding price process, which can be referred to as an exponential au­ toregressive moving average-type process, is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.70)

The log-price process Y n is not a Markov chain. However, this process can be rep­ resented as the first component of some (p + q)-dimensional Markov chain. Let us introduce the (p + q)-dimensional vector process n = (Y n,1 , . . . , Y n,p , Y n,p+1 , . . . , Y n,p+q) Y = ( Y n , . . . , Y n − p +1 , W n , . . . , W n − q +1 ) ,

(1.71)

n = 1, 2, . . . , and the sequence of (p + q)-dimensional i.i.d. random vectors  n = (W n,1 , W n,2 , . . . , W n,p , W n,p+1 , W n,p+2 , . . . , W n,p+q ) W = (W n , 0, . . . 0, W n , 0, . . . , 0) ,

(1.72)

n = 1, 2, . . . . Using this formula and relations (1.69), (1.71), and (1.72), we get the following vec­ tor stochastic transition dynamic relation (written as a system of stochastic transition

24 | 1 Multivariate modulated Markov log-price processes (LPP) dynamic relations): ⎧ ⎪ Y n,1 = a0 + a1 Y n−1,1 + · · · + a p Y n−1,p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + b 1 Y n−1,p+1 + · · · + b q Y n−1,p+q + σW n,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Y n,p = Y n−1,p−1 + W n,p , ⎪ ⎪ Y n,p+1 = W n,p+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+2 = Y n−1,p−1 + W n,p+2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+q = Y n−1,p+q−1 + W n,p+q , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

(1.73)

This dynamic relation can be rewritten in the equivalent form of vector stochastic difference equation (written as a system of stochastic difference equations), ⎧ ⎪ Y n,1 − Y n−1,1 = a0 + (a1 − 1)Y n−1,1 + · · · + a p Y n−1,p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + b 1 Y n−1,p+1 + · · · + b q Y n−1,p+q + σW n,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 − Y n−1,2 = Y n−1,1 − Y n−1,2 + W n,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Y n,p − Y n−1,p = Y n−1,p−1 − Y n−1,p + W n,p , (1.74) ⎪ ⎪ Y n,p+1 − Y n −1,p+1 = − Y n −1,p+1 + W n,p+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+2 − Y n−1,p+2 = Y n−1,p−1 − Y n−1,p+2 + W n,p+2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y − Y = Y n,p+ q n −1,p+ q n −1,p+ q −1 − Y n −1,p+ q + W n,p+ q , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . . The vector stochastic difference equation (1.74) is a particular case of the vector stochastic difference equation (1.32). n , n = 0, 1, . . . , is also a discrete time homogeneous mul­ Thus, the process Y 0 and, re­ tivariate Ornstein–Uhlenbeck-type process, with the initial random value Y λ and Λ, Σ determined by relations (1.69) and spectively, vector and matrix parameters  (1.74).

1.3.5 Nonlinear autoregressive moving average-type log-price and price processes Let us consider a model that can be interpreted as a nonlinear autoregressive mov­ ing average-type process. For simplicity, we consider a univariate log-price process

1.3 Autoregressive LPP

| 25

defined by the following stochastic transition dynamic relation: Y n = A n ( Y n −1 , . . . , Y n − p , W n −1 , . . . , W n − q , W n ) ,

n = 1, 2, . . . ,

(1.75)

0 = (Y 0 , . . . , Y −p+1 , W0 , . . . , W−q+1 ) is a (p + q)-dimensional ran­ where (a) Y dom vector with real-valued components, (b) W1 , W2 , . . . , is a sequence of real0 and the random valued independent random variables, (c) the random vector Y sequence W1 , W2 , . . . , are independent, (d) p and r are positive integer numbers, and (e) A n (y1 , . . . , y p , w1 , . . . , w q , w), n = 1, 2, . . . , are the measurable functions acting from the space Rp+q+1 to R1 . The corresponding price process, which can be referred to as a nonlinear exponen­ tial autoregressive moving average-type process, is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.76)

The log-price process Y n is not a Markov chain. However, this process can be con­ sidered as the first component of some (p + q)-dimensional Markov chain. As given in Subsection 1.3.4, let us introduce the (p + q)-dimensional vector pro­ cess n = (Y n,1 , . . . , Y n,p , Y n,p+1 , . . . , Y n,p+q) Y = ( Y n , . . . , Y n − p +1 , W n , . . . , W n − q +1 ) ,

(1.77)

n = 1, 2, . . . , and the sequence of (p + q)-dimensional independent random vectors  n = (W n,1 , W n,2 , . . . , W n,p , W n,p+1 , W n,p+2 , . . . , W n,p+q ) W = (W n , 0, . . . 0, W n , 0, . . . , 0) ,

(1.78)

n = 1, 2, . . . . Using relations (1.75), (1.77), and (1.78), we get the following vector stochastic tran­ sition dynamic relation (written as a system of stochastic transition dynamic rela­ tions), which is a particular case of the vector transition dynamic relation (1.4): ⎧ ⎪ ⎪ ⎪ Y n,1 = A n (Y n−1,1 , . . . , Y n−1,p+q , W n,1 ) ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p = Y n−1,p−1 + W n,p , ⎪ ⎪ ⎨ Y n,p+1 = W n,p+1 , (1.79) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+2 = Y n−1,p−1 + W n,p+2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+q = Y n−1,p+q−1 + W n,p+q , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

26 | 1 Multivariate modulated Markov log-price processes (LPP) n , n = 0, 1, . . . , is an inhomogeneous in time multivariate Thus, the process Y 0 and the corresponding vector dynamic Markov chain with the initial random value Y transition functions determined by relations (1.75) and (1.79). A particular case of the above model is an inhomogeneous in time autoregres­ sive-type model, where the log-price process Y n is given by the following transition dynamic relation:

Y n = a n,0 + a n,1 Y n−1 + · · · + a n,p Y n−p + b n,1 W n−1 + · · · + b n,q W n−q + σ n W n ,

n = 1, 2, . . . ,

(1.80)

0 = (Y 0 , . . . , Y −p+1 , W0 , . . . , W−q+1 ) is a (p + q)-dimensional random where (a) Y vector with real-valued components, (b) W1 , W2 , . . . , is a sequence of real-valued 0 and the random sequence independent random variables, (c) the random vector Y W1 , W2 , . . . , are independent, (d) p and q are the positive integer numbers, and (e) a n,0 , a n,1 , . . . , a n,p , b n,1 , . . . , b n,q , σ n , n = 1, 2, . . . , are the real-valued constants.

1.3.6 Modulated mixed autoregressive moving average-type log-price and price processes Let us consider a model that can be interpreted as a modulated nonlinear autore­ gressive moving average-type process. For simplicity, we consider a log-price process Z n = (Y n , X n ) with a univariate log-price component Y n . The log-price process Z n is defined by the following stochastic transition dynamic relation: ⎧ ⎪ ⎪ ⎪ Y n = A n ( Y n −1 , . . . , Y n − p , W n −1 , . . . , W n − q , X n −1 , . . . , X n − r , W n ) , ⎨ X n = C n ( X n −1 , . . . , X n − r , U n ) , (1.81) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 = (Y 0 , . . . , Y −p+1 , W0 , . . . , W−q+1 ) is the (p + q)-dimensional random where (a) Y  0 = (X0 , . . . , X−r+1 ) is a r-dimensional vector with real-valued components, (b) X random vector with components taking values in space X(r) , (c) (W1 , U1 ), (W2 , U2 ), . . . , is a sequence of independent random vectors taking values in the space R1 × U, where U is a measurable space with a σ-algebra of measurable subsets BU , (d) the 0 , X  0 ) and the random sequence (W1 , U1 ), (W2 , U2 ), . . . , Z0 = (Y random vector  are independent, (e) p, q, and r are the positive integer numbers, (f) A n (y1 , . . . , y p , w1 , . . . , w q , x1 , . . . , x r , w), n = 1, 2, . . . , are the measurable functions acting from the space Rp+q × X(r) × R1 to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X. The corresponding price process, which can be referred to as a modulated nonlin­ ear exponential autoregressive moving average-type process, is defined by the follow­ ing relation: V n = (S n , X n ) = (e Y n , X n ) , n = 0, 1, . . . . (1.82)

1.3 Autoregressive LPP

| 27

The log-price process Y n is not a Markov chain. However, this process can be rep­ resented as the first component of some (p + q)-dimensional Markov chain. As given in Subsection 1.3.5, let us introduce the (p + q)-dimensional vector pro­  n = (Y n,1 , . . . , Y n,p , Y n,p+1 , . . . , Y n,p+q) = (Y n , . . . , Y n−p+1 , W n , . . . , W n−q+1 ), cess Y n = n = 1, 2, . . . , and the sequence of (p + q)-dimensional i.i.d. random vectors, W (W n,1 , W n,2 , . . . , W n,p , W n,p+1 , W n,p+2 , . . . , W n,p+q) = (W n , 0, . . . 0, W n , 0, . . . , 0), n = 1, . . . , according, respectively, relations (1.77) and (1.78), and, also, the r-dimensional vector process,  n = (X n,1 , . . . , X n,r ) = (X n , . . . , X n−r+1 ) , X

n = 1, 2, . . . .

(1.83)

Using relations (1.77), (1.78), (1.81), and (1.83), we get the following vector stochas­ tic transition dynamic relation (written as a system of stochastic transition dynamic relations): ⎧ Y n,1 = A n (Y n−1,1, . . . , Y n−1,p+q , X n−1,1 , . . . , X n−1,r , W n,1) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p = Y n−1,p−1 + W n,p , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+1 = W n,p+1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = Y n−1,p−1 + W n,p+2 , Y ⎪ ⎪ ⎨ n,p+2 ... ... (1.84) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+q = Y n−1,p+q−1 + W n,p+q , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,1 = C n (X n−1,1, . . . , X n−1,r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,2 = X n−1,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,r = X n−1,r−1 , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . . The vector transition dynamic relation (1.84) is a particular case of the vector tran­ sition dynamic relation (1.4). n , X n = (Y  n ), n = 0, 1, . . . , is an inhomogeneous in time Thus, the vector process Z multivariate Markov chain with the initial random value  Z0 and the corresponding vector dynamic transition functions determined by relations (1.81) and (1.84). A particular case of the above model is an inhomogeneous in time autoregressive moving average-type model, where the modulated log-price process Z n = (Y n , X n ) is

28 | 1 Multivariate modulated Markov log-price processes (LPP) given by the following stochastic transition dynamic relation: ⎧ Y = a n,0 (X n−1 , . . . , X n−r ) + a n,1(X n−1 , . . . , X n−r )Y n−1 ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ + · · · + a n,p (X n−1 , . . . , X n−r )Y n−p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + b n,1(X n−1 , . . . , X n−r )W n−1 ⎪ ⎪ ⎨ + · · · + b n,q (X n−1 , . . . , X n−r )W n−q ⎪ ⎪ ⎪ ⎪ ⎪ + σ n ( X n −1 , . . . , X n − r ) W n , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X = C n ( X n −1 , . . . , X n − r , U n ) , ⎪ ⎪ ⎪ n ⎪ ⎩ n = 1, 2, . . . ,

(1.85)

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y 0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector with com­ components, (b) X ponents taking values in the space X(r) , (c) (W1 , U1 ), (W2 , U2 ) . . . , is a sequence of independent random vectors taking values in the space R1 × U, (d) the random vec­ 0 , X 0 ) and the random sequence (W1 , U1 ), (W2 , U2 ), . . . , are indepen­ tor  Z0 = (Y dent, (e) p, q, and r are positive integer numbers, (f) a n,i (x1 , . . . , x r ), b n,j(x1 , . . . , x r ), σ n (x1 , . . . , x r ), i = 1, . . . , p, j = 1, . . . , q, n = 1, 2, . . . , are the measurable functions acting from the space X(r) to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the mea­ surable functions acting from the space X(r) × U to the space X.

1.4 Autoregressive stochastic volatility LPP In this section, we consider a number of typical examples of discrete time modulated Markov log-price and price processes of autoregressive stochastic volatility type.

1.4.1 Centered autoregressive conditional heteroskedastic-type log-price and price processes Let us consider a model known as a centered autoregressive conditional heteroskedas­ tic ARCH(p) type model. For simplicity, we consider a univariate log-price process, which is defined by the following stochastic difference equation: Y n − Y n −1 = g κ ( σ n ) W n ,

n = 1, 2, . . . ,

(1.86)

where  σ n = d 0 + d 1 ( Y n −1 − e 1 Y n −2 )2 + · · · + d p −1 ( Y n − p +1 − e p −1 Y n − p )2

21

,

n = 1, 2, . . . ,

(1.87)

1.4 Autoregressive stochastic volatility LPP |

29

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued and (a) Y components, (b) W1 , W2 , . . . , is a sequence of real-valued i.i.d. random variables, 0 and the random sequence W1 , W2 , . . . , are independent, (c) the random vector Y (d) p is a positive integer number, (e) d0 , d1 , . . . , d p−1 are nonnegative constants, (f) e1 , . . . , e p−1 are the constants taking values in the interval [0, 1], and (g) g κ (·) is a function from the class Gκ , for some κ ≥ 0. Here and henceforth, Gκ = {g κ (·)} is, for every κ ≥ 0, the class of measurable g κ (y) functions acting from [0, ∞) to [0, ∞) such that supy≥0 1+ y κ < ∞. The case with the parameter κ = 1 corresponds to ARCH (p)-type models. In par­ ticular, the standard model corresponds to the case where function g κ (y) ≡ y and constants e i = 1, i = 1, . . . , p − 1. The case with the parameter κ = 21 corresponds to CIR(p) (Cox, Ingersoll, Ross) autoregressive-type models. In particular, the standard model corresponds to the case √ where function g κ (y) ≡ y, parameter p = 2 and the constants d0 = 0, d1 = 1, e1 = 0. The standard assumption is also that the random variables W1 , W2 , . . . , have the standard normal distribution with parameters 0 and 1. The volatility transformation g κ (·) penetrating relation (1.86) plays a role of smoothing transformation. It prevents the volatility of log-price process Y n from tak­ ing too large values that can cause in some cases appearance of infinite values for expected log-rewards. In particular, such problems can appear for the standard ARCH (p) models with Gaussian noise variables W n , but they do not appear for standard CIR(p) models with Gaussian noise variables W n . The corresponding examples are given in the second volume of the book. The corresponding centered autoregressive conditional heteroskedastic price pro­ cess is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.88)

The log-price process Y n is not a Markov chain. However, this process can be rep­ resented as the first component of some p-dimensional Markov chain. Let us introduce the p-dimensional vector process n = (Y n,1 , . . . , Y n,p ) = (Y n , . . . , Y n−p+1 ) , Y

n = 1, 2, . . . ,

(1.89)

n = 1, 2, . . . .

(1.90)

and the sequence of p-dimensional i.i.d. random vectors  n = (W n,1 , . . . , W n,p ) = (W n , 0, . . . , 0) , W

Using relations (1.86), (1.87), (1.89), and (1.90), we get the following vector stochas­ tic transition dynamic relation (written as a system of stochastic transition dynamic

30 | 1 Multivariate modulated Markov log-price processes (LPP) relations):

⎧ ⎪ Y n,1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y ⎪ ⎪ ⎨ n,2 ... ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p ⎪ ⎪ ⎪ ⎪ ⎩ n

= Y n−1,1 + g κ (σ (Y n−1,1 , . . . , Y n−1,p))W n,1 , = Y n−1,1 + W n,2 ,

...

(1.91)

= Y n−1,p−1 + W n,p , = 1, 2, . . . ,

where the functions σ (y1 , . . . , y p ) defined for y1 , . . . , y p ∈ R1 by the following for­ mula: 1  σ ( y 1 , . . . , y p ) = d 0 + d 1 ( y 1 − e 1 y 2 )2 , . . . , d p −1 ( y p −1 − e p −1 y p )2 2 . (1.92) The vector transition dynamic relation (1.91) is a particular case of the vector tran­ sition dynamic relation (1.4).  n , n = 0, 1, . . . , is a homogeneous in time multivariate Thus, the vector process Y 0 and the corresponding vector transition Markov chain with the initial random value Y dynamic functions determined by relations (1.91) and (1.92).

1.4.2 Autoregressive conditional heteroskedastic-type log-price and price processes Let us consider a model known as the autoregressive conditional heteroskedastic AR(p )/ARCH(p) type model. For simplicity, we consider an univariate log-price pro­ cess, which is defined by the following stochastic autoregressive difference equation: Y n − Y n −1 = a 0 + a 1 ( Y n −1 − f 1 Y n −2 ) + · · · + a p  −1 ( Y n − p  +1 − f p −1 Y n − p  ) + g κ ( σ n ) W n ,

n = 1, 2, . . . ,

(1.93)

where  σ n = d 0 + d 1 ( Y n −1 − e 1 Y n −2 )2 + · · · + d p −1 ( Y n − p +1 − e p −1 Y n − p )2

21

,

n = 1, 2, . . . ,

(1.94)

0 = (Y 0 , . . . , Y − max(p ,p)+1 ) is a max(p, p )-dimensional random vector with and (a) Y real-valued components, (b) W1 , W2 , . . . , is a sequence of real-valued i.i.d. random 0 and the random sequence W1 , W2 , . . . , are inde­ variables, (c) the random vector Y pendent, (d) p and p are the positive integer numbers, (e) a0 , a1 , . . . , a p −1 are the real-valued constants, (f) d0 , d1 , . . . , d p−1 are nonnegative constants, (g) e1 , . . . , e p−1 , f1 , . . . , f p−1 are constants taking values in the interval [0, 1], and (h) g κ (·) is a function from the class Gκ , for some κ ≥ 0. Respectively, the corresponding autoregressive conditional heteroskedastic-type price process is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.95)

1.4 Autoregressive stochastic volatility LPP |

31

Note first of all that the consideration can be restricted by the case p = p. Indeed, this can be achieved by choosing coefficients d p , . . . , d p−1 = 0 if p < p or coefficients a p , . . . , a p −1 = 0 if p < p . The log-price process Y n is not a Markov chain. However, this process can be rep­ resented as the first component of some p-dimensional Markov chain. As given in Subsection 1.4.1, let us introduce the p-dimensional vector process n = (Y n,1 , . . . , Y n,p ) = (Y n , . . . , Y n−p+1 ) , Y

n = 1, 2, . . . ,

(1.96)

n = 1, 2, . . . .

(1.97)

and the sequence of p-dimensional i.i.d. random vectors  n = (W n,1 , . . . , W n,p ) = (W n , 0, . . . , 0) , W

Using relations (1.93), (1.94), (1.96), and (1.97), we get the following vector stochas­ tic transition dynamic relation (written as a system of stochastic transition dynamic relations) ⎧ Y n,1 = Y n−1,1 + a0 + a1 (Y n−1,1 − f1 Y n−1,2) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + · · · + a p−1 (Y n−1,p−1 − f p−1 Y n−1,p) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + g κ (σ (Y n−1,1 , . . . , Y n−1,p ))W n,1 , ⎪ ⎪ ⎨ Y n,2 = Y n−1,1 + W n,2 , (1.98) ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p = Y n−1,p−1 + W n,p , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , where  σ n ( y 1 , . . . , y p ) = d 0 + d 1 ( y 1 − e 1 y 2 )2 + · · · + d p −1 ( y p −1 − e p −1 y p )2

21

,

n = 1, 2, . . . .

(1.99)

The vector stochastic transition dynamic relation (1.98) is a particular case of the vector transition dynamic relation (1.4). n , n = 0, 1, . . . , is a homogeneous in time multivariate Thus, the vector process Y 0 and the corresponding vector dynamic Markov chain with the initial random value Y transition functions determined by relations (1.98) and (1.99).

1.4.3 Nonlinear autoregressive conditional heteroskedastic-type log-price and price processes For simplicity, we consider a univariate log-price process, which is defined by the fol­ lowing stochastic transition dynamic relation: Y n = An (Y n−1 , . . . , Y n−p ) + A n ( Y n −1 , . . . , Y n − p ) W n ,

n = 1, 2, . . . ,

(1.100)

32 | 1 Multivariate modulated Markov log-price processes (LPP) 0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y components, (b) W1 , W2 , . . . , is a sequence of real-valued independent random vari­ 0 and the random sequence W1 , W2 , . . . , are indepen­ ables, (c) the random vector Y dent, (d) p is a positive integer number, and (e) An (y1 , . . . , y p ), A n ( y 1 , . . . , y p ), n = 1, 2, . . . , are the measurable functions acting from the space Rp to R1 . This model is, in fact, the particular case of the model introduced in the Subsec­ tion 1.3.2, with the transition dynamic functions A n (y1 , . . . , y p , w) = An (y1 , . . . , y p )+ A n ( y 1 , . . . , y p ) w, n = 1, 2, . . . . The particular variant of this model is an inhomogeneous in time autoregressive conditional heteroskedastic-type model given by the following stochastic difference equation:

Y n − Y n−1 = a n,0 + a n,1(Y n−1 − f n,1 Y n−2 ) + · · · + a n,p−1 (Y n−p+1 − f n,p−1 Y n−p ) + g κ (σ n )W n ,

n = 1, 2, . . . ,

(1.101)

where  σ n = d n,0 + d n,1(Y n−1 − e n,1 Y n−2 )2 + · · · + d n,p−1(Y n−p+1 − e n,p−1 Y n−p )2

21

,

n = 1, 2, . . . ,

(1.102)

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued com­ and (a) Y ponents, (b) W1 , W2 , . . . , is a sequence of real-valued independent random variables, 0 and the random sequence W1 , W2 , . . . , are independent, (c) the random vector Y (d) p is a positive integer number, (e) a n,0 , a n,1, . . . , a n,p−1, n = 1, 2, . . . , are the realvalued constants, (f) d n,0 , d n,1 , . . . , d n,p−1, n = 1, 2, . . . , are nonnegative constants, (g) e n,1 , . . . , e n,p−1 , f n,1 , . . . , f n,p−1, n = 1, 2, . . . , are constants taking values in the in­ terval [0, 1], and (h) g κ (·) is a function from the class Gκ for some κ ≥ 0.

1.4.4 Modulated nonlinear autoregressive conditional heteroskedastic-type log-price and price processes Let us consider a process that can be interpreted as a modulated nonlinear autoregres­ sive conditional heteroskedastic-type process. For simplicity, we consider a log-price process Z n = (Y n , X n ) with a univariate log-price component Y n . The process Z n is defined by the following stochastic transition dynamic relation: ⎧ ⎪ Y n = An (Y n−1 , . . . , Y n−p , X n−1 , . . . , X n−r ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ + A n ( Y n −1 , . . . , Y n − p , X n −1 , . . . , X n − r ) W n , (1.103) ⎪ ⎪ X n = C n ( X n −1 , . . . , X n − r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . ,

1.4 Autoregressive stochastic volatility LPP

| 33

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued where (a) Y  0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector with compo­ components, (b) X nents taking values in the space X(r) , (c) (W1 , U1 ), (W2 , U2 ), . . . , is a sequence of in­ dependent random vectors taking values in the space R1 ×U, where U is a measurable 0 , X 0 ) space with a σ-algebra of measurable subsets BU , (d) the random vector  Z0 = (Y and the random sequence (W1 , U1 ), (W2 , U2 ), . . . , are independent, (e) p and r are the positive integer numbers, (f) An (y1 , . . . , y p , x1 , . . . , x r ), A n ( y 1 , . . . , y p , x 1 , . . . , x r ), n = 1, 2, . . . , are the measurable functions acting from the space Rp × X(r) to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to X. The corresponding modulated price process, which can be referred to as a modu­ lated nonlinear exponential autoregressive-type process, is defined by the following relation: V n = (S n , X n ) = (e Y n , X n ) , n = 0, 1, . . . . (1.104)

This model is, in fact, the particular case of the model introduced in Subsec­ tion 1.3.3, with the transition dynamic functions A n (y1 , . . . , y p , x1 , . . . , x r , w) = An (y1 , . . . , y p , x1 , . . . , x r ) + A n ( y 1 , . . . , y p , x 1 , . . . , x r ) w and C n ( x 1 , . . . , x r , u ), n = 1, 2, . . . . A particular case of the above model is an inhomogeneous in time autoregres­ sive-type model, where the modulated log-price process Z n = (Y n , X n ) is given by the following vector stochastic difference equation: ⎧ Y n − Y n−1 = a n,0 (X n−1 , . . . , X n−r ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + a n,1 (X n−1 , . . . , X n−r )(Y n−1 − f n,1 (X1 , . . . , X r )Y n−2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + · · · + a n,p−1 (X n−1 , . . . , X n−r )(Y n−p+1 ⎪ ⎪ ⎨ − f n,p−1(X1 , . . . , X r )Y n−p ) ⎪ ⎪ ⎪ ⎪ ⎪ + g κ (σ n )W n , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n = C n ( X n −1 , . . . , X n − r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . ,

(1.105)

where  σ n = d n,0(X n−1 , . . . , X n−r ) + d n,1(X n−1 , . . . , X n−r )(Y n−1 − e n,1(X1 , . . . , X r )Y n−2 )2 + · · · + d n,p−1(X n−1 , . . . , X n−r )(Y n,p+1 1 −e n,p−1(X1 , . . . , X r )Y n−p )2 2 ,

n = 1, 2, . . . ,

(1.106)

0 = (Y 0 , . . . , Y −p+1 ) is a p-dimensional random vector with real-valued com­ and (a) Y  0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector takin values in ponents, (b) X

34 | 1 Multivariate modulated Markov log-price processes (LPP) the space X(r) , (c) (W1 , U1 ), (W2 , U2 ) . . . , is a sequence of independent random vec­ 0 , X 0 ) and the tors taking values in the space R1 × U, (d) the random vector  Z0 = (Y random sequence (W1 , U1 ), (W2 , U2 ) . . . , are independent, (e) p and r are the positive integer numbers, (f) a n,i (x1 , . . . , x r ), i = 0, . . . , p − 1, n = 1, 2, . . . , are the measur­ able functions acting from the space X(r) to R1 , (g) d n,i (x1 , . . . , x r ), i = 0, . . . , p − 1, n = 1, 2, . . . , are the measurable functions acting from the space X(r) to the interval [0, ∞), (h) e n,i (x1 , . . . , x r ), f n,i (x1 , . . . , x r ), i = 0, . . . , p − 1, n = 1, 2, . . . , are the mea­ surable functions acting from the space X(r) to the interval [0, 1], (i) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X, and (j) g κ (·) is a function from the class G κ for some κ ≥ 0.

1.4.5 Generalized autoregressive conditional heteroskedastic-type log-price and price processes Let us consider a model that can be referred to as a autoregressive conditional het­ eroskedastic model GARCH(p, q) type model. For simplicity, we consider a univariate log-price process Y n defined by the stochastic difference equation Y n − Y n −1 = g κ ( σ n ) W n ,

n = 1, 2, . . . ,

(1.107)

where  σ n = d 0 + d 1 ( Y n −1 − e 1 Y n −2 )2 + · · · + d p −1 ( Y n − p +1 − e p −1 Y n − p )2 1 2 +b 1 σ 2n−1 + · · · + b q σ 2n−q , n = 1, 2, . . . ,

(1.108)

0 = (Y 0 , . . . , Y −(p+q)+1 ) is a (p + q)-dimensional random vector with realand (a) Y valued components, (b) W1 , W2 , . . . , is a sequence of real-valued i.i.d. random vari­ 0 and the random sequence W1 , W2 , . . . , are indepen­ ables, (c) the random vector Y dent, (d) p and q are the positive integer numbers, (e) d0 , d1 , . . . , d p−1 and b 1 , . . . , b q are nonnegative constants, (f) e1 , . . . , e p−1 are constants taking values in the interval [0, 1], and (g) g κ (·) is a function from the class G κ , for some κ ≥ 0. The corresponding price process is defined by the following relation:

Sn = e Yn ,

n = 0, 1, . . . .

(1.109)

The standard case is where random variables W0 , W1 , . . . , have the standard nor­ mal distribution with parameters 0 and 1. Let us introduce the (p + q)-dimensional vector process  n = (Y n,1 , . . . , Y n,p , Y n,p+1 , . . . , Y n,p+q) Y = ( Y n , . . . , Y n − p +1 , σ n , . . . , σ n − q +1 ) ,

n = 1, 2, . . . ,

(1.110)

1.4 Autoregressive stochastic volatility LPP

| 35

and the sequence of (p + q)-dimensional i.i.d. random vectors,  n = (W n,1 , . . . , W n,p+q ) W = (W n , 0, . . . , 0) ,

n = 1, 2, . . . .

(1.111)

Using relations (1.107), (1.108) (1.110), and (1.111), we get the following vector stochastic transition dynamic relation (written as a system of stochastic transition dynamic relations): ⎧ ⎪ Y n,1 = Y n−1,1 + g κ (σ (Y n−1,1, . . . , Y n−1,p+q))W n,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎨ ... ... (1.112) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+q = Y n−1,p+g −1 + W n,p+q , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , where the functions σ (y1 , . . . , y p+q ) are defined for y1 , . . . , y p ∈ R1 , y p+1 , . . . , y p+q ∈ [0, ∞) by the following formula:  σ ( y 1 , . . . , y p + q ) = d 0 + d 1 ( y 1 − e 1 y 2 )2 + · · · + d p −1 ( y p −1 − e p −1 y p )2 1 2 +b 1 y2p+1 + · · · + b q y2p+q . (1.113)

The vector transition dynamic relation (1.112) is a particular case of the vector tran­ sition dynamic relation (1.4). n , n = 0, 1, . . . , is a homogeneous multivariate Markov Thus, the vector process Y 0 and the corresponding vector dynamic transi­ chain with the initial random value Y tion functions determined by relations (1.107), (1.108), and (1.112).

1.4.6 Nonlinear generalized autoregressive conditional heteroskedastic-type log-price and price processes Let us consider a model that can be referred to as a nonlinear autoregressive condi­ tional heteroskedastic-type model. For simplicity, we consider a univariate log-price process Y n defined by the following stochastic transition dynamic relation: Yn = σn Wn ,

n = 1, 2, . . . ,

(1.114)

where σ n = A n ( Y n −1 , . . . , Y n − p , σ n −1 , . . . , σ n − q ) ,

n = 1, 2, . . . ,

(1.115)

0 = (Y 0 , . . . , Y −(p+q)+1 ) is a (p + q)-dimensional random vector with real-val­ and (a) Y ued components, (b) W1 , W2 , . . . , is a sequence of real-valued i.i.d. random variables, 0 and the random sequence W1 , W2 , . . . , are independent, (d) p (c) the random vector Y

36 | 1 Multivariate modulated Markov log-price processes (LPP) and q are the positive integer numbers, and (e) A n (y1 , . . . , y p+q ), n = 1, 2, . . . , are the measurable functions acting from the space Rp+q to R1 . As given in Subsection 1.4.5, let us introduce the (p + q)-dimensional vector pro­ cess  n = (Y n,1 , . . . , Y n,p , Y n,p+1 , . . . , Y n,p+q) Y = ( Y n , . . . , Y n − p +1 , σ n , . . . , σ n − q +1 ) ,

n = 1, 2, . . . ,

(1.116)

and the sequence of (p + q)-dimensional independent random vectors  n = (W n,1 , . . . , W n,p+q ) = (W n , 0, . . . , 0) , W

n = 1, 2, . . . .

(1.117)

Using relations (1.114)–(1.117), we get the following vector stochastic transition dy­ namic relation (written as a system of stochastic transition dynamic relations): ⎧ ⎪ ⎪ ⎪ Y n,1 = A n (Y n−1,1 , . . . , Y n−1,p+q)W n,1 , ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎨ ... ... (1.118) ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+q = Y n−1,p+g −1 + W n,p+q , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . . The vector stochastic transition dynamic relation (1.118) is a particular case of the vector transition dynamic relation (1.4).  n , n = 0, 1, . . . , is a homogeneous multivariate Markov Thus, the vector process Y 0 and the corresponding vector dynamic transi­ chain with the initial random value Y tion functions determined by relations (1.114) and (1.118). The particular variant of the above model is an inhomogeneous in time autore­ gressive conditional heteroskedastic model GARCH(p, q)-type log-price process given by the following stochastic difference equation: Y n − Y n −1 = g κ ( σ n ) W n , where

n = 1, 2, . . . ,

(1.119)

 σ n = d n,0 + d n,1(Y n−1 − e n,1 Y n−2 )2 + · · · + d n,p−1(Y n−p+1 − e n,p−1 Y n−p )2 1 2 + b n,1 σ 2n−1 + · · · + b n,q σ 2n−q ,

(1.120) n = 1, 2, . . . ,

0 = (Y 0 , . . . , Y −(p+q)+1 ) is a (p + q)-dimensional random vector with realand (a) Y valued components, (b) W1 , W2 , . . . , is a sequence of real-valued i.i.d. random vari­ 0 and the random sequence W1 , W2 , . . . , are indepen­ ables, (c) the random vector Y dent, (d) p and q are the positive integer numbers, (e) d n,0 , . . . , d n,p−1, n = 1, 2, . . . , and b n,1 , . . . , b n,q , n = 1, 2, . . . , are nonnegative constants, (f) e n,1 , . . . , e n,p−1, n = 1, 2, . . . , are constants taking values in the interval [0, 1], and (g) g κ (·) is a function from the class Gκ , for some κ ≥ 0.

1.4 Autoregressive stochastic volatility LPP |

37

1.4.7 Modulated nonlinear generalized autoregressive conditional heteroskedastic-type log-price and price processes Let us consider a model that can be referred to as a modulated nonlinear autoregres­ sive conditional heteroskedastic-type model. For simplicity, we consider a log-price process Z n = (Y n , X n ) with a univariate log-price component Y n . The process Z n is defined by the following stochastic transition dynamic relation: ⎧ ⎪ ⎪ ⎪ Yn = σn Wn ⎨ X n = C n ( X n −1 , . . . , X n − r , U n ) , (1.121) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , where σ n = A n ( Y n −1 , . . . , Y n − p , σ n −1 , . . . , σ n − q , X n −1 , . . . , X n − r ) ,

n = 1, 2, . . . ,

(1.122)

0 = (Y 0 , . . . , Y −(p+q)+1 ) is a (p + q)-dimensional random vector with realand (a) Y  0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector tak­ valued components, (b) X ing values in the space X(r) , (c) (W1 , U1 ), (W2 , U2 ) . . . , is a sequence of independent random vectors taking values in the space R1 × U, where U is a measurable space 0 , X 0 ) and Z0 = (Y with a σ-algebra of measurable subsets BU , (d) the random vector  the random sequence (W1 , U1 ), (W2 , U2 ) . . . , are independent, (e) p, q and r are the positive integer numbers, (f) A n (y1 , . . . , y p+q , x1 , . . . , x r ), n = 1, 2, . . . , are the mea­ surable functions acting from the space Rp+q × X(r) to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X. The corresponding modulated price process, which can be referred to as a mod­ ulated nonlinear exponential generalized autoregressive-type process, is defined by the following relation:

V n = (S n , X n ) = (e Y n , X n ) ,

n = 0, 1, . . . .

(1.123)

As given in Subsection 1.4.6, let us introduce the (p + q)-dimensional vector pro­  n = (Y n,1 , . . . , Y n,p , Y n,p+1 , . . . , Y n,p+q) = (Y n , . . . , Y n−p+1 , σ n , . . . , σ n−q+1), cess, Y n = 1, 2, . . . , and the sequence of (p + q)-dimensional independent random vectors  n = (W n,1 , . . . , W n,p+q ) = (W n , 0, . . . , 0) , n = 1, 2, . . . , defined, respectively, by W relations (1.116) and (1.117), as well as the r-dimensional vector process,  n = (X n,1 , . . . , X n,r ) = (X n , . . . , X n−r+1 ) , X

n = 1, 2, . . . .

(1.124)

Using relations (1.116), (1.117), and (1.121)–(1.124), we get the following vector stochastic transition dynamic relation (written as a system of stochastic transition

38 | 1 Multivariate modulated Markov log-price processes (LPP) dynamic relations): ⎧ ⎪ ⎪ ⎪ Y n,1 = A n (Y n−1,1 , . . . , Y n−1,p+q , X n−1,1 , . . . , Y n−1,r)W n,1 , ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 = Y n−1,1 + W n,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y = Y n−1,p+g −1 + W n,p+q , ⎪ ⎪ ⎨ n,p+q X n,1 = C n (X n−1,1 , . . . , X n−1,r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎪ X n,2 = X n−1,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,r = X n−1,r−1 , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

(1.125)

The vector transition dynamic relation (1.125) is a particular case of the vector tran­ sition dynamic relation (1.4). n , X  n ), n = 0, 1, . . . , is a homogeneous multi­ Z n = (Y Thus, the vector process  Z0 and the corresponding vector variate Markov chain with the initial random value  dynamic transition functions determined by relations (1.121) and (1.125). The particular variant of the above model is a inhomogeneous in time modu­ lated autoregressive conditional heteroskedastic GARCH (p, q, r)-type log-price pro­ cess given by the following autoregressive stochastic difference equation: Y n − Y n −1 = g κ ( σ n ) W n , where

n = 1, 2, . . . ,

(1.126)

 σ n = d n,0(X n−1 , . . . , X n−r ) + d n,1(X n−1 , . . . , X n−r )(Y n−1 − e n,1 (X n−1 , . . . , X n−r )Y n−2)2 + · · · + d n,p−1(X n−1 , . . . , X n−r )(Y n−p+1

(1.127)

− e n,p−1(X n−1 , . . . , X n−r )Y n−p )2 + b n,1 (X n−1 , . . . , X n−r )σ 2n−1 1 2 + · · · + b n,q(X n−1 , . . . , X n−r )σ 2n−q , n = 1, 2, . . . , 0 = (Y 0 , . . . , Y −(p+q)+1 ) is a (p + q)-dimensional random vector with realand (a) Y 0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector tak­ valued components, (b) X ing values in the space X(r) , (c) (W1 , U1 ), (W2 , U2 ) . . . , is a sequence of independent 0 , X 0) random vectors taking values in the space R1 ×U, (d) the random vector  Z0 = (Y and the random sequence (W1 , U1 ), (W2 , U2 ) . . . , are independent, (e) p, q and r are the positive integer numbers, (f) d n,i (x1 , . . . , x r ), i = 0, . . . , p − 1, n = 1, 2, . . . , and b n,j (x1 , . . . , x r ), j = 1, . . . , q, n = 1, 2, . . . , are the measurable functions acting from the space X(r) to the interval [0, ∞), (g) e n,i (x1 , . . . , x r ), i = 0, . . . , p − 1, n = 1, 2, . . . , are the measurable functions acting from the space X(r) to the interval [0, 1], and (h) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X; (i) g κ (·) is a function from the class G κ for some κ ≥ 0.

1.4 Autoregressive stochastic volatility LPP

| 39

1.4.8 Stochastic volatility models of log-price and price processes Let us consider a model known as a stochastic volatility SV (p)-type model. For sim­ plicity, we consider a univariate log-price process Y n defined by the following stochas­ tic difference equation: Y n − Y n−1 = g κ,ν (σ n )W n ,

n = 1, 2, . . . ,

(1.128)

where ln σ n = a0 + a1 ln σ n−1 + · · · + a p ln σ n−p + cW n ,

n = 1, 2, . . . ,

(1.129)

σ 0 = (ln σ 0 , . . . , ln σ −p+1 ) is a p-dimensional random vector with real-val­ and (a) ln  ued components, (b) (W1 , W1 ), (W2 , W2 ), . . . , is a sequence of i.i.d. random vectors σ 0 and the random sequence with real-valued components, (c) the random vector ln      (W1 , W1 ), (W2 , W2 ) . . . , are independent, (d) p is a positive integer number, and (e) a0 , a1 , . . . , a p , c are the real-valued constants, (f) g κ,ν (·) is a function from the class Gκ,ν , for some κ, ν ≥ 0. Here and henceforth, Gκ,ν = {g κ,ν (·)} is, for every κ, ν ≥ 0, the class of functions g (e y ) acting from R1 to [0, ∞) such that supy∈R1 1+|κ,νy|κ e νy < ∞. The standard assumption is also that random vectors (W1 , W1 ), (W2 , W2 ), . . . , have normal distributions. As for autoregressive conditional heteroskedastic-type models considered in pre­ vious subsections, the volatility transformation g κ,ν (·) penetrating relation (1.86) plays a role of smoothing transformation. It prevents the volatility of log-price process Y n from taking too large values that can cause in some cases appearance of infinite values for expected log-rewards. In particular, such problems can appear even for the standard variant of the above SV (p), where the parameters κ = 0, ν = 1 and the function g κ,ν (y) = y. The corre­ sponding examples will be given in the second volume of the book. The corresponding price process is defined by the following relation: Sn = e Yn ,

n = 0, 1, . . . .

(1.130)

Let us show in which way this model can be imbedded in a model of multivariate Markov modulated log-price process. Let us define the (p + 1)-dimensional discrete time process  n = (Y n,1 , . . . , Y n,p+1) Y = (Y n , ln σ n , ln σ n−1 , . . . , ln σ n−p+1 ) ,

n = 1, 2, . . . ,

(1.131)

and the (p + 1)-dimensional sequence of i.i.d. random vectors,  n = (W n,1 , . . . , W n,p+1) = (W n , W n , 0, . . . , 0) , W

n = 1, 2, . . . .

(1.132)

40 | 1 Multivariate modulated Markov log-price processes (LPP) Using relations (1.128), (1.129), (1.131), and (1.132), we get the following vector stochastic transition dynamic relation (written as a system of stochastic transition dynamic relations), ⎧  ⎪ Y n,1 = Y n−1,1 + g κ,ν (e a0 +a1 Y n−1,2 +···+a p Y n−1,p+1 +cW n )W n , ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ Y n,2 = a0 + a1 Y n−1,2 + · · · + a p Y n−1,p+1 + cW n , ⎪ ⎪ ⎪ ⎪ ⎨ Y n,3 = Y n−1,2 , (1.133) ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+1 = Y n−1,p , ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . The vector stochastic transition dynamic relation (1.133) is a particular case of the vector transition dynamic relation (1.4).  n , n = 0, 1, . . . , is a homogeneous multivariate Markov Thus, the vector process Y 0 and the corresponding vector transition dy­ chain with the initial random value Y namic functions determined by relations (1.128), (1.129), and (1.133).

1.4.9 Nonlinear autoregressive stochastic volatility models of log-price and price processes Let us consider a model that can be referred to as a nonlinear autoregressive stochas­ tic volatility-type model. For simplicity, we consider a univariate log-price process Y n defined by the following stochastic transition dynamic relation: Y n = σ n W n ,

n = 1, 2, . . . ,

(1.134)

where ln σ n = A n (ln σ n−1 , . . . , ln σ n−p , W n ) ,

n = 1, 2, . . . ,

(1.135)

σ 0 = (ln σ 0 , . . . , ln σ −p+1 ) is a p-dimensional random vector with real-val­ and (a) ln  ued components, (b) (W1 , W1 ), (W2 , W2 ), . . . , is a sequence of independent random vectors with real-valued components, (c) the random vector ln  σ 0 and the random se­ quence (W1 , W1 ), (W2 , W2 ) . . . , are independent, (d) p is a positive integer number, and (e) A n (y1 , . . . , y p , w), n = 1, 2, . . . , are the measurable functions acting from the space Rp × R1 to R1 . The standard assumption is also that random vectors (W1 , W1 ), (W2 , W2 ), . . . , have normal distributions. The corresponding price process is defined by the following relation: Sn = e Yn ,

n = 0, 1, . . . .

(1.136)

Let us show in which way this model can be imbedded in the model of multivariate Markov log-price process.

1.4 Autoregressive stochastic volatility LPP |

41

Let us define the (p + 1)-dimensional discrete time process,  n = (Y n,1 , . . . , Y n,p+1) Y = (Y n , ln σ n , ln σ n−1 , . . . , ln σ n−p+1 ) ,

n = 1, 2, . . . ,

(1.137)

and the (p + 1)-dimensional sequence of independent random vectors  n = (W n,1 , . . . , W n,p+1) W = (W n , W n , 0, . . . , 0) ,

n = 1, 2, . . . .

(1.138)

Using relations (1.134), (1.135), (1.137), and (1.138), we get the following vector stochastic transition dynamic relation (written as a system of stochastic transition dynamic relations): ⎧  ⎪ Y n,1 = e A n (Y n−1,2 ,...,Y n−1,p+1 ,W n ) W n , ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ Y n,2 = A n (Y n−1,2 , . . . , Y n−1,p+1 , W n ) , ⎪ ⎪ ⎪ ⎪ ⎨ Y n,3 = Y n−1,2 , (1.139) ⎪ ⎪ . . . . . . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,p+1 = Y n−1,p , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . The vector stochastic transition dynamic relation (1.139) is a particular case of the vector stochastic transition dynamic relation (1.4). n , n = 0, 1, . . . , is a homogeneous multivariate Markov Thus, the vector process Y 0 and the corresponding vector dynamic transi­ chain with the initial random value Y tion functions determined by relations (1.134), (1.135), and (1.139). The particular variant of the above model is an inhomogeneous in time stochastic volatility-type log-price process given by the following stochastic difference equation: Y n − Y n−1 = g κ,ν (σ n )W n ,

n = 1, 2, . . . ,

(1.140)

where ln σ n = a n,0 + a n,1 ln σ n−1 + · · · + a n,p ln σ n−p + c n W n ,

n = 1, 2, . . . ,

(1.141)

and (a) ln  σ 0 = (ln σ 0 , . . . , ln σ −p+1 ) is a p-dimensional random vector with real-val­ ued components, (b) (W1 , W1 ), (W2 , W2 ) . . . , is a sequence of independent random vectors with real-valued components, (c) the random vector ln  σ 0 and the random se­     quence (W1 , W1 ), (W2 , W2 ) . . . , are independent, (d) p is a positive integer number, and (e) a n,0 , a n,1 , . . . , a n,p , c n , n = 1, 2, . . . , are the real-valued constants, (f) g κ,ν (·) is a function from the class Gκ,ν , for some κ, ν ≥ 0.

42 | 1 Multivariate modulated Markov log-price processes (LPP)

1.4.10 Modulated nonlinear autoregressive stochastic volatility log-price and price processes Let us consider a model that can be referred to as a modulated nonlinear autore­ gressive stochastic volatility-type model. For simplicity, we consider a process Z n = (Y n , X n ) with a univariate log-price component Y n . The process Z n is defined by the following stochastic transition dynamic relation: ⎧ ⎪ Y n = σ n W n , ⎪ ⎪ ⎨ X n = C n ( X n −1 , . . . , X n − r , U n ) , (1.142) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , where ln σ n = A n (ln σ n−1 , . . . , ln σ n−p ,

X n−1 , . . . , X n−r , W n ) ,

n = 1, 2, . . . ,

(1.143)

σ 0 = (ln σ 0 , . . . , ln σ −p+1 ) is a p-dimensional random vector with real-val­ and (a) ln  0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector tak­ ued components, (b) X ing values in the space X(r) , (c) (W1 , W1 , U1 ), (W2 , W2 , U2 ), . . . , is a sequence of independent random vectors taking values in the space R1 × R1 × U, where U is a measurable space with a σ-algebra of measurable subsets BU , (d) the random vector 0 ) and the random sequence (W1 , W1 , U1 ), (W2 , W2 , U2 ), . . . , are indepen­ (ln  σ0 , X dent, (e) p and r are the positive integer numbers, (f) A n (y1 , . . . , y p , x1 , . . . , x r , w), n = 1, 2, . . . , are the measurable functions acting from the space Rp × X(r) × R1 to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X. The corresponding modulated price process is defined by the following relation: (S n , X n ) = (e Y n , X n ) ,

n = 0, 1, . . . .

(1.144)

Let us show in which way this model can be imbedded in a model of modulated multivariate Markov log-price process. As given in Subsection 1.4.9, let us define the (p + 1)-dimensional discrete n = (Y n,1 , . . . , Y n,p+1) = (Y n , ln σ n , ln σ n−1 , . . . , ln σ n−p+1), n = time process Y 0, 1, . . . , and the (p + 1)-dimensional sequence of independent random vectors  n = (W n,1 , . . . , W n,p+1 ) = (W n , W n , 0, . . . , 0), n = 0, 1, . . . , using, respectively, W relations (1.137) and (1.138), as well as define the r-dimensional vector process,  n = (X n,1 , . . . , X n,r ) = (X n , . . . , X n−r+1) , X

n = 0, 1, . . . .

(1.145)

Using relations (1.137), (1.138), (1.142), and (1.145), we get the following vector stochastic transition dynamic relation (written as a system of stochastic transition

1.4 Autoregressive stochastic volatility LPP | 43

dynamic relations), ⎧ A n ( Y n−1,2 ,...,Y n−1,p+1 ,X n−1,1 ,...,X n−1,r ,W n)  ⎪ Wn , ⎪ ⎪ Y n,1 = e ⎪ ⎪ ⎪ ⎪ ⎪ Y n,2 = A n (Y n−1,2 , . . . , Y n−1,p+1 , X n−1,1 , . . . , X n−1,r , W n ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Y n,3 = Y n−1,2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Y n,p+1 = Y n−1,p , ⎪ ⎪ X n,1 = C n (X n−1,1 . . . , X n−1,r , U n ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,2 = X n−1,1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ... ... ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ X n,r = X n−1,r−1 , ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . .

(1.146)

The vector stochastic transition dynamic relation (1.146) is a particular case of the vector stochastic transition dynamic relation (1.4). n , X  n ), n = 0, 1, . . . , is a multivariate modulated Z n = (Y Thus, the vector process  Markov chain with the initial random value  Z0 and the corresponding vector transition dynamic functions determined by relations (1.142), (1.143), and (1.146). The particular variant of the above model is an inhomogeneous in time autore­ gressive stochastic volatility-type log-price process given by the following stochastic difference equation: ⎧ ⎪ Y − Y n−1 = g κ,ν (σ n ) · W n , ⎪ ⎪ ⎨ n X n = C n ( X n −1 , . . . , X n − r , U n ) , (1.147) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , where ln σ n = a n,0(X n−1 , . . . , X n−r ) + a n,1 (X n−1 , . . . , X n−r ) ln σ n−1 + · · · + a n,p (X n−1 , . . . , X n−r ) ln σ n−p + c n (X n−1 , . . . , X n−r )W n ,

n = 1, 1, . . . ,

(1.148)

and (a) ln  σ 0 = (ln σ 0 , . . . , ln σ −p+1 ) is a p-dimensional random vector with real-val­ 0 = (X0 , . . . , X−r+1 ) is a r-dimensional random vector taking ued components, (b) X values in the space X(r) , (c) (W1 , W1 , U1 ), (W2 , W2 , U2 ), . . . , is a sequence of inde­ pendent random vectors taking values in the space R1 × R1 × U, (d) the random vector 0 ) and the random sequence (W1 , W1 , U1 ), (W2 , W2 , U2 ), . . . , are indepen­ (ln  σ0 , X dent, (e) p and r are the positive integer numbers, (f) a n,i (x1 , . . . , x r ), I = 0, . . . , p, n = 1, 2, . . . , and c n (x1 , . . . , x r ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) to R1 , and (g) C n (x1 , . . . , x r , u ), n = 1, 2, . . . , are the measurable functions acting from the space X(r) × U to the space X .

2 American-type options In this chapter, we define American-type options and introduce basic objects con­ nected with American-type options for multivariate modulated Markov price and log-price processes. In Section 2.1, we introduce American-type options for multivariate modulated Markov price processes with general pay-off functions depending on both price and index components. We also formulate basic optimization problems for American-type options con­ nected with finding optimal expected rewards and optimal stopping times represent­ ing optimal stopping strategies for buyers of American options. In Section 2.2, we present basic examples of American-type options including call and put-type options, exchange of assets options, digital options as well as option-­ type contracts represented by portfolios of options. In Section 2.3, we investigate the properties of reward and log-reward functions and optimal expected rewards and give backward recurrence relations for reward and log-reward functions. Such recurrence relations play an important role in stochastic approximation algorithms for the option rewards of American-type options. In Section 2.4, we define optimal stopping times for American-type options and investigate the hitting structure of optimal stopping times. We also give proofs of the basic theorems about backward recurrence relations for option rewards and hitting structure of optimal stopping times. In Section 2.5, we introduce American-type knockout options and show how these options can be imbedded in the model of usual American-type options by extending the underlying price process by an additional modulating index component. The main results are given in Theorems 2.3.1–2.3.4, which present backward recur­ rence relations for option rewards, and Theorems 2.4.1–2.4.4, which describe hitting structure of optimal stopping times for American-type options. These results are variants of classical results by Chow, Robbins, and Siegmund (1971) and Shiryaev (1976) on optimal stopping for Markov processes.

2.1 American-type options In this section, we define American-type option contracts for discrete time multivari­ ate modulated Markov price and log-price processes.

2.1 American-type options

|

45

2.1.1 American-type options for discrete time price processes  n = ( Let us consider a discrete time multivariate modulated price process V S n , X n ),  n , X n ), n =  n = 0, 1, . . . , and the corresponding modulated log-price process Z n = (Y 0, 1, . . . , defined in Section 1.1. n are connected by the Recall that the price process  S n and the log-price process Y n Y  relation S n = e , n = 0, 1, . . . .  n is a Markov chain with a phase space We always assume that the price process V 0 ∈ B} and transition probabilities P˙ n ( V, an initial distribution P˙ 0 (B) = P{V v , B ). Z n is also a Markov chain with a phase space Z, In this case, the log-price process  an initial distribution P(A) and transition probabilities P n ( z , A). The corresponding  n and  relations between initial and transition probabilities for the processes V Z n are given in Subsection 1.1.3. 1 , . . . , V  n ], n = 0, 1, . . . , be a natural filtration generated by the Let also Fn = σ [V  process V n . It coincides with the natural filtration Fn = σ [ Z1 , . . . ,  Z n ], n = 0, 1, . . . ,  generated by the process Z n . We also consider a pay-off function g(n, s , x) which is assumed to be a real-valued function defined for (n, s , x) ∈ N × R+ k × X, where N = {0, 1, . . . }. We also assume s , x) ∈ V for every n ∈ N. that g(n, s , x) is a measurable function in argument ( The pay-off function can also be expressed in the equivalent form as g(n, ey , x), and considered as a function defined for (n,  y , x) ∈ N × Rk × X. The American-type option contract is an agreement between two parties, a seller and a buyer. The contract has a price C payed by the buyer to the seller. The option contract guarantees to the buyer the possibility of executing the option contract in any S n , X n ) = g(n, e Yn , X n ). moment 0 ≤ n ≤ N and of getting in this case the pay-off g(n,  The parameter N ∈ N is called a maturity.

2.1.2 Buyer aims and timing for analysis of an option contract We analyze the contract from the position of the buyer. In this case, the American option contract can be analyzed either in the situation, when the contract is not yet bought by the buyer, or in the situation, when the contract was already bought by the buyer. In the first case, one of the important questions is what should be the fair price of the contract? In the second case, the goal of the buyer is to execute the contract in an optimal way according to some reasonable criteria of optimality. One of such natural criterion is to execute the contract in the way that would max­ imize the expected pay-off (reward).

46 | 2 American-type options

2.1.3 Stopping buyer’s strategies It is natural to assume that the decision of the buyer to execute or do not execute the contract at some moment 0 ≤ n ≤ N should be based only on the information about the price or log-price process and the modulating index up to moment n. This corresponds to the assumption that the “stopping” strategy of the buyer is defined by a random stopping time τ that is a Markov moment for the price process  n , X n ).  n = ( V S n , X n ) or, that is equivalent, for the log-price process  Z n = (Y This means that τ is an integer-valued random variable taking values 0, 1, . . . , such that the random event {τ = n} ∈ Fn for every n = 0, 1, . . . . According the definition of the option contract, the Markov moment τ defining the stopping strategy of the buyer should also satisfy the inequality 0 ≤ τ ≤ N.  n such that Let Mmax,N be the class of all Markov moments for the price process V 0 ≤ τ ≤ N.  n and  The processes V Z n generate the same natural filtration Fn , n = 0, 1, . . . , and, therefore, the same family Mmax,N of all Markov moments 0 ≤ τ ≤ N. If the buyer decides to execute the option contract at a stopping moment τ ∈ Sτ , Xτ ) = Mmax,N , the corresponding pay-off (reward) will be a random variable g(τ,  τ Y g(τ, e , X τ ). The following quantity represents the expected reward corresponding to the stop­ ping time τ:  ϕ τ = Eg(τ,  S τ , X τ ) = Eg(τ, e Y τ , X τ ) . (2.1)

2.1.4 Optimization problems connected with expected rewards Let MN ⊆ Mmax,N be some class of Markov moments τ from the class Mmax,N . Objects of our interest are the functionals Φ(MN ) = sup ϕ τ .

(2.2)

τ ∈MN

The first main optimization problem connected with the American option is to find or estimate the optimal expected reward Φ(Mmax,N ). But we shall also study functionals Φ(MN ) for different classes MN . In particular, the class of hitting times is the subject of our special interest. In order to simplify notations we denote Φ = Φ(Mmax,N ) . We assume that the following condition holds: S n , X n )| = E|g(n, e Yn , X n )| < ∞ , n = 0, 1, . . . , N . A1 : E|g(n, 

(2.3)

2.2 Pay-off functions |

47

In Chapter 4, we give different conditions, expressed in terms of one-step transition probabilities for price or log-price processes and pay-off functions, which are sufficient for the condition A1 to hold. The following inequality holds for any random moment 0 ≤ τ ≤ N: |g(τ,  S τ , X τ )| ≤ max |g(n,  S n , X n )| ≤ 0≤ n ≤ N

N

|g(n,  S n , X n )| .

(2.4)

n =0

The condition A1 guarantees that for any class of Markov moments MN ⊆ Mmax,N , |Φ(MN )| ≤ sup E|g(τ,  S τ , X τ )| τ ∈MN

S n , X n )| ≤ ≤ E max |g(n,  0≤ n ≤ N

N

E|g(n,  S n , X n )| < ∞ .

(2.5)

n =0

Thus, under the condition A1 , the functionals ϕ τ and Φ(MN ), in particular Φ = Φ(Mmax,N ), are well defined. Another important problem is to answer the question if there exists – and, if this the case, to find – the optimal stopping time that is the Markov moment τ∗ ∈ Mmax,N such that Φ = sup ϕ τ = ϕ τ∗ . (2.6) τ ∈Mmax,N

2.2 Pay-off functions In this section, we formulate general conditions, which we impose on pay-off func­ tions of American-type options, and present a number of typical examples of pay-off functions.

2.2.1 General conditions imposed on pay-off functions In this book, we study American-type options with the pay-off functions g(n, s , x) that have not more than polynomial rate of growth in the price argument  s = ( s1 , . . . , s k ) ∈ and are bounded in the time and index arguments, respectively, 0 ≤ n ≤ N and R+ k x ∈ X. This means that the following condition is assumed to hold for some vector pa­ rameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components: | g ( n, s,x )| B˙ 1 [𝛾¯]: sup0≤n≤N,(s,x)∈V 1+k K (s ∨s −1 )𝛾i < K1 , for some and 0 ≤ K1 , K2,1 , . . . , K2,k 0, or f n (s) = a n s𝛾, where a n > 0, 𝛾 > 0. It is also worth to note that models with nonlinear functions f n (s) can also appear as a result of transformation of a price process or a log-price process. Let h n (s) be, for every n = 0, 1, . . . , a strictly monotonic function, which maps − + − + one-to-one the interval R+ 1 to some open interval ( s n , s n ), where 0 ≤ s n < s n ≤ ∞. 1 In this case, there exists the inverse function f n (s) = h− n ( s) such that f n ( h n ( s)) = s, + s ∈ R1 . In some cases, the price process S n , n = 0, 1, . . . , can be simplified if one (h ) transforms it in the process S n = h n (S n ), n = 0, 1, . . . . The random pay-off process g(n, S n , X n ), n = 0, 1, . . . , is transformed in this case as g(n, S n , X n ) = (h ) g(n, f n (h n (S n )), X n ) = g(f ) (n, S n , X n ), n = 0, 1, . . . , where g(f ) (n, s, x) = g(n, f n (s), x) is a transformed pay-off function. Generalizations described above, namely, portfolios of option contracts with in­ homogeneous in time and strike and waiting coefficients depending on the index ar­ gument can also be considered for the above model. In this case, a pay-off function is defined for n = 0, 1, . . . , N, s ∈ R+ k , x ∈ X by the following formula: g(n, s, x) = e−R n

m

(ai (n, x)[f i (n, s, x) − K i (n, x)]+ i =1

+

a i ( n,

 x)[K  i ( n, x ) − f i ( n, s, x )]+ ) ,

(2.18)

where (a) 0 ≤ K i (n, x), K  i ( n, x ) ≤ K < ∞, x ∈ X, n = 1, . . . , N, i = 1, . . . , m are strike prices, (b) R n = r0 + · · · + r n ≥ 0, n = 0, 1, . . . , N are accumulated free inter­ est rates and r n ≥ 0, n = 0, 1, . . . , N are instant free interest rates, (c) f i (n, s, x), f i (n, s, x) ≥ 0, n = 0, 1, . . . , N, s ∈ R+ 1 , x ∈ X, i = 1, . . . , m are the measur­ able transformation functions such that f i (n, s, x) ≤ K1 + K2 s𝛾, n = 0, 1, . . . , N,   s ∈ R+ 1 , x ∈ X, i = 1, . . . , m, where 0 ≤ K 1 , K 2 < ∞, 0 ≤ 𝛾 < ∞, (d) −A ≤ ai (n, x), a i ( n, x ) ≤ A < ∞, x ∈ X, n = 1, . . . , N, i = 1, . . . , m are waiting coef­ ficients for the corresponding option subcontracts, and (e) N is a maturity. Analogous examples of options with nonlinear pay-off functions and portfolios of such options for multivariate modulated price processes can also be given.

2.2.4 Pay-off functions for exchange of assets option-type contracts Another type of option-type contracts is connected with the model of exchange of as­ sets.  n = ( S n , X n ), where  S n = (S n,1 , Let us first consider the simplest case, where V S n,2), is a two-dimensional price process without an index component.

52 | 2 American-type options In this case, a pay-off function is defined for  s = (s1 , s2 ) ∈ R+ 2 , x ∈ X, n = 0, 1, . . . , N by the following formula: g(n, s ) = e−rn (s1 − s2 ) ,

(2.19)

where (a) r ≥ 0 is a free interest rate and (b) N is a maturity. A natural generalization is connected with the model of exchange of k assets,  n = ( S n , X n ) has a k-dimensional price compo­ where a modulated price process V nent  S n = (S n,1 , . . . , S n,k ). In the simplest case, a pay-off function does not depend on the index argument s = (s1 , . . . , s k ) ∈ R+ and is defined for  k , x ∈ X, n = 0, 1, . . . , N by the following formula: k

g(n, s , x) = e−rn ai si , (2.20) i =1

where (a) r ≥ 0 is a free interest rate, (b) a i ∈ R1 , i = 1, . . . , k are exchange waiting coefficients, and (c) N is a maturity. The further generalization is connected with the model of exchange of k assets, where the waiting coefficients depend on the index argument. In this case, a pay-off function is defined for n = 0, 1, . . . , N,  s = (s1 , . . . , s k ) ∈ R+ k , x ∈ X, by the following formula: k

a i (x)s i , (2.21) g(n, s , x) = e−rn i =1

where (a) r ≥ 0 is a free interest rate,, (b) −A ≤ a i (x) ≤ A < ∞, x ∈ X, i = 1, . . . , k are exchange waiting coefficients, and (c) N is a maturity. Also the generalization connected with inhomogeneous in time variant of such contracts is possible. In this case, a pay-off function is defined for n = 0, 1, . . . , N,  s = (s1 , . . . , s k ) ∈ R+ k , x ∈ X, by the following formula: g(n, s , x) = e−R n

k

a i (n, x)s i ,

(2.22)

i =1

where (a) R n = r0 + · · · + r n , n = 0, 1, . . . , N are accumulated free interest rates and r n ≥ 0, n = 0, 1, . . . , N are instant free interest rates, (b) −A ≤ a i (n, x) ≤ A < ∞, x ∈ X, i = 1, . . . , k, n = 0, 1, . . . , N are exchange waiting coefficients, and (c) N is a maturity. In exchange asset contracts, pay-off functions are again linear functions in argu­ ment  s. The nonnegativity and monotonicity properties may not hold. In the above models, the condition B˙ 1 [¯1] obviously holds for pay-off functions. As above, generalizations described above namely exchange of assets option con­ tracts inhomogeneous in time, options with parameters depending on the index argu­ ment, and portfolios of such options for multivariate modulated price processes can also be considered.

2.3 Reward and log-reward functions

| 53

2.2.5 Pay-off functions for digital-type options The further examples are connected with the more exotic pay-off functions g(n, s , x). For example, so-called digital-type contracts are based on stepwise pay-off functions. Let us restrict consideration by the simplest case, a price process V n = (S n , X n ) has a univariate price component S n . In this case, a pay-off function is defined for n = 0, 1, . . . , N, s ∈ R+ 1 , x ∈ X, by the following formula: (2.23) g(n, s, x) = e−rn d(s, x) , m where (a) r ≥ 0 is a free interest rate, (b) d(s, x) = i=1 d i (x)I (e i−1 (x) < s ≤ e i (x)) is a nondiscounted stepwise (in the price argument s) pay-off function, where −D ≤ d i (x) ≤ D < ∞, i = 1, . . . , m, 0 = e0 (x) < e1 (x) < · · · < e m (x) = ∞, and (c) N is a maturity. ¯ ] holds, since the pay-off functions for dig­ In the above model, the condition B˙ 1 [0 ital option contracts are always bounded. As above, generalizations described above namely digital option contracts inho­ mogeneous in time, options with parameters of stepwise pay-off functions depending on the index argument, and portfolios of digital options for multivariate modulated price processes can also be considered for the above model. Taking into account the great variety of different option-type contracts, we shall try to get results for general pay-off functions g(n, s, x) satisfying minimal continu­ ity conditions as well as conditions on the rate of growth for these functions in price arguments.

2.3 Reward and log-reward functions In this section, we introduce so-called log-reward functions for American-type options for discrete time multivariate modulated Markov price and log-price processes.

2.3.1 Conditional expectations for random functionals defined on trajectories of Markov price processes  n = ( Let V S n , X n ) be a discrete time multivariate modulated Markov price process with v , B ). a phase space V, initial distribution P˙ 0 (B) and the transition probabilities P˙ n ( n , . . . , V  m ], 0 ≤ n ≤ m < ∞. Let us introduce σ-algebras Fn,m = σ [V ˙ n be a real-valued random variable adapted to the σ-algebra Let 0 ≤ n ≤ N and W ˙ n ∈ C} ∈ Fn,N , for any C ∈ B1 . Fn,N , i.e. random variable such that events {W ˙ n can be represented in the form W ˙n = As is known, the random variable W n , . . . , V  N ), where f˙n ( vn , . . . ,  v N ) is a measurable real-valued function acting f˙n (V

54 | 2 American-type options from the space V(N −n+1) = V × · · · × V to R1 (here V(r) = V × · · · × V is r-times ˙ n can be considered as a random functional defined product of the space V). Thus, W on trajectories of the price process V l . ˙ n ∈ C} of the random In this case, there exists the conditional distribution P˙ v ,n {W ˙ n under the condition V n =  variable W v, for  v ∈ V. It is defined for C ∈ B1 by the following relation: ˙ n ∈ C } = P{ W ˙ n ∈ C/V n =  v} P˙ v ,n {W

= P˙ n+1 ( v , d v n+1 ) P˙ n+2 ( v n+1 , d v n +2 ) V

V

I (f˙n ( v,  v n +1 , . . . ,  v N ) ∈ C)P˙ N ( v N −1 , d vN ) .

···

(2.24)

V

Here and henceforth, we use the notation P˙ v ,n for conditional probabilities under n =  v. the condition V This conditional distribution is a probability measure, as a function of C ∈ B1 , for every  v ∈ V, and a measurable function of argument  v ∈ V, for every C ∈ B1 . Let us consider the set of integrability for the random variable W n , ⎧ ⎫ ⎪

⎪ ⎨ ⎬ ˙ n | = |w|P˙ v ,n {W ˙ n ∈ dw} < ∞ . I˙ W˙ n = ⎪ v ∈ V : E˙ v ,n |W (2.25) ⎪ ⎩ ⎭ R1

Here and henceforth we use the notation E˙ v ,n for conditional expectations under n =  the condition V v. The set I˙ W˙ n ∈ BV . In particular, the set I˙ W˙ n can be empty or coincide with V. ˙ n under the condition V n = The conditional expectation of the random variable W ˙ v is defined for  v ∈ I W˙ n , 

˙ n = wP˙ v ,n {W ˙ n ∈ d n , . . . , V N ) v } = E˙ v ,n f˙n (V E˙ v ,n W R1



P˙ n+1 ( v , d v n +1 )

= V

P˙ n+2 ( v n+1 , d v n +2 ) V

v,  v n +1 , . . . ,  v N )P˙ N ( v N −1 , d vN ) . f˙n (

···

(2.26)

V

In fact, the assumption  v ∈ I˙ W˙ n enables one to first obtain the analog of relation ˙ n | = |f˙n (V n , . . . , V  N )|. This makes it possible to ob­ (2.26) for the random variable |W ˙ n given by the expression on the tain relation (2.26) and to get the finite value for E˙ v ,n W right-hand side of this relation. ˙ n | < ∞. Let us assume that E|W This condition does not imply that the set I˙ W˙ n = V. But this assumption does ˙ n | should be equal to ∞.  n ∈ I˙ W˙ } = 1. Otherwise, the expectation E|W imply that P{V n

2.3 Reward and log-reward functions

| 55

˙ n | < ∞. Note also that the assumption I˙ W˙ n = V does not imply that E|W ˙ n | < ∞, the following relation takes place: Under the assumption E|W

˙ n P{ V ˙ n = E˙ v ,n W  n ∈ d v} EW V



P˙ 0 (d v0 )

= V

˙ n P˙ v0 ,0 {V  n ∈ d E˙ v ,n W v} .

(2.27)

V

˙ n | < ∞ enables one first to obtain the analog of As above, the assumption E|W ˙ n | = |f˙n (V n , . . . , V  N )|. This makes it possible relation (2.27) for the random variable |W ˙ n given by the expression on the to get relation (2.27) and to get the finite value for E˙ W right-hand side of this relation. ˙ n for  Note also that one should additionally define the function E˙ v ,n W v ∈ I˙ W˙ n , in  n ∈ I˙ W˙ } = 0, this can be done in an order to be able to get relation (2.27). Since P˙ {V n

arbitrary “measurable” way because of the Lebesgue integral over the set I˙ W˙ n will be ˙ n = 0, for  any way equal to 0. The standard way is to define E˙ v ,n W v ∈ I˙ W˙ . n

2.3.2 Reward functions for discrete time multivariate modulated Markov price processes Let us define reward functions of an American option for a multivariate modulated  n = ( Markov price process V S n , X n ). Let us introduce, for 0 ≤ n ≤ N, the class Mmax,n,N of all Markov moments τ for  n such that (a) n ≤ τ ≤ N and (b) event {τ = m} ∈ Fn,m = the Markov process V n , . . . , V  m ], n ≤ m ≤ N. σ[V Obviously, the class Mmax,0,N = Mmax,N . Let us assume the following condition: S r , X r )| < ∞,  v ∈ V, 0 ≤ n ≤ r ≤ N . A2 : E˙ v ,n |g(r,  In Chapter 4, we give a number of conditions, expressed in terms of one-step transition probabilities of price or log-price processes and pay-off functions, which are sufficient for the condition A2 to hold. The following inequality obviously holds, for every Markov moment τ n ∈ Mmax,n,N , n = 0, 1, . . . , N: |g( τ n ,  S τ n , X τ n )| ≤ max |g(r,  S r , X r )| ≤ n≤r≤N

N

|g(r,  S r , X r )| .

(2.28)

r=n

v ∈ V, n = and, thus, by the condition A2 , the following inequality holds, for  0, 1, . . . , N: N

E˙ v ,n |g(r,  S τ n , X τ n )| ≤ S r , X r )| < ∞ . (2.29) E˙ v ,n |g(τ n ,  r=n

56 | 2 American-type options Due to (2.29), the following finite measurable functions can be defined for n = 0, 1, . . . , N: ϕ˙ n,τ n ( v) = ϕ˙ n,τ n ( s , x) = E˙ v ,n g(τ n ,  S τn , X τn ) ,

 v ∈ V.

(2.30)

S τ n , X τ n ) is a random variable adapted to the σ-algebra Fn,N . Thus, Note that g(τ n ,  n , . . . , V  N ), where f˙n ( it can be represented in the form g(τ n ,  S τ n , X τ n ) = f˙n (V vn , . . . , ( N − n +1 )  v N ) is a measurable real-valued function defined on the space V . This func­ tion has, in this case, a specific form. Since τ n is a Markov moment taking values n, . . . , N, there exists the sequence of sets B n,r ∈ BV(r−n+1) , r = n, . . . N such that n , . . . , V  r ) ∈ B n,r }, for every r = n, . . . , N, and ∪Nr=n {τ n = the event {τ n = r} = {(V N   r} = ∪r=n {(V n , . . . , V r ) ∈ B n,r } = Ω (here  Ω, F , P is a probability space where the  n , . . . , V  l is defined). Therefore, g(τ n ,  r ) ∈ process V S , X ) = Nr=n g(r,  S r , X r )I ((V N τ n τ n B n,r ), and, thus, f n ( vn , . . . ,  v N ) = r=n g(r, s r , x r )I (( vn , . . . ,  v r ) ∈ B n,r ), where  vr = ( s r , x r ), r = n, . . . , N. The above remarks and relation (2.26) imply that the functions ϕ˙ n,τ n ( v ), n = 0, . . . , N are indeed measurable and that the following relation takes place, for  v ∈ V, n = 0, . . . , N: v) = ϕ˙ n,τ n ( s , x) = E˙ v ,n g(τ n ,  S τn , X τn ) ϕ˙ n,τ n ( =

N

r=n

n , . . . , V  r ) ∈ B n,r ) . S r , X r )I ((V E˙ v ,n g(r, 

(2.31)

Let us also define, for every n = 0, 1, . . . , N, the reward function for the American option v) = ϕ˙ n ( s , x) = sup ϕ˙ n,τ n (n,  v) ,  v = ( s , x) ∈ V . (2.32) ϕ˙ n ( τ n ∈Mmax,n,N

v), n = 0, . . . , N are finite functions. Indeed, inequali­ The reward functions ϕ˙ n ( v ∈ V, n = 0, 1, . . . , N, ties (2.29) also imply that for every  ˙ n ( |ϕ v)| ≤ ≤

sup τ n ∈Mmax,n,N N

r=n

S τ n , X τ n )| E˙ v ,n |g(τ n , 

E˙ v ,n |g(r,  S r , X r )| < ∞ .

(2.33)

The following theorem plays an important role in what follows. v) = ϕ˙ n ( s , x ), Theorem 2.3.1. Let the condition A2 hold. Then, the reward functions ϕ˙ n ( n = 0, 1, . . . , N are the measurable functions of  v = ( s , x) ∈ V and the following backward recurrence relations hold, for every  v n = ( s n , x n ) ∈ V, n = 0, 1, . . . , N: ⎧ ⎪ ϕ˙ N ( v ) = g(N, s , x) ,  v = ( s , x) ∈ V , ⎪ ⎪ ⎪ ⎪ ⎪ ˙ r ( ⎨ ϕ  r+1 )) ,  v ) = max(g(r, s , x), E˙ v ,r ϕ˙ r+1 (V v = ( s , x) ∈ V , (2.34) ⎪ ⎪ r = N − 1, . . . , n + 1 , ⎪ ⎪ ⎪ ⎪ ⎩ ˙  n+1 )) . ϕ n ( v n ) = max(g(n, s n , x n ) , E˙ v n ,n ϕ˙ n+1 (V

2.3 Reward and log-reward functions |

57

Theorem 2.3.1 gives a general backward reward algorithm for finding the reward func­ tions for discrete time price processes. The algorithm is based on the recurrence rela­ tions (2.34). This algorithm includes two rather complicated operations that are maximization and integration. This make the algorithm not too effective for general Markov price processes. However, for some simple Markov price processes such as Bernoulli and other types of discrete random walks, the algorithm can be effectively used for finding re­ ward functions. We present the corresponding results in Section 1.3. Moreover, some more general discrete time Markov price processes can be approx­ imated by simpler price processes of random walk types. The corresponding conver­ gence results for options rewards are presented in Sections 1.5 and 1.6. This makes it possible to use the corresponding backward algorithms for finding approximations for reward functions for various discrete time Markov price processes. Let the conditions A1 and A2 hold. Recall the optimal expected reward Φ = Φ(Mmax,N ) introduced in Subsec­ tion 1.2.1, S τ0 , X τ0 ) . (2.35) Φ = sup Eg(τ0 ,  τ 0 ∈Mmax,N

Relations (2.27) and (2.35) imply that the following relation takes place:

Φ=

sup τ 0 ∈Mmax,N

E˙ v ,n g(τ0 ,  S τ0 , X τ0 )P˙ 0 (d v) .

(2.36)

V

The following theorem shows that one can exchange operation of maximization and integration in formula (2.36) when computing the optimal expected reward Φ. After this exchange, computing of the expectation and then the supremum under the v). Thus, sign of integral yields, according to relation (2.32), the reward function ϕ˙ 0 ( the optimal expected reward Φ can be computed by integration of this reward function by the initial distribution P˙ 0 (B). Theorem 2.3.2. Let the conditions A1 and A2 hold. Then, the optimal expected reward Φ v) are connected by the following formula: and the reward function ϕ0 ( 0 ) = Φ = Eϕ˙ 0 (V

ϕ˙ 0 ( v)P˙ 0 (d v) . V

The proofs of Theorems 2.3.1 and 2.3.2 are given in Section 2.4.

(2.37)

58 | 2 American-type options

2.3.3 Conditional expectations for random functional defined on trajectories of discrete time log-price processes n , X n ) be a discrete time multivariate modulated Markov log-price pro­ Let  Z n = (Y cess with a phase space Z, an initial distribution P0 (B) and transition probabilities P n ( z , B ). The relations given in Subsection 2.3.1 can be rewritten in terms of the log-price n , X n ) connected with the price process V  n = ( Z n = (Y S n , X n ) by the relation process  n Y  S n = e , n = 0, 1, . . . . Recall that the initial distribution and the transition probabilities for the log-price   n are connected by the following relations, for  Z n and the price process V z = ( y , x) ∈ Z, A ∈ BZ , n = 1, 2, . . . :

P0 (A) = P˙ 0 (B A ) ,

P n ( z , A) = P˙ n ( vz , B A ) ,

(2.38)

vz = (ey , x) ∈ V, B A = { vz :  z ∈ A} ∈ BV . where   n obviously generate the same The log-price process  Z n and the price process V n , . . . , V  m ], 0 ≤ n ≤ m ≤ N. Zn , . . . ,  Z m ] = σ[V σ-algebras Fn,m = σ [ Let us again assume that 0 ≤ n ≤ N and W n be a real-valued random variable adapted to the σ-algebra Fn,N . In this case, the random variable W n can be represented in the form W n = f n ( Zn , . . . ,  Z N ), where f n ( zn , . . . ,  z N ) is a measurable real-valued function acting from the space Z(N −n+1) = Z × · · · × Z to R1 (here Z(r) = Z × · · · × Z is the r-times product of the space Z). Thus, W n can be considered as a random functional defined on trajectories of the log-price process Z l . In this case, there exists the conditional distribution Pz ,n {W n ∈ C} of the random Zn =  z , for  z ∈ Z. It is defined for C ∈ B1 by the variable W n under the condition  following relation: Pz ,n {W n ∈ C} = P{W n ∈ C/ Zn =  z}

z , d z n+1 ) P n+2 ( z n+1 , d z n +2 ) = P n+1 ( Z

···

Z

I (f n ( z,  z n +1 , . . . ,  z N ) ∈ C)P N ( z N −1 , d zN ) .

(2.39)

Z

Here and henceforth we use the notation Pz ,n for conditional probabilities under the condition  Zn =  z. This conditional distribution is a probability measure, as a function of C ∈ B1 , for every  z ∈ Z, and a measurable function of argument  z ∈ Z, for every C ∈ B1 . ˙ n = f˙n (V n , . . . , V  N ) intro­ The random functionals W n = f n ( Zn , . . . ,  Z N ) and W ˙ n if f n ( zn , . . . ,  z N ) ≡ f˙n ( vn , . . . ,  v N ), duced in Subsection 2.3.1 coincide, i.e. W n ≡ W  yr where  v r = ( s r , x r ) = (e , x r ),  z r = ( y r , x r ) ∈ Z, r = n, . . . , N.

2.3 Reward and log-reward functions

| 59

In this case, relations (2.24), (2.38), and (2.39) imply that the following relation z ∈ Z, C ∈ B1 , n = 0, 1, . . . , N: holds, for  ˙ n ∈ C} . Pz ,n {W n ∈ C} = P˙ vz ,n {W

(2.40)

Zn , . . . ,  Z N ) is interpreted as a On the left-hand side of relation (2.40), W n = f n ( random functional defined on trajectories of the log-price process Z l . On the right-­ ˙ n = f˙n (V n , . . . , V  N ) is in­ hand side of relation (2.40), in fact, the same functional W terpreted as a random functional defined on trajectories of the price process V l . Let us consider the set of integrability for the random variable W n ⎧ ⎫ ⎪ ⎪

⎨ ⎬ I W n = ⎪ z ∈ Z : Ez ,n |W n | = |w|Pz ,n {W n ∈ dw} < ∞⎪ . (2.41) ⎩ ⎭ R1

Here and henceforth we use the notation Ez ,n for conditional expectations under the condition  Zn =  z. Set I W n ∈ BZ . In particular, the set I W n can be empty or coincide with Z. Relations (2.38) and (2.40) imply that the following relations hold, for 0 ≤ n ≤ N: I˙ W˙ n = { vz :  z ∈ IWn } ,

I W n = { zv :  v ∈ I˙ W˙ n } .

(2.42)

Zn = The conditional expectation of the random variable W n under the condition  v is defined for  z ∈ IWn , 

Ez ,n W n = wPz ,n {W n ∈ d w  } = Ez ,n f n ( Zn , . . . ,  ZN ) R1

=

P n+1 ( z , d z n +1 ) Z

···

P n+2 ( z n+1 , d z n +2 ) Z

f n ( z,  z n +1 , . . . ,  z N )P N ( z N −1 , d zN ) .

(2.43)

Z

z ∈ I W n enables one first to get the analog of relation In fact, the assumption  (2.43) for the random variable |W n | = |f n ( Zn , . . . ,  Z N )|. This makes it possible to write relation (2.43) and to get the finite value for Ez ,n W n given by the expression on the right-hand side of this relation. z ∈ Relations (2.38) and (2.40) also imply that the following relation holds, for  I W n , n = 0, 1, . . . , N: ˙n. (2.44) Ez ,n W n = E˙ vz ,n W As in relation (2.40), W n = f n ( Zn , . . . ,  Z N ) is interpreted as a random functional defined on trajectories of the log-price process Z l , on the left-hand side of relation ˙ n = f˙n (V n , . . . , V  N ) is interpreted as a (2.44), while, in fact, the same functional W random functional defined on on trajectories of the price process V l , on the right-hand side of relation (2.44).

60 | 2 American-type options Let us assume that E|W n | < ∞. This condition does not imply that the set I W n = Z. But this assumption does imply n ∈ I W n } = 1. Otherwise, the expectation E|W n | should be equal to ∞. Note that P{Z also that the assumption I W n = Z does not imply that E|W n | < ∞. Under the assumption E|W n | < ∞, the following relation takes place:

n ∈ d EW n = Ez ,n W n P{Z z} Z

=

P0 (d z0 ) Z

n ∈ d Ez ,n W n Pz0 ,0 {Z z} .

(2.45)

Z

As above, the assumption E|W n | < ∞ enables one first to write the analog of Zn , . . . ,  Z N )|. This makes it possible relation (2.45) for the random variable |W n | = |f n ( to obtain relation (2.45) and to get the finite value for EW n given by the expression on the right-hand side of this relation. z ∈ I W n , in Note also that one should additionally define the function Ez ,n W n for  n ∈ I W n } = 0, this can be done in order to be able to obtain relation (2.45). Since P{Z an arbitrary “measurable” way because of the Lebesgue integral over the set I W n will z ∈ I Wn . be any way equal to 0. The standard way is to define Ez ,n W n = 0, for  Note that, in this case, relation (2.44) holds for any  z ∈ Z, n = 0, 1, . . . , N, since ˙ n = 0, for  we defined Ez ,n W n = E˙ vz ,n W z ∈ I Wn . Also, relations (2.27), (2.38), (2.44), and (2.45) imply, by the theorem about ex­ change of variables in Lebesgue integral, that the following relation holds:

n ∈ d EW n = Ez ,n W n P{Z z} Z

=

P0 (d z0 ) Z

˙n= = EW

Z

˙ n P{ V  n ∈ d v} E˙ v ,n W

V

P˙ 0 (d v0 )

= V

n ∈ d Ez ,n W n Pz0 ,0 {Z z}

˙ n P˙ v0 ,0 {V  n ∈ d E˙ v ,n W v} .

(2.46)

V

2.3.4 Log-reward functions for discrete time log-price processes The above definitions and results concerned log-reward functions can also be refor­ mulated in terms of log–log-reward functions based on the log-price processes  Zn = n Y   (Y n , X n ). This can be done using the relation S n = e , n = 0, 1, . . . , that connect the n . S n and the log-price process Y price process  Let us recall the class Mmax,n,N of all Markov moments τ for the Markov process  Z n such that (a) n ≤ τ ≤ N and (b) event {τ = m} ∈ Fn,m , n ≤ m ≤ N.

2.3 Reward and log-reward functions

| 61

As above, we assume that the condition A2 holds. This condition can also be ex­ pressed in terms of log-price processes: A2 : Ez ,n |g(r, e Yr , X r )| < ∞ ,  z ∈ Z, 0 ≤ n ≤ r ≤ N. The condition A2 implies that the following inequality analogous to (2.29) holds, for  z ∈ Z, n = 0, 1, . . . , N: 

Ez ,n |g(τ n , e Y τn , X τ n )| ≤

N

r=n



Ez ,n |g(r, e Y r , X r )| < ∞ .

(2.47)

Due to (2.47), the following finite measurable functions can be defined for n = 0, 1, . . . , N: 

ϕ n,τ n ( z ) = ϕ n,τ n ( z , x) = Ez ,n g(τ n , e Y τn , X τ n ) ,

 z ∈ Z.

(2.48)

As was pointed out in Subsection 2.3.2, g(τ n , e Yn τn , X τ n ) is a variable adapted to the σ-algebra Fn,N . Thus, it is can be represented in the form g(τ n , e Yτn , X τ n ) Zn , . . . ,  Z N ), where f n ( zn , . . . ,  z N ) is a measurable real-valued function defined = f n ( on the space Z(N −n+1). This function has, in this case, a specific form. Since τ n is a Markov moment taking values n, . . . , N, there exists the sequence of sets A n,r ∈ Zn , . . . ,  Z r ) ∈ A n,r }, for every BZ(r−n+1) , r = n, . . . N such that event {τ n = r} = {( N N   r = n, . . . , N, and ∪r=n {τ n = r} = ∪r=n {(Z n , . . . , Z r ) ∈ A n,r } = Ω (here  Ω, F , P is a probability space where the process  Z l is defined). Therefore, g(τ n , e Yτn , X τ n )  N r Y n , . . . ,  Z r ) ∈ A n,r ), and, thus, f n ( zn , . . . ,  z N ) = Nr=n g(r, ey r , = r=n g(r, e , X r )I ((Z zn , . . . ,  z r ) ∈ A n,r ), where  z r = ( y r , x r ), r = n, . . . , N. x r )I (( The above remarks and relation (2.43) imply that the functions ϕ n,τ n ( z ), n = 0, . . . , N are indeed measurable and that the following relation takes place, for  z ∈ Z, n = 0, . . . , N: 

ϕ n,τ n ( z) = ϕ n,τ n ( y , x) = Ez ,n g(τ n , e Y τn , X τ n ) =

N



r=n

n , . . . ,  Ez ,n g(r, e Y r , X r )I ((Z Z r ) ∈ A n,r ) .

(2.49)

z= Relations (2.44), (2.31), and (2.49) imply that the following relation holds, for  y , x) ∈ Z, n = 0, 1, . . . , N: ( ϕ n,τ n ( z) = ϕ n,τ n ( y , x) = ϕ˙ n,τ n ( vz ) = ϕ˙ n,τ n (ey , x) .

(2.50)

Let us also define, for every n = 0, 1, . . . , N, the log-reward function for American option, ϕ n ( z) = ϕ n ( y , x) =

sup τ n ∈Mmax,n,N

ϕ n,τ n (n,  z) ,

 z = ( y , x) ∈ Z .

(2.51)

62 | 2 American-type options The log-reward functions ϕ n ( z ), n = 0, 1, . . . , N are finite functions. Indeed, in­ equalities (2.29) also imply that for every  z ∈ Z, n = 0, 1, . . . , N, |ϕ n ( z)| ≤ ≤



sup τ n ∈Mmax,n,N N

r=n

Ez ,n |g(τ n , e Y τn , X τ n )|

Ez ,n |g(r,  S r , X r )| < ∞ .

(2.52)

Lemma 2.3.1. Let the condition A2 holds. Then the following relation holds for  z = ( y , x) ∈ Z, n = 0, 1, . . . , N: z) = ϕ n ( y , x) = ϕ˙ n ( vz ) = ϕ˙ n (ey , x) . ϕ n (

(2.53)

Proof. Relations (2.32), (2.50), and (2.51) imply that the following relation holds, for  z = ( y , x) ∈ Z, n = 0, 1, . . . , N: ϕ n ( y , x) = =

sup τ n ∈Mmax,n,N

sup τ n ∈Mmax,n,N

ϕ n,τ n (n,  y , x) ϕ˙ n,τ n (n, ey , x) = ϕ˙ n (ey , x) .

(2.54)

Relation (2.54) implies relation (2.53) to hold. Theorems 2.3.1, expressed in terms of log-reward functions, takes the following form. z) = Theorem 2.3.3. Let the condition A2 holds. Then, the log-reward functions ϕ n ( ϕ n ( y , x), n = 0, 1, . . . , N are the measurable functions of  z = ( y , x) ∈ Z, and the following backward recurrence relations hold, for every  z n = ( y n , x n ) ∈ Z, n = 0, 1, . . . , N: ⎧ ⎪ ϕ N ( z ) = g(N, ey , x) ,  z = ( y , x) ∈ Z , ⎪ ⎪ ⎪ ⎪ ⎪  y ⎨ ϕ r ( z ) = max(g(r, e , x), Ez ,r ϕ r+1 ( Z r+1)) ,  z = ( y , x) ∈ Z , (2.55) ⎪ ⎪ r = N − 1, . . . , n + 1 , ⎪ ⎪ ⎪ ⎪ ⎩ ϕ n ( z n ) = max(g(n, ey n , x n ), Ez n ,n ϕ n+1 ( Z n+1)) . The following lemma shows that the optimal expected reward Φ can be computed via  n or via the log-reward functions for the the reward functions for the price process V log-price process  Zn. Lemma 2.3.2. Let the conditions A1 and A2 hold. Then the following relation takes place: Φ=

sup τ 0 ∈Mmax,N



Eg(τ0 , e Y τ0 , X τ0 ) =

sup τ 0 ∈Mmax,N

Eg(τ0 ,  S τ0 , X τ0 ) .

(2.56)

S n = e Yn , n = 1, 2, . . . , that Proof. Relation (2.56) follows directly from the relation  links the price and the log-price processes and the fact that these processes generate the same class of Markov moments Mmax,N .

2.4 Optimal stopping times |

Relations (2.46) and (2.56) imply that the following relation holds:

 Ez ,n g(τ0 , e Y τ0 , X τ0 )P0 (d z) . Φ = sup τ 0 ∈Mmax,N

63

(2.57)

Z

One can exchange operation of maximization and integration in formula (2.57) when computing the optimal expected reward Φ. After this exchange, computing the expectation and the supremum under the sign of integral will yield according to re­ z). Thus, the optimal expected reward can lation (2.51) the log-reward function ϕ0 ( be computed by integration of the corresponding log-reward function by the initial distribution P0 (A). Theorems 2.3.2, expressed in terms of log-reward functions, takes the following form. Theorem 2.3.4. Let the conditions A1 and A2 hold. Then, the optimal expected reward Φ z ) are connected by the following formula: and the log-reward function ϕ0 (

Z0 ) = Ez ,0 ϕ0 ( z )P0 (d z) . (2.58) Φ = Eϕ0 ( Z

The proofs of Theorems 2.3.3 and 2.3.4 are given in Subsection 2.4.4.

2.4 Optimal stopping times In this section, we describe the structure of optimal stopping times for American-type options for discrete time multivariate modulated Markov price and log-price processes.

2.4.1 Optimal stopping times for Markov price-processes Let D˙ =  D˙ n , n = 0, 1, . . . , N  be a sequence of measurable subsets of the space V. We refer to D˙ as a time–space domain. Let us define for the multivariate, modulated Markov process Vr the hitting time to the time–space domain D˙ in the discrete time interval [n, N ],  r ∈ D˙ r ) ∧ N . τ˙ n, D˙ = min(r ≥ n : V

(2.59)

˙ n,N the class of ˙ we denote the family of all time–space domains D˙ and by M By D ˙. all hitting times τ˙ n, D˙ such that D˙ ∈ D ˙ n,N and Mmax,n,N are connected by the relation M ˙ n,N ⊂ Obviously, the classes M Mmax,n,N . Let us assume that the condition A2 holds. Relation (2.33) implies that, in this case, the reward function ϕ˙ n ( s , x) defined by relation (2.32) is a finite function, i.e. ˙ n ( |ϕ s , x)| < ∞, for any  v = ( s , x ) ∈ V.

64 | 2 American-type options A Markov moment τ˙ ∗ n from the class Mmax,n,N is called an optimal stopping time if it satisfies the following relation:  , X τ˙ ∗ ) , s , x) = ϕ˙ n, τ˙ ∗n ( s , x) = Ev ,n g(τ˙ ∗ ϕ˙ n ( n , S τ˙ ∗ n n

 v = ( s , x) ∈ V .

(2.60)

Relation (2.60) requires optimality of stopping time τ˙ ∗ n simultaneously for all pos­ sible initial states  v = ( s , x) ∈ V, at moment n. The question arises: does such mo­ ment exist? The answer is affirmative. Moreover, the optimal stopping time is the hit­ ting moment to the special time–space domain determined by the reward functions. Since τ n ≡ n is a Markov moment from the class Mmax,n,N the reward functions s , x) satisfy the following inequalities, for every n = 0, 1, . . . , N: ϕ˙ n ( g(n, s , x) ≤ ϕ˙ n ( s , x) , D˙ ∗ n

 v = ( s , x) ∈ V .

(2.61)

Let us denote by D˙ ∗ =  D˙ ∗ n , n = 0, 1, . . . , N  the time–space domain with the sets defined in the following way: D˙ ∗ v = ( s , x) ∈ V : g(n, s , x) = ϕ˙ n ( s , x)} , n = {

n = 0, 1, . . . , N .

(2.62)

The following theorem shows that the time–space domain D˙ ∗ can be referred to as ˙ n, D˙ ∗ , the optimal stopping time–space domain since it defines the hitting times τ˙ ∗ n = τ ∗ n = 0, 1, . . . , N such that τ˙ n is the optimal stopping time in the class Mmax,n,N , for every n = 0, 1, . . . , N. Theorem 2.4.1. Let the condition A2 holds. Then, for every n = 0, . . . , N, the hitting  r ∈ D˙ ∗ ˙ n, D˙ ∗ = min(r ≥ n : V time τ˙ ∗ n = τ r ) ∧ N is the optimal stopping time in the class Mmax,n,N , for every n = 0, 1, . . . , N, i.e. the following optimality relation holds for every  v = ( s , x) ∈ V, n = 0, 1, . . . , N:  , X τ˙ ∗ ) . ϕ˙ n ( s , x) = ϕ˙ n, τ˙ ∗n ( s , x) = E˙ v ,n g(τ˙ ∗ n , S τ˙ ∗ n n

(2.63)

Note that Theorem 2.4.1 states that the hitting time τ˙ n, D˙ ∗ is the optimal stopping time. However, this theorem does not guaranty that this optimal stopping time is unique. s , x ), It is worth to note that relation (2.63) implies that the reward functions ϕ˙ n ( n = 0, 1, . . . , N are the measurable functions in argument ( s , x). This does not follow directly from relation (2.32), which defines the reward functions. ˙ 0, D˙ ∗ is an optimal stopping time Theorem 2.4.1 shows that the hitting time τ∗ 0 = τ simultaneously for all possible initial states  v = ( s , x) ∈ V, at moment 0. Thus, it is natural to expect that this hitting time should also be an optimal stopping time for the case where the initial distribution P˙ 0 (A) is not concentrated in a point. ˙ 0, D˙ ∗ = Theorem 2.4.2. Let the conditions A1 and A2 hold. Then, the hitting time τ˙ ∗ 0 = τ  r ∈ D˙ ∗ min(r ≥ 0 : V ) ∧ N is the optimal stopping time which satisfies the following r optimality relation: ∗ ∗ S0 , X0 ) = Eg(τ˙ ∗ Φ(Mmax,N ) = Eϕ˙ 0 ( 0 , S τ˙ 0 , X τ˙ 0 ) .

The proofs of Theorems 2.4.1 and 2.4.2 are given in Subsection 2.4.4.

(2.64)

2.4 Optimal stopping times |

65

2.4.2 Optimal stopping times for Markov log-price processes The results presented in Theorems 2.4.1 and 2.4.2 can be reformulated in terms of log-price processes. Let D =  D n , n = 0, 1, . . . , N  be a sequence of measurable subsets of the space Z. We refer to D as a time–space domain. Z n the hitting Let us define for the multivariate modulated Markov price process  time to the time–space domain D in the discrete time interval [n, N ], τ n,D = min(r ≥ n :  Zr ∈ Dr ) ∧ N .

(2.65)

By D we denote the family of all time–space domains D and by Mn,N the class of all hitting times τ n,D such that D ∈ D. There exists the one-to-one mapping between the families of time–space domains ˙ and D given by the following relation connecting the corresponding time–space D domains D˙ =  D˙ n , n = 0, 1, . . .  and D =  D n , n = 0, 1, . . . : s , x) = (ey , x) : ( y , x) ∈ D n } , D˙ n = {(

n = 0, 1, . . . , N .

(2.66)

This one-to-one mapping generates, due to the relation  S n = e Yn , connecting the  n = ( S n , X n ) and the modulated log-price price component of the modulated process V   process Z n = (Y n , X n ), the one-to-one mapping between the classes of hitting time ˙ n,N and Mn,N . This mapping is given by the following relation, which connects the M hitting times τ˙ n, D˙ and τ n,D for time space domains D˙ and D connected by relation (2.66), τ˙ n, D˙ = τ n,D . (2.67)  n and  Recall that the processes V Z n generate the same class of Markov moments Mmax,n,N . Let us assume that the condition A2 holds. Relation (2.52) implies that, in this y , x) defined by relation (2.51) is a finite function, case, the log-reward function ϕ n ( i.e. |ϕ n ( y , x)| < ∞, for any  z = ( y , x) ∈ Z. A Markov moment τ∗ n from the class Mmax,n,N is called an optimal stopping time if it satisfies the following relation: 

Y τ∗ n , X ∗) , y , x) = ϕ n,τ∗n ( y , x) = Ez ,n g(τ∗ ϕ n ( τn n,e

 z = ( y , x) ∈ Z .

(2.68)

Lemma 2.4.1. Let the condition A2 holds. In this case, the optimal stopping time τ∗ n in the sense of relation (2.60) will also be an optimal in the sense of relation (2.68) and vice versa, for every n = 0, 1, . . . , N. Proof. Recall the one-to-one mapping between spaces Z and V given by the function  vz = ( sz , xz ) = (ey , x),  z = ( y , x) ∈ Z and its inverse one-to-one mapping given by the function  zv = ( yv , xv ) = (ln  s , x),  v = ( s , x ) ∈ V.

66 | 2 American-type options Relations (2.50) and (2.53) imply the following relation, which holds for  z ∈ Z, n = 0, 1, . . . , N:  τ∗ Y ˙ n ( ˙ vz ,n g(τ∗  , X τ∗ ) = ϕ n , X ∗) = E z) = Ez ,n g(τ∗ vz ) . ϕ n ( τn n,e n , S τ∗ n n

(2.69)

Since  vz runs over the whole space V when  z runs over the whole space Z, relation (2.69) is equivalent to the following relation, which holds for  v ∈ V, n = 0, 1, . . . , N:  , X τ∗ ) . v ) = E˙ v ,n g(τ∗ ϕ˙ n ( n , S τ∗ n n

(2.70)

Thus the optimality relation (2.60) holds for the Markov moment τ∗ n. Analogously, relations (2.50) and (2.53) imply the following relation, which holds v ∈ V, n = 0, 1, . . . , N: for   τ˙ ∗ Y  , X τ˙ ∗ ) = Ez ,n g(τ˙ ∗ n , X ∗ ) = ϕ ( v) = E˙ v ,n g(τ˙ ∗ ϕ˙ n ( n z τ˙ n v) . n , S τ˙ ∗ n,e  v n n

(2.71)

zv runs over the whole space Z when  v runs over the whole space V, relation Since  (2.71) is equivalent to the following relation, which holds for  z ∈ Z, n = 0, 1, . . . , N: 

Y τ˙ ∗ n , X ∗) . z ) = Ez ,n g(τ˙ ∗ ϕ n ( τ˙ n n,e

(2.72)

Thus, the optimality relation (2.68) holds for the Markov moment τ˙ ∗ n , for every n = 0, 1, . . . , N. Since τ n ≡ n is a Markov moment from the class Mmax,n,N , the log-reward functions ϕ n ( s , x) satisfy the following inequalities, for every n = 0, 1, . . . , N: y , x) , g(n, ey , x) ≤ ϕ n ( D∗ n

 z = ( y , x) ∈ Z .

(2.73)

Let us denote by D∗ =  D∗ n , n = 0, 1, . . . , N  the time–space domain with the sets defined in the following way: D∗ z = ( y , x) ∈ Z : g(n, ey , x) = ϕ n ( y , x)} , n = {

n = 0, 1, . . . , N .

(2.74)

It follows from relations (2.53) that the time–space domains D˙ ∗ and D∗ are con­ nected by relation (2.66), i.e. D˙ ∗ s , x) = (ey , x) : ( y , x) ∈ D∗ n = {( n},

n = 0, 1, . . . , N .

(2.75)

This implies that the hitting times τ˙ n, D˙ ∗ and τ n,D ∗ coincide for every n = 0, 1, . . . , N, i.e. τ˙ n, D˙ ∗ = τ n,D ∗ . (2.76) The following theorem shows that the time–space domain D∗ defines the hitting ∗ times τ∗ n = τ n,D ∗ , n = 0, 1, . . . , N such that τ n is the optimal stopping time in the class Mmax,n,N , for every n = 0, 1, . . . , N.

2.4 Optimal stopping times

| 67

Theorem 2.4.3. Let the condition A2 holds. Then, for every n = 0, . . . , N, the hitting ∗  ˙ n, D˙ ∗ = time τ∗ n = τ n,H ∗ = min( r ≥ n : Z r ∈ D r ) ∧ N coincides with the hitting time τ ∗ ˙  min(r ≥ n : V r ∈ D r ) ∧ N, and it is the optimal stopping time in the class Mmax,n,N , z = ( y , x) ∈ Z, n = 0, 1, . . . , N: i.e. the following optimality relations hold for every  

Y τ∗ n , X ∗) . ϕ n ( y , x) = ϕ n,τ∗n ( y , x) = Ez ,n g(τ∗ τn n,e

(2.77)

y , x ), It is worth to note that relation (2.77) implies that the log-reward functions ϕ n ( y , x). This does not follow n = 0, 1, . . . , N are the measurable functions in argument ( directly from relation (2.51), which defines the log-reward functions. Note also that the measurability of log-reward functions also follows from the measurability of reward functions, pointed out in the remarks following after Theorem 2.4.1, and relation (2.53) connecting reward and log-reward functions. The following theorem, which is an analog of Theorem 2.4.2, also takes place. Theorem 2.4.4. Let the conditions A1 and A2 hold. Then, the hitting time τ∗ 0 = τ 0,H ∗ = ∗  min(r ≥ 0 : Z r ∈ D r ) ∧ N satisfies the following optimality relation: 0 , X0 ) = Eg(τ ∗ Φ = Eϕ0 (Y 0,e

 τ∗ Y 0

, X τ∗0 ) .

(2.78)

2.4.3 A hitting structure of Markov stopping times In this subsection, we give proofs of Theorem 2.3.1–2.3.4 and 2.4.1–2.4.4, which describe the structure of log-reward functions and optimal stopping times for American-type options for multivariate Markov modulated price and log-price processes.  n = ( Let V S n , X n ), n = 0, 1, . . . , be a Markov modulated price process with a ˙ phase space V = R+ k × X, an initial distribution P 0 ( B ), and transition probabilities P˙ n (v, A). Let τ˙ n be a Markov stopping time from the class Mmax,n,N , i.e. a random vari­ able which take integer values n ≤ r ≤ N such that the event {τ˙ n = r} ∈ Fn,r = n , . . . , V  r ] for every n ≤ r ≤ N. σ[V Let Vl = V × · · · × V be l-times product of the space V and BVl be the mini­ mal σ-algebra of subsets of space Vl , which contains all rectangle sets B1 × · · · × B l , B i ∈ BV , i = 1, . . . , l. It follows from the above definition of the Markov stopping time τ˙ n that there exist n , . . . , V  r ) ∈ B n,r }, for every sets B n,r ∈ BVr−n+1 such that the event {τ˙ n = r} = {(V n ≤ r ≤ N. Thus let one represent the Markov stopping time τ˙ n in the following hitting form: n , . . . , V  r ) ∈ B n,r ) ∧ N . τ˙ n = min(r ≥ n : (V

(2.79)

This representation let one interpret any Markov stopping time τ˙ n as some kind of generalized hitting time.

68 | 2 American-type options Let us connect with the Markov stopping time τ˙ n ∈ Mmax,n,N the shifted Markov stopping times τ˙ n ( v),  v ∈ V, 0 ≤ n < N defined by the following relation:  n +1 , . . . , V  r ) ∈ B n,r ) ∧ N . τ˙ n ( v) = min(r ≥ n + 1 : ( v, V

(2.80)

By the definition, the Markov stopping times τ˙ n (v) ∈ Mmax,n+1,N , v ∈ V, 0 ≤ n < N. Let D˙ =  D˙ n , n = 0, 1, . . . , N  be a sequence of measurable subsets of the space V. Let us now consider the hitting time to the time–space domain D˙ in the discrete time interval [n, N ],  r ∈ D˙ r ) ∧ N . τ˙ n, D˙ = min(r ≥ n : V (2.81) It is obvious that τ˙ n, D˙ is a Markov stopping time from the class Mmax,n,N . n , . . . , V  r ) ∈ B n,r }, where B n,r = V r−n × In this case, the event {τ˙ n, D˙ = r} = {(V ˙D r } for 0 ≤ n ≤ r ≤ N, and, therefore, the corresponding Markov stopping times τ˙ n, D˙ ( v),  v ∈ V, 0 ≤ n < N take the following form:  r ∈ D˙ r ) ∧ N = τ˙ n+1, D˙ . τ˙ n, D˙ ( v) = min(r ≥ n + 1 : V

(2.82)

v ) = τ˙ n+1, D˙ ,  v ∈ V, 0 ≤ n < N themselves Thus, the Markov stopping times τ˙ n, D˙ ( are ordinary hitting times.

2.4.4 Proof of Theorems 2.3.1, 2.3.3, 2.4.1, and 2.4.3 Recall the time–space domains D˙ ∗ =  D˙ ∗ n , n = 0, 1, . . . , N , where D˙ ∗ v = ( s , x) : g(n, s , x) = ϕ˙ n ( v )}, 0 ≤ n ≤ N . n = {

(2.83)

 r ∈ D˙ ∗ ˙ n, D˙ ∗ = min(r ≥ n : V The hitting times τ˙ ∗ n = τ r ) ∧ N, 0 ≤ n ≤ N possess the property given by relation (2.82). Let us assume that the condition A2 holds. This condition and relation (2.29) let one define the finite measurable functions, for n = 0, 1, . . . , N,  , X τ˙ ∗ ),  v ) = E˙ v ,n g(τ˙ ∗ v ∈ V. (2.84) ϕ˙ n ( n , S τ˙ ∗ n n

Note that we are going to prove that the functions defined in relation (2.84) coin­ cide with the reward functions defined in relation (2.32) and, that is why, we use for them the same notations. Lemma 2.4.2. Let the condition A2 holds. Then, the functions ϕ˙ n ( v),  v ∈ V, n = 0, 1, . . . , n satisfy the following backward recurrence relations: ⎧ ϕ˙ ( v ) = g(N, s , x) ,  v = ( s , x) ∈ V , ⎪ ⎪ ⎪ N ⎪ ⎛ ⎞ ⎪ ⎪ ⎪

⎨ ⎜ ⎟ ϕ˙ ( v ) = max ⎝g(m, s , x), P˙ m+1 ( v , d v )ϕ˙ m+1 ( v  )⎠ ,  v = ( s , x) ∈ V , (2.85) ⎪ m ⎪ ⎪ ⎪ V ⎪ ⎪ ⎪ ⎩ m = N − 1, . . . , 0 .

2.4 Optimal stopping times | 69

˙ v) = g(N, s , x),  v = ( s , x ) ∈ V. Proof. By the definition, τ˙ ∗ N = N and, therefore, ϕ N ( Thus, the first series of equalities in (2.85), for functions ϕ˙ N ( v), holds. Let us now prove the inductive proposition that the assumption of holding equal­ v = ( s , x) ∈ V, m = n + 1, for some 0 ≤ n < N implies that these ities (2.85) for  equalities also hold for  v = ( s , x) ∈ V and m = n. Indeed, using the above induction assumption, the Markov property of the pro­ v ∈ V, cess V n and relations (2.82) and (2.83), we get, for   , X τ˙ ∗ ) = I ( v ∈ D∗ s , x) E˙ v ,n g(τ˙ ∗ n , S τ˙ ∗ n ) g ( n,  n n

 , X τ˙ ∗ ) + I ( P˙ n+1 ( v ∉ D∗ v , d v )E˙ v ,n+1 g(τ˙ ∗ n) n +1 , S τ˙ ∗ n +1 n +1 V

= I ( v∈

D∗ s, n ) g ( n, 

x) + I ( v∉

D∗ n)

P˙ n+1 ( v , d v )ϕ˙ n+1 ( v )

V

⎛ ⎜ s , x), = max ⎝g(n, 

⎞

⎟ P˙ n+1 ( v , d v )ϕ˙ n+1 ( v )⎠ = ϕ˙ n ( v) .

(2.86)

V

v ∈ V, 0 ≤ m ≤ N. By induction, equalities (2.84) hold for any  In fact, first one should realize the above induction proof, in particular to get re­ lation analogous to (2.86) for finite (due to the condition A2 and relation (2.29)), and  , X τ˙ ∗ )|,  measurable functions ϕ˙ + v ) = E˙ v ,n |g(τ˙ ∗ v ∈ V, n = 0, 1, . . . , N. Then, n ( n , S τ˙ ∗ n n one can repeat the induction proof, in particular to get relation (2.86) for the functions  , X τ˙ ∗ ),  v ) = E˙ v ,n g(τ˙ ∗ v ∈ V, n = 0, 1, . . . , N. ϕ˙ n ( n , S τ˙ ∗ n n Now, we should prove that the functions ϕ˙ n ( v), n = 0, 1, . . . , N are, indeed, reward v ∈ V, n = 0, 1, . . . , N: functions, i.e. they satisfy the following optimality relation, for  ϕ˙ n ( v) =

sup τ˙ n ∈Mmax,n,N

S τ˙ n , X τ˙ n ) . E˙ v ,n g(τ˙ n , 

(2.87)

˙ v ) = E˙ v ,n g(τ˙ ∗  , X τ˙ ∗n ), for  Since τ˙ ∗ v∈ n ∈ Mmax,n,N , n = 0, 1, . . . , N and ϕ n ( n , S τ˙ ∗ n V, n = 0, 1, . . . , N, relation (2.87) follows from the following lemma. v= Lemma 2.4.3. Let the condition A2 holds. Then, the following inequality holds, for  s , x) ∈ V, 0 ≤ n ≤ N, ( sup τ˙ n ∈Mmax,n,N

S τ˙ n , X τ˙ n ) ≤ ϕ˙ n ( v) . E˙ v ,n g(τ˙ n , 

(2.88)

Proof. Let us take an arbitrary sequence of Markov stopping times τ˙ n ∈ Mmax,n,N , n = 0, 1, . . . , N and use for these Markov stopping times, their representation in the form of generalized hitting time given by relation (2.79) and let also τ˙ n ( v),  v ∈ V, n = 0, 1, . . . , N − 1 be the corresponding shifted Markov stopping times defined in relation (2.80). v ) = g(N, s , x),  v = ( s , x) ∈ V. Thus, relation By the definition, τ˙ N = N and ϕ˙ N ( (2.88) holds and takes, in this case, the form of equality.

70 | 2 American-type options Let us now prove the inductive proposition that the assumption of holding the inequalities (2.88) for  v = ( s , x) ∈ V, m = n + 1 for some 0 ≤ n < N implies that these inequalities also hold for  v = ( s , x) ∈ V and m = n. Indeed, using the above induction assumption and the Markov property of the v ∈ V, process V n , we get, for any Markov stopping time τ˙ n ∈ Mmax,n,N and  S τ˙ n , X τ˙ n ) = I ( v ∈ B n,n )g(n, s , x) E˙ v ,n g(τ˙ n , 

+ I ( v ∉ B n,n ) P˙ n+1 ( v , d v )E˙ v ,n+1 g(τ˙ n ( v),  S τ˙ n (v) , X τ˙ n (v) ) V

≤ I ( v ∈ B n,n )g(n, s , x) + I ( v ∉ B n,n ) ⎛ ⎜ ≤ max ⎝g(n, s, x),

V

P˙ n+1 ( v , d v )ϕ˙ n+1 ( v ) ⎞

⎟ P˙ n+1 ( v , d v )ϕ˙ n+1( v )⎠ = ϕ˙ n ( v) .

(2.89)

V

v∈V Since an arbitrary choice of the Markov stopping time τ˙ n ∈ Mmax,n,N and  in the above relation, inequality (2.88) holds for n = m. By induction, inequalities (2.88) hold for any  v ∈ V, 0 ≤ m ≤ N. In fact, first one should realize the above induction proof, in particular to get relation analogous to (2.89) for finite (due to the condition A2 and relation (2.29)) and measurable functions E˙ v ,n |g(τ˙ n ,  S τ˙ n , X τ˙ n )|,  v ∈ V, n = 0, 1, . . . , N. Then, one can repeat the induction proof, in particular to get relation (2.89) for functions S τ˙ n , X τ˙ n ),  v ∈ V, n = 0, 1, . . . , N. E˙ v ,n g(τ˙ n ,  So, we proved that the finite measurable functions ϕ˙ n ( v),  v ∈ V, n = 0, 1, . . . , N defined by relation (2.84) satisfy the optimality relation (2.87) and, therefore, they co­ incide with the reward functions defined in relation (2.32).  r ∈ D˙ ∗ ˙ n, D˙ ∗ = min(r ≥ n : V Also, we proved that the hitting times τ˙ ∗ n = τ r ) ∧ N is the optimal stopping time in the class Mmax,n,N , for every n = 0, 1, . . . . v),  v ∈ V, n = 0, 1, . . . , N satisfy the Finally, we proved that the functions ϕ˙ n ( backward recurrence relations (2.85), which are equivalent to the backward recurrence relations given in Theorem 2.3.1. Both Theorems 2.3.1 and 2.4.1 are proved. Theorems 2.3.3 and 2.4.3 just reformulate Theorems 2.3.1 and 2.4.1 in terms of log-price processes. The proof analogous to those given for Theorems 2.3.1 and 2.4.1 in terms of the  n can be rewritten in terms of the log-price process  price process V Zn. However, Theorems 2.3.3 and 2.4.3 can also be considered as corollaries of The­ orems 2.3.1 and 2.4.1. The proofs can be obtained by application to relations given in Theorems 2.3.1 and 2.4.1, namely, relation (2.50), which connects log-reward functions ϕ n ( z) and reward functions ϕ˙ n ( v); relation (2.38), which connects transition proba­ z , A) for the log-price process  Z n and transition probabilities P˙ n ( v , B) for bilities P n (

2.4 Optimal stopping times

| 71

 n ; relation (2.44), which connects the expectations of random func­ the price process V ˙ defined on trajectories, respectively, of log-price and price processes; tionals W ≡ W relation (2.75), which connects the optimal stopping time–space domains D∗ and D˙ ∗ ; and relation (2.76), which connects the corresponding stopping times τ n,D ∗ and τ˙ n, D˙ ∗ . The measurability of the log-reward functions ϕ n ( z) follows, due to relation (2.50), from the measurability of the log-reward functions ϕ˙ n ( v ). The corresponding substitution, based on the above relations (2.50), (2.38), and (2.44), transforms the backward recurrence relations (2.34), given for reward functions v ) in Theorem 2.3.1, in the backward recurrence relations (2.55), given for log-re­ ϕ˙ n ( z ) in Theorem 2.3.3. ward functions ϕ n ( The substitution, based on the above relations (2.50), (2.44), (2.75), and (2.76), transforms optimality relation (2.63) given for stopping times τ˙ n, D˙ ∗ in Theorem 2.4.1 in the analogous optimality relation (2.77) given for stopping times τ n, D˙ ∗ in Theo­ rem 2.4.3. The above remarks complete the proofs of Theorems 2.3.3 and 2.4.3.

2.4.5 Proof of Theorems 2.3.2, 2.3.4, 2.4.2, and 2.4.4 We assume that both the conditions A1 and A2 hold. These conditions imply that the ˙ < ∞. optimal expected reward Φ  n and relation (2.45) that It follows from the Markov property of the price process V for any Markov stopping time τ˙ 0 ∈ Mmax,0,N ,

˙ (τ˙ 0 ,  S τ˙ 0 , X τ˙ 0 ) = E˙ v ,0 g(τ˙ 0 ,  S τ˙ 0 , X τ˙ 0 )P˙ 0 (d v) . (2.90) Eg V

Relations (2.90) and (2.32) defining the reward functions ϕ˙ 0 ( v ) imply that the fol­ lowing inequality holds: Φ=

sup τ 0 ∈Mmax,0,N

˙ (τ˙ 0 ,  S τ˙ 0 , X τ˙ 0 ) Eg

≤

sup V

τ˙ 0 ∈Mmax,0,N

E˙ v ,0 g(τ˙ 0 ,  S τ˙ 0 , X τ˙ 0 )P˙ 0 (d v) =

ϕ˙ 0 ( v)P˙ 0 (d v) .

(2.91)

V

Also, relations (2.45), (2.64) given in Theorem 2.4.2, and (2.90) imply that the fol­ lowing equality to holds: Φ=

sup τ˙ 0 ∈Mmax,0,N

= V

˙ (τ˙ 0 ,  ˙ (τ˙ ∗  , X τ˙ ∗ ) S τ˙ 0 , X τ˙ 0 ) ≥ Eg Eg 0 , S τ˙ ∗ 0 0

∗ ∗ E˙ v ,0 g(τ˙ ∗ v) = 0 , S τ˙ 0 , X τ˙ 0 ) P 0 ( d

ϕ˙ 0 ( v)P˙ 0 (d v) .

(2.92)

V

Relations (2.37) and (2.64) given in Theorems 2.3.2 and 2.4.2 follow in an obvious way from relations (2.91) and (2.92).

72 | 2 American-type options Theorems 2.3.4 and 2.4.4 just reformulate Theorems 2.3.2 and 2.4.2 in terms of log-price processes and do not require separate proofs.

2.5 American-type knockout options In this section, we extend the class of American-type options presented in this chap­ ter. We show that so-called American-type knockout options can be imbedded into the model of ordinary American-type options by extending the corresponding price or log-price processes by a special modulating index component. The American-type knockout options include as a particular case various American-type barrier options.

2.5.1 American-type knockout options  n = ( Let V S n , X n ) be a Markov modulated price process with a phase space V = + 0 ∈ A}, and the transition probabilities Rk × X, an initial distribution P˙ 0 (A) = P˙ {V ˙P n ((   ˙ s , x), A) = P{V n ∈ A/V n−1 = ( s , x)}. Let also g(n, s , x) be a pay-off function, which is assumed to be a real-valued mea­ surable function defined for (n,  s , x) ∈ N × R+ k × X. ˙ ˙ ˙ Let also H =  H0 , . . . , H N  be a so-called time–space knockout domain, and ˙ r ) ∧ N, 0 ≤ n ≤ N be knockout stopping times. r ∈ H τ˙ n, H˙ = min(r ≥ n : V We would like to define an American-type knockout option for the price process ˙ V n with the pay-off function g(n, s , x) and the time–space knock-out domain H.

Let us assume that the conditions A1 and A2 hold. These conditions guarantee that the reward functions and optimal expected rewards defined below are finite. If such an option is executed at the Markov stopping time τ˙ n ∈ Mmax,n,N then the S τ˙ n , X τ˙ n )I (τ˙ n < τ˙ n, H˙ ). corresponding random reward is g(τ˙ n ,  The corresponding knockout reward function is defined by the following relation, for  v ∈ V, 0 ≤ n ≤ N: v) = ϕ˙ n (

sup τ˙ n ∈Mmax,n,N

S τ˙ n , X τ˙ n )I (τ˙ n < τ˙ n, H˙ ) , E˙ v ,n g(τ˙ n , 

(2.93)

and the optimal knockout expected reward by the relation ˙ = Φ

sup τ˙ 0 ∈Mmax,0,N

˙ (τ˙ 0 ,  Eg S τ˙ 0 , X τ˙ 0 )I (τ˙ 0 < τ˙ 0, H˙ ) .

(2.94)

The above American-type knockout option can also be redefined in terms of the n , X n ) = (ln  Z n = (Y S n , X n ), which is also a Markov mod­ Markov log-price process  ulated price process with a phase space Z = Rk × X, an initial distribution P0 (B) = 0 ∈ B}, and transition probabilities P n (( P{ Z y , x ) , B ) = P{  Z n ∈ B/ Z n−1 = ( y , x)}.

2.5 American-type knockout options

|

73

In this case, one should consider a time–space knockout domain H =  H0 , . . . , H N , where H n ∈ BZ , n = 0, . . . , N and define knockout stopping times in terms of the log-price process  Z n as τ n,H = min(n ≥ 0 :  Z n ∈ H n ) ∧ N, 0 ≤ n ≤ N. The following relation define a natural one-to-one correspondence between knockout domains H˙ and H, for n = 0, 1, . . . , N, v = ( s , x) :  s = ey ,  z = ( y , x) ∈ H n } , H˙ n = {

(2.95)

which, obviously, implies the following relation for knockout times, for n = 0, 1, . . . , N: τ˙ n, H˙ = τ n,H . (2.96) The corresponding knockout log-reward function is defined by the following re­ lation, for  z ∈ Z, 0 ≤ n ≤ N: z) = ϕ n (



sup τ n ∈Mmax,n,N

Ez ,n g(τ n , e Y τn , X τ n )I (τ n < τ n,H ) .

(2.97)

and the optimal knockout expected log-reward by the relation, Φ=



sup τ 0 ∈Mmax,0,N

Eg(τ0 , e Y τ0 , X τ0 )I (τ0 < τ0,H ) .

(2.98)

In this case, where relation (2.95) holds, the knockout options defined via the price process or the log-price process define, in fact, the same option contract. Correspond­ v ) and the log-reward functions ϕ n ( z) are connected ingly, the reward functions ϕ˙ n ( by the following relation, for  z = ( y , x) ∈ Z,  vz = (ey , x), n = 0, 1, . . . , N: ϕ n ( z ) = ϕ˙ n ( vz ) ,

(2.99)

˙ and Φ are connected by the following while the optimal knockout expected rewards Φ relation, ˙ . Φ=Φ (2.100)

2.5.2 Imbedding into the model of ordinary American-type options In the model of the American-type knockout options, the random reward  r , n ≤ r ≤ τ n ∧ N and g(τ n ,  S τ n , X τ n )I (τ n < τ˙ n, H˙ ) depends on the whole trajectory V τn Y the random log-reward g(τ n , e , X τ n )I (τ n < τ˙ n, H˙ ) depends on the whole trajectory  Z r , n ≤ r ≤ τ n ∧ N. However, this model can be imbedded in the model of the ordinary American n or the log-price type option using an appropriate extension of the price process V process  Z n . Let us describe the corresponding imbedding transformation in terms of the log-price process  Zn.

74 | 2 American-type options Let us introduce the additional index component X n =

n 

I ( Zr ∉ Hr ) ,

n = 0, 1, . . . .

(2.101)

r =0

Z n = ( Z n , X n , X n ), n = 0, 1, . . . , is also a Markov It is obvious that the process  chain, with the extended phase space Z = Rk × X × {0, 1}. n = In this case, the role of the modulated index process is played by the process X  (X n , X n ), with the phase space X × {0, 1}. Let us define the domains G n , 0 ≤ n ≤ N by the following relation:   G n = ( y , x, e) : I (( y , x) ∈ H n , e = 1) + I (( y , x) ∈ H n , e = 0) . (2.102) The definition of the additional index component X n implies that the following relation holds: n ∈ G n , 0 ≤ n ≤ N } = 1 . (2.103) P{ Z The initial distribution for the Markov chain  Z n is defined by the following rela­ tion, for A ∈ BZ , e = 0, 1: ⎧ ⎨P0 (A ∩ H 0 ) if e = 1 , P0 (A × {e}) = ⎩ (2.104) P0 (A ∩ H0 ) if e = 0 . Z n are defined by the following re­ The transition probabilities for Markov chain    lation, for ( y , x, e) ∈ Z , A ∈ BZ , e = 0, 1, n = 1, 2, . . ., y , x, e), A × {e }) Pn (( ⎧ ⎪ P n (( y , x), A ∩ H n ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y , x), A ∩ H n ) ⎪P n (( ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨P n (( y , x), A) = ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P n (( y , x), A) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎩ P n (( y , x), A)

if ( y , x ) ∈ H n −1 , e = 1 , e  = 1 , if ( y , x ) ∈ H n −1 , e = 1 , e  = 0 , if ( y , x ) ∈ H n −1 , e = 0 , e  = 1 , if ( y , x ) ∈ H n −1 , e = 0 , e  = 0 ,

(2.105)



if ( y , x ) ∈ H n −1 , e = 1 , e = 1 , if ( y , x ) ∈ H n −1 , e = 1 , e  = 0 , if ( y , x ) ∈ H n −1 , e = 0 , e  = 1 , if ( y , x ) ∈ H n −1 , e = 0 , e  = 0 .

Let us also introduce the transformed pay-off function defined by the following y , x, e) ∈ Z , 0 ≤ n ≤ N: relation, for ( g(n, ey , x, e) = g(n, ey , x)I (( y , x, e) ∈ G n ) .

(2.106)

It is readily seen that the class of Markov stopping times Mmax,n,N coincides for the processes  Z n and  Z n , since the corresponding σ-algebras Fn,m = σ [ Zn , . . . ,  Zm]   Zn , . . . ,  Z m ] for any 0 ≤ n ≤ m ≤ N. = σ[

2.5 American-type knockout options

|

75

Also, the following relation holds for the corresponding random rewards, for any τ n ∈ Mmax,n,N , 0 ≤ n ≤ N, 



g(τ n , e Y τn , X τ n )I (τ n < τ n,H ) = g(τ n , e Y τn , X τ n , X τ n ) .

(2.107)

Let us now introduce the log-reward functions for the ordinary American-type options for the price process  Z n with the pay-off function g(n, ey , x, e), defined for    z = ( y , x, e) ∈ Z , 0 ≤ n ≤ N, ϕn ( s , x, e) =



sup τ n ∈Mmax,n,N

Ez ,n g(τ n , e Y τn , X τ n , X τ n ) ,

(2.108)

and the optimal expected reward by the relation Φ =

sup τ 0 ∈Mmax,0,N



Eg(τ0 , e Y τ0 , X τ0 , X τ0 ) .

(2.109)

Relation (2.107) implies that the log-reward functions for the American-type knockout option for the log-price process  Z n (with the pay-off function g(n, ey , x) and the stopping knockout time–space domain H) are connected with the log-reward functions for the ordinary American-type option for the log-price process  Z n (with the pay-off function g(n, ey , x, e)) by the following relation, for  z  = ( y , x, e) ∈ Z , 0 ≤ n ≤ N: ⎧ ⎪ y , x) if ( y , x) ∈ H n , e = 1 , ⎪ϕ n ( ⎪ ⎨  ϕ n ( (2.110) y , x, e) = ⎪0 if ( y , x) ∈ H n , e = 0 , ⎪ ⎪ ⎩0 if ( y , x, e) ∈ G . n

while the corresponding optimal expected rewards are connected by the following relation, (2.111) Φ = Φ .

3 Backward recurrence reward algorithms In this chapter, we investigate the structure of backward recurrence relations for log-price processes represented by atomic Markov chains, whose transition probabil­ ities are concentrated on finite sets. In Sections 3.1 and 3.2, we consider binomial and trinomial models, univariate and multivariate, homogeneous and inhomogeneous in time and space as well as models with Markov modulation. We study the structure of binomial and trinomial trees of nodes generated by binomial and trinomial random walks. A special attention is paid to the so-called recombining condition. This condition guarantees that the number of time–space nodes generated by trajectories of such random walks has not more than polynomial rate of growth as a function of the number of time steps. We also present recurrence systems of linear equations for log-reward functions of American-type op­ tions. Also, we present recurrent algorithms for the effective computing of binomial and trinomial probabilities penetrating the above-mentioned recurrence system of lin­ ear equations. In Section 3.3, we consider a general random walk models with distributions of jumps concentrated on finite sets. We describe trees of nodes generated by trajecto­ ries of such random walks, formulate the corresponding variants of recombining con­ ditions and give recurrence systems of linear equations for log-reward functions of American-type options. In Section 3.4, we consider a general model of multivariate modulated atomic Markov chains, including space-skeleton Markov chains and space-skeleton Markov random walks. Recombining conditions are given for these models as well as recur­ rence system of linear equations for log-reward functions of American-type options. The characteristic property of the models satisfying recombining conditions is that the corresponding algorithms for finding of log-reward functions have backward recurrence structure with not more than polynomial rate of growth of number of un­ known values of log-reward functions as the number of the corresponding time steps. This makes these systems computationally effective. The corresponding recurrence systems of linear equations for the log-reward functions are given for all above-listed models of atomic Markov chains in Lem­ mas 3.1.1–3.4.4. These lemmas are particular variants of Theorem 2.3.3 and, therefore, do not require special proofs.

3.1 Binomial tree reward algorithms In this section, we describe backward tree algorithms for finding log-reward functions and optimal expected rewards for the binomial tree models of log-price processes. We consider classical homogeneous models, model inhomogeneous in time and space as

3.1 Binomial tree reward algorithms

|

77

well as models with Markov modulation. Binomial models can be effectively used for approximation of more general log-price processes represented by Gaussian random walks homogeneous and inhomogeneous in time and space.

3.1.1 A classical homogeneous binomial tree model This is the simplest model, where a log-price Y n is a univariate random walk, i.e. Y n = Y n −1 + W n ,

n = 1, 2, . . . .

(3.1)

where (a) Y0 is a real-valued random variable with the distribution P0 (A), (b) W n , n = 1, 2, . . . , is a sequence of i.i.d. binary random variables, which take two values δ+ and δ− with probabilities, respectively, p+ and p− = 1 − p+ , (c) the random variable Y0 and the random sequence W n , n = 1, 2, . . . , are independent. Usual assumptions also are (d) δ− < δ+ , and (e) 0 ≤ p+ ≤ 1. In this case, the index variable X n is not included in the model. The pay-off func­ tion g(n, e y ) and the corresponding log-reward functions ϕ n (y) do not depend on the index argument. Let us assume that the initial state of the log-price process Y0 = y0 with proba­ bility 1, i.e. the initial distribution P0 (A) = I (y0 ∈ A) is concentrated at the point y0 ∈ R1 . The tree structure of trajectories is well visible from the recurrent algorithm, which describes sequential transitions of the log-price process Y n . If Y n−1 = y for some n = 1, 2, . . . , then, either Y n = y + δ+ or Y n = y + δ− . The initial point y0 determines the unique tree T(y0 ,0) , with the root in the node (y0 , 0) in the space–time domain R1 × {0, 1, . . . }. According to the algorithm described above the tree T(y0 ,0) has one initial node (y0 , 0). It includes two additional nodes after the first step, then four additional nodes after the second step, etc. It seems that the tree may include 2n additional nodes after the nth step. If this is the case, i.e. the tree T(y0 ,0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithm based on such a tree would not be effective from the computational point of view. However, this is not the case. The tree T(y0 ,0) includes only n + 1 additional nodes after n steps. In fact, the tree T(y0 ,0) satisfies the so-called recombining condition. This condition means that the random walk Y n , which is in a position Y n−1 = y for some n = 1, 2, . . . , and makes from this position first jump δ+ and then jump δ− should end up at the same point as if it would make the first jump δ− and then the second jump δ+ . This should hold for any possible position y ∈ R1 and n = 1, 2, . . . . Indeed, if Y n−1 = y for some n = 1, 2, . . . , then Y n+1 = y + δ+ + δ− if W n = δ+ , W n+1 = δ− or Y n+1 = y + δ− + δ+ if W n = δ− , W n+1 = δ+ . In both cases Y n+1 takes the same value.

78 | 3 Backward recurrence reward algorithms The tree T(y0 ,0) obviously includes one initial node (y0 , 0) and two additional nodes (y0 + δ+ , 1) and (y0 + δ− , 1) after the first step. Due to the recombining prop­ erty, the tree includes three additional nodes (y0 + 2δ+ , 2), (y0 + δ+ + δ− , 2) and (y0 + 2δ− , 2) after the second step, etc. The tree T(y0 ,0) includes n + 1 additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 + 2 + · · · + (n + 1) =

(n + 2)(n + 1) . 2

(3.2)

This means that the tree possesses a quadratic rate of growth for the number of nodes in the tree after n steps. This makes algorithms based on such a tree effective from the computational point of view. The nodes appearing in the tree after the nth step are determined by the number l of jumps δ+ in n steps but does not depend of moments when these jumps occurred. The number l also determines the number n − l of jumps δ− in n steps. The states that can be reached by the random walk at moments n = 0, 1, . . . , are given by the following formula: y n,l = y0 + δ+ l + δ− (n − l) ,

l = 0, 1, . . . , n , n = 0, 1, . . . .

(3.3)

It is worth to note that nodes appearing in the tree after n steps can alternatively be indexed by the number r = 2l − n that is the difference between the number of jumps δ+ and δ− in n steps. We, however, do prefer to use the former indexation by the num­ ber l of jumps δ+ in n steps. As a matter of fact, in this case of a binomial tree model, index r either takes only even values if n is an even number or only odd values if n is an odd number. This does not improve computational aspects for the corresponding backward algorithms, and is not convenient for writing the corresponding formulas. It should be mentioned that the procedure for the construction of the tree T(y0 ,0) described above does not take into account the actual values of probabilities for the appearance nodes in the tree generated by trajectories of the random walk Y n . The probability that a node y n,l will be reached after n steps is given by the formula (n−l)

l p− p n,l = C ln p+

,

l = 0, 1, . . . , n , n = 0, 1, . . . ,

(3.4)

where 00 should be counted as 1. Here and henceforth, C ln = l!(nn!−l)! . Note also that numbers y n,l satisfy the inequalities, y n,0 < y n,1 < · · · < y n,n ,

n = 0, 1, . . . .

(3.5)

It is useful to note that all these points are located on the grid of points G[y0 , δ+ , δ− ] = {y0 + δ+ l+ + δ− l− , l± = 0, 1, . . . }. Moreover, in the standard symmetric case, where ±δ± = δ > 0, this grid reduces to the simple grid of points, G[y0 , δ] = {y0 + δl, l = 0, ±1, . . . }.

3.1 Binomial tree reward algorithms

| 79

According to formula (3.3), the tree T(y0 ,0) is the following set of nodes: T(y0 ,0) = (y n,l , n), l = 0, 1, . . . , n, n = 0, 1, . . .  .

(3.6)

The tree T(y0 ,0) includes a particular point (y, n) if and only if y n,l = y0 + δ+ l + δ− (n − l) = y for some l = 0, 1, . . . , n. The condition A2 holds since, in this case, the following inequality take place for any y ∈ R1 , n = 0, 1, . . . , N:       Ey,n max g(r, e Y r ) ≤ max max g(r, e y r,l ) < ∞ , (3.7) n≤r≤N

n≤r≤N m≤l≤m+r−n

where the points y r,l are given by formulas (3.3), with y0 and 0 ≤ m ≤ n chosen such that y n,m = y. Theorem 2.3.3 takes in this case the following form. Lemma 3.1.1. Let a log-price process Y n be represented by the binomial random walk defined by the stochastic transition dynamic relation (3.1). Then the log-reward functions ϕ n (y n,l ), for points y n,l = y0 + δ+ l + δ− (n − l), l = 0, 1, . . . , n, n = 0, 1, . . . , N given by formulas (3.3) are the unique solution for the following finite recurrence system of linear equations: ⎧ y ⎪ ⎪ ϕ N (y N,l ) = g(N, e N,l ) , l = 0, 1, . . . , N , ⎪ ⎪  ⎪ ⎪ ⎨ ϕ n (y n,l ) = max g(n, e y n,l ), ϕ n+1 (y n+1,l)p− (3.8) ⎪ ⎪ + ϕ n+1 (y n+1,l+1)p+ , l = 0, 1, . . . , n , ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 . This system lets one find the values of the log-reward function ϕ n (y) at points y n,l , l = 0, 1, . . . , n given by formulas (3.3), recursively, for n = N, . . . , 0. The particular value ϕ n (y) can be calculated using this system if the initial value y0 is chosen so that y n,l = y0 + δ+ l + δ− (n − l) = y for some l = 0, 1, . . . , n. It is useful to mention that the states y n,l = y0 + δ+ l + δ− (n − l) are, in fact, functions of the initial state y0 . Let us use, for the moment, the notation y n,l = y n,l (y0 ) to show this explicitly. Let us now consider the trees T(y0 +δ+ l+ +δ− l− ,0) , 0 ≤ l± ≤ r± , where 0 ≤ r± ≤ N. These trees have joint nodes since the following relation hold for 0 ≤ l± ≤ r± , 0 ≤    l ± ≤ r± , 0 ≤ l , l ≤ n ≤ N:      y n,l (y0 + δ+ l+ + δ− l− ) = y n,l (y0 + δ+ l + + δ − l − ) if l ± ± l = l ± ± l .

(3.9)

Thus, one can aggregate the backward recurrence relations (3.8) for initial nodes (y0 + δ+ l+ + δ− l− , 0), 0 ≤ l± ≤ r± in one system, and, in this way, reduce computa­ tions for simultaneous funding values of reward functions ϕ n (y) at points y n,l (y0 + δ+ l+ + δ− l− ), 0 ≤ l± ≤ r± , 0 ≤ l ≤ n ≤ N.

80 | 3 Backward recurrence reward algorithms Also it is useful to note that every node (y n,m , n), 0 ≤ m ≤ n < ∞, in the tree T(y0 ,0) generates its own subtree which is the following sets of nodes: T(y n,m ,n) = (y n,m + δ+ l + δ− (r − n − l), r), l = 0, . . . , r − n, r = n, n + 1, . . .  .

(3.10)

Moreover, it is easily seen that the backward recurrence relations in the system (3.8) written down for points in the above-reduced tree T(y n,m ,n) create the reduced system of the backward recurrence relations that let one to compute values of the logreward function ϕ r (y) at points (y r,l , r), l = m, . . . , m + r − n, r = n, . . . , N. As far as the optimal expected reward Φ(Mmax,N ) is concerned, it is given, ac­  cording to Theorem 2.3.4 by the integral, Φ(Mmax,N ) = R1 ϕ0 (y)P0 (dy). In the case, where P0 (A) is a discrete distribution with a finite number of atoms, i.e. P0 (A) = j i =1 p i I ( y i ∈ A ) the above integral takes the form of the sum Φ(Mmax,N ) =

j

p i ϕ0 (y i ) .

(3.11)

i =1

3.1.2 A binomial tree model inhomogeneous in time and space In this model, the log-price process is given by the following stochastic transition dy­ namic relation: ⎧ ⎪ Y n = Y n −1 + W n ⎪ ⎪ ⎨ W n = δ+ (3.12) n ( Y n −1 ) I ( U n ≤ p n ( Y n −1 )) ⎪ ⎪ ⎪ ⎩ + δ− (Y )I (U > p (Y )) , n = 1, 2, . . . , n

n −1

n

n

n −1

where (a) Y0 is a real-valued random variable with a distribution P0 (A), (b) U n , n = 1, 2, . . . , is a sequence of i.i.d. random variables uniformly distributed in the interval [0, 1], (c) the random variable Y 0 and the random sequence U n , n = 1, 2, . . . , are − independent, (d) δ+ n ( y ) , δ n ( y ), n = 1, 2, . . . , are measurable functions acting from R1 to R1 , and (e) p n (y), n = 1, 2, . . . , are the measurable functions acting from R1 to [0, 1]. + A usual assumption also is (e) δ− n ( y ) < δ n ( y ), y ∈ R1 , n = 1, 2, . . . . In the above model, the index variable X n is not included in the model. Corre­ spondingly, a pay-off function g(n, e y ) and the corresponding log-reward functions ϕ n (y) do not depend on the index argument. Let us assume that the initial state of the log-price process Y0 = y0 with proba­ bility 1, i.e. the initial distribution P0 (A) = I (y0 ∈ A), is concentrated at the point y0 ∈ R1 . The tree structure of trajectories is analogous to those described above for the simplest binomial tree model. It is well visible from the following recurrent algorithm

3.1 Binomial tree reward algorithms

| 81

that describes sequential transitions of the log-price process Y n . If Y n−1 = y for some − n = 0, 1, . . . , then, either Y n = y + δ+ n ( y ) with probability p n ( y ) or Y n = y + δ n ( y ) with probability 1 − p n (y). The initial point y0 determines the unique tree T(y0 ,0) with the root in the node (y0 , 0) in the space–time domain R1 × {0, 1, . . . }. According to the algorithm described above, the tree T(y0 ,0) may include two ad­ ditional nodes after the first step, four nodes after the second steps, etc. The tree may include 2n additional nodes after the nth step. If this is the case, i.e. the tree T(y0 ,0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithm based on such a tree would not be effective from the computational point of view. A recombining condition should hold in order to prevent such a case. As above, this condition means that the random walk Y n , which is in a position y at a moment n − − + 1, and makes first jump δ+ n ( y ) and then the second jump δ n +1 ( y + δ n ( y )) should end up at the same point as if it would make the first jump δ− n ( y ) and then the second jump − δ+ n +1 ( y + δ n ( y )). In this case, the recombining condition does not hold automatically. This should hold for any possible position of the random walk. − + If Y n−1 = y for some n = 1, . . . ,, then Y n+1 = y + δ+ n ( y ) + δ n +1 ( y + δ n ( y )) if − + + − − W n = δ+ n ( y ) , W n +1 = δ n +1 ( y + δ n ( y )), while Y n +1 = y + δ n ( y ) + δ n +1 ( y + δ n ( y )) if + − − W n = δ n (y), W n+1 = δ n+1 (y + δ n (y)). In order for the recombining condition to hold, Y n+1 should take the same value in both cases, for any y ∈ R1 , n = 1, . . . . Thus, the recombining condition requires the following relation to hold for any y ∈ R1 , n = 1, . . . : − + − + − δ+ n ( y ) + δ n +1 ( y + δ n ( y )) = δ n ( y ) + δ n +1 ( y + δ n ( y )) .

(3.13)

This condition obviously holds for the model with values of jumps homogeneous in time and space, i.e. where there exists δ− < δ+ such that δ± n ( y ) = δ ± , y ∈ R1 , n = 1, 2, . . . . In the case, of the model with values of jumps inhomogeneous in time but ho­ ± mogeneous in space, i.e. where δ± n ( y ) = δ n , y ∈ R1 , n = 1, . . . , the recombining condition takes the form of the following relation: − − + δ+ n + δ n +1 = δ n + δ n +1 ,

n = 1, 2, . . . .

(3.14)

− This relation means that the difference δ+ n − δ n does not depend on n, i.e. there exists δ > 0 such that − n = 1, 2, . . . . (3.15) δ+ n − δ n = 2δ ,

Let us denote

− δ+ n + δn , n = 1, 2, . . . . 2 Then the recombining condition can be rewritten in the following form:

δ◦n =

◦ δ+ n = δ + δn ,

◦ δ− n = −δ + δ n ,

n = 1, 2, . . . .

(3.16)

(3.17)

82 | 3 Backward recurrence reward algorithms As an example of the model with values of jumps inhomogeneous in time and space, let us consider a model, where values of jumps are linear functions of y, i.e. ± δ± n ( y ) = δ n y, y ∈ R1 , n = 1, 2, . . . . In this case, the recombining condition takes the form of the following relation: − − + − + + − δ+ n + δ n +1 + δ n +1 δ n = δ n + δ n +1 + δ n +1 δ n ,

n = 1, 2, . . . .

(3.18)

± This condition holds, for example, if coefficients δ± n = δ , n = 1, 2, . . . , do not depend on n. The tree T(y0 ,0) obviously contains one initial node (y0 , 0) and two nodes (y0 + − δ+ ( y 0 ) , 1) and ( y 0 + δ n ( y 0 ) , 1) after the first step. If the recombining condition holds, n + + + the tree contains three nodes (y0 + δ+ 1 ( y 0 ) + δ 2 ( y 0 + δ 1 ( y 0 )) , 2), ( y 0 + δ 1 ( y 0 ) + − + − + − − δ2 (y0 + δ1 (y0 )), 2) = (y0 + δ1 (y0 )+ δ2 (y0 + δ1 (y0 )), 2), and (y0 + δ1 (y0 )+ δ− 2 (y0 + δ− ( y )) , 2 ) , after the second step, etc. 0 1 If the recombining condition holds, the node with l jumps δ+ · (·) in n steps does not depend of the order of such jumps in n steps. This makes it possible to index these nodes by parameter l and to use the same notation (y n,l , n), l = 0, 1, . . . , n for nodes appearing in the tree after n steps. The states y n,l n , l n = 0, 1, . . . , n, that can be reached by the log-price process Y n with the initial value Y0 = y0 at moments n = 0, 1, . . . , are given by the following recurrence relation, for 0 ≤ l n ≤ n, n = 0, 1, . . . : ⎧ ⎨ δ+ (y n,l ) if l n+1 = l n + 1 , n +1 (3.19) y n+1,l n+1 = y n,l n + ⎩ − δ (y n,l ) if l n+1 = l n − 1 ,

n +1

where y0,0 = y0 . Note that numbers y n,l n satisfy the inequalities y n,0 < y n,1 < · · · < y n,n ,

n = 0, 1, . . . .

(3.20)

According to the above remarks, the tree T(y0 ,0) is the following set of nodes: T(y0 ,0) = (y n,l , n), l = 0, 1, . . . , n, n = 0, 1, . . .  .

(3.21)

If the recombining condition holds, the number of additional nodes appearing in the tree T(y0 ,0) after the nth step is n + 1. The total number of nodes in the tree T(y0 ,0) n +1 ) after n steps is again L n = 1 + 2 · · · + (n + 1) = (n+2)( . This means that the tree 2 has a quadratic rate of growth for the number of nodes in the tree as a function of the number of steps n. This makes algorithms based on such a tree effective from the computational point of view. The tree T(y0 ,0) includes a particular point (y, n) if and only if y n,l = y for some l = 0, 1, . . . , n. The condition A2 holds since, in this case, the following inequality take place for any y ∈ R1 , n = 0, 1, . . . , N:       max g(r, e y r,l ) < ∞ , (3.22) Ey,n max g(r, e Y r ) ≤ max n≤r≤N

n≤r≤N m≤l≤m+r−n

3.1 Binomial tree reward algorithms

| 83

where the points y r,l are given by formulas (3.19), with y0 and 0 ≤ m ≤ n chosen such that y n,m = y. Theorem 2.3.3 takes in this case the following form. Lemma 3.1.2. Let the log-price process Y n be represented by the inhomogeneous in space and time binomial random walk defined by the stochastic transition dynamic relation (3.12). Then the log-reward functions ϕ n (y n,l ), for points y n,l , l = 0, 1, . . . , n, n = 0, 1, . . . , N given by formulas (3.19), are the unique solution for the following finite recurrence system of linear equations: ⎧ y N,l ⎪ ⎪ ⎪ ϕ N (y N,l ) = g(N, e ) , l = 0, 1, . . . , N , ⎪  ⎪ ⎪ ⎨ ϕ n (y n,l) = max g(n, e y n,l ), ϕ n+1 (y n+1,l)p− n ( y n,l ) (3.23) + ⎪ ⎪ +ϕ n+1 (y n+1,l+1)p n (y n,l) , l = 0, 1, . . . , n , ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 . This system lets one find values of the log-reward function ϕ n (y) at points y n,l , l = 0, 1, . . . , n given by formulas (3.19), recursively, for n = N, . . . , 0. The particular value ϕ n (y) can be calculated by using this system if the initial value y0 is chosen so that y n,l = y for some l = 0, 1, . . . , n. The extension and reduction of the system of backward recurrence relations (3.23) can be realized in the same way as it is described in Subsection 3.1.1, for the system of backward recurrence relations (3.8). Also, the remarks concerned finding of optimal expected reward, analogous to those made in Subsection 3.1.1 can be made.

3.1.3 A multivariate binomial tree model n is a k-dimensional ran­ This is the model where the multivariate Markov log-price Y dom walk,  n −1 + W  n , n = 1, 2, . . . . n = Y Y (3.24) 0 = (Y 0,1 , . . . , Y 0,k ) is a random vector, which takes values in Rk and where (a) Y  n = (W n,1 , . . . , W n,k ), n = 1, 2, . . . , is a sequence of has a distribution P0 (A), (b) W k-dimensional independent random vectors taking values δ¯¯ı = (δ1,ı1 , . . . , δ k,ı k ) with probabilities p¯ı for ¯ı ∈ Ik , where Ik = {¯ı = (ı1 , . . . , ı k ), ı1 , . . . , ı k = ±}, and (c) the 0 and the random sequence W  n , n = 1, 2, . . . , are independent. random vector Y Also usual assumptions are (d) δ j,− < δ j,+, j = 1, . . . , k, and (e) p¯ı ≥ 0, ¯ı ∈ Ik ,  ¯ı∈I p¯ı = 1. In this case, the index component X n is not included in the model. Correspond­ ingly, a pay-off function g(n, ey ) and the corresponding log-reward functions ϕ n ( y) do not depend on the index argument.

84 | 3 Backward recurrence reward algorithms 0 =  Let us assume that the initial state of the log-price process Y y0 with proba­ bility 1, i.e. the initial distribution P0 (A) = I ( y0 ∈ A), is concentrated at the point  y0 ∈ Rk . The multivariate tree structure of trajectories is analogous to those described above for the simplest binomial tree model in the one-dimensional case. It is well vis­ ible from the following recurrent algorithm that describes the sequential transitions n . If Y n =   n +1 =  y for some n = 0, 1, . . . , then Y y + δ¯¯ı with of the log-price process Y k probability p¯ı , where ¯ı ∈ Ik . Note that the set Ik contains 2 elements. The initial point  y0 = (y0,1 , . . . , y0,k ) determines the unique tree T(y0 ,0) with the y0 , 0) in the space–time domain Rk × {0, 1, . . . }. root in the node ( According to the algorithm described above the tree T(y0 ,0) includes one initial node and may include 2k additional nodes after the first step, 22k nodes after the sec­ ond step, etc. The tree may include 2nk additional nodes after the nth step. If this is the case, i.e. the tree T(y0 ,0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithm based on such a tree would not be effective from the computational point of view. However, this is not the case. In fact, the tree includes only (n + 1)k additional nodes after the nth step. This is because the tree T(y0 ,0) satisfies the recombining con­ dition. This property is inherited by the multivariate tree T(y0 ,0) generated by the vec­ n from its one-dimensional components T(y0,j ,0) generated by onetor random walk Y dimensional random walks Y n,j, for every j = 1, . . . , k. Let us denote ¯ı n = (ı n,1 , . . . , ı n,k ), n = 1, 2, . . . . The tree T(y0 ,0) obviously includes one initial node ( y0 , 0) and 2k additional nodes ¯ ( y0 + δ¯ı1 , 1), ¯ı1 ∈ Ik after the first step. Due to the recombining property, the tree in­ cludes 3k additional nodes (y0 + δ¯¯ı1 + δ¯¯ı2 , 2), ¯ı1 , ¯ı2 ∈ Ik after the second step. Indeed, the jth component of the vector δ¯¯ı1 + δ¯¯ı2 = (δ1,ı1,1 + δ2,ı2,1 , . . . , δ k,ı1,k + δ k,ı2,k ) can take three different values 2δ j,+ , δ j,+ + δ j,−, and 2δ j,−. Subsequently, the tree includes 4k additional nodes after the third step, etc. The tree T(y0 ,0) includes (n + 1)k additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 + 2k + · · · + (n + 1)k . Obviously, (n + 1)k = k+1

n +1

n +2 k

x k dx =

x dx ≤ L n ≤ 0

1

(n + 2)k − 1 . k+1

(3.25)

Therefore, the tree possesses a polynomial of the order k + 1 rate of growth for the number of nodes in the tree after n steps as a function of n. This makes algorithms based on such a tree effective from the computational point of view. The states appearing after n steps are determined by the vector ¯l = (l1 , . . . , l k ), where l j is the number of jumps δ j,+ in n steps, for the component Y n,j of the random n . But these states do not depend of moments when these jumps occur. States walk Y that can be reached by this random walk at moments n = 0, 1, . . . , N are given by the

3.1 Binomial tree reward algorithms

| 85

following relation:  y n,¯l = (y n,1,l1 , . . . , y n,k,l k ) ,

¯l = (l1 , . . . , l k ) ∈ Ln ,

n = 0, 1, . . . ,

(3.26)

where the set Ln = {¯l : 0 ≤ l1 , . . . , l k ≤ n} and components y n,j,l j = y0,j + δ j,+ l j + δ j,−(n − l j ), j = 1, . . . , k. Note also that numbers y n,j,l j satisfy the inequalities y n,j,0 < y n,j,1 < · · · < y n,j,n ,

j = 1, . . . , k ,

n = 0, 1, . . . .

(3.27)

According to the above remarks, the tree T(y0 ,0) is the following set of nodes: T(y0 ,0) = ( y n,¯l , n), ¯l = (l1 , . . . , l k ) ∈ Ln , n = 0, 1, . . .  .

(3.28)

The tree T(y0 ,0) includes a particular point ( y , n) if  y n,l = y for some ¯l ∈ Ln . The condition A2 holds since, in this case, the following inequality takes place for any y ∈ R1 , n = 0, 1, . . . , N:          g(r, ey r,¯l ) < ∞ , Ey ,n max g(r, e Y r ) ≤ max max (3.29) n≤r≤N

n≤r≤N m j ≤l j ≤m j +r−n ,

j =1,...,k

y r,¯l are given by formulas (3.26), with  y0 and m  = ( m 1 , . . . , m k ), where the points  0 ≤ m j ≤ n, j = 1, . . . , k chosen such that  y n, m¯ =  y. Theorem 2.3.3 takes the following form. Lemma 3.1.3. Let the log-price process Y n be represented by the multivariate binomial random walk defined by the stochastic transition dynamic relation (3.24). Then the logreward functions ϕ n ( y n,l ), for points  y n,¯l , ¯l = (l1 , . . . , l k ) ∈ Ln , n = 0, 1, . . . , N given by formulas (3.26), are the unique solution for the following finite recurrence system of linear equations: ⎧ ϕ N ( y N,¯l ) = g(N, ey N,l ) , ¯l ∈ LN , ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ ⎨

 y ¯ (3.30) ϕ n ( y n,¯l ) = max ⎝g(n, e n,l ), ϕ n+1 ( y n+1,l+e¯¯ı )p¯ı ⎠ , ¯l ∈ Ln , ⎪ ⎪ ⎪ ¯ı∈Ik ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 . where vectors e¯¯ı = (e ı1 , . . . , e ı k ), e ı j = 1 if ı j = + or e ı j = 0 if ı j = −, for j = 1, . . . , k. y) This system of equations (3.30) let one find the values of the log-reward function ϕ n ( ¯ at points  y n,¯l , l ∈ Ln given by formulas (3.26), recursively, for n = N, . . . , 0. The particular value ϕ n ( y) can be found using this system if the initial value  y0 is y n,¯l =  y for some ¯l ∈ Ln . chosen so that  The extension and reduction of the system of backward recurrence relations (3.30) can be realized in the same way as it is described in Subsection 3.1.1 for the system of backward recurrence relations (3.8). Also, the remarks concerning finding of optimal expected reward analogous to those made in Subsection 3.1.1 can be made.

86 | 3 Backward recurrence reward algorithms

3.1.4 A binomial tree model modulated by a Markov chain We consider a univariate, homogeneous in time and space log-price processes repre­ sented by the binomial random walk modulated by a homogeneous in time Markov chain. We restrict consideration by the model where values of jumps do not depend while the corresponding jump probabilities do depend on states of the modulating Markov chain. We also assume that the modulating Markov chain has a finite phase space X = {1, . . . , m}. In this model, the log-price process Y n is given by the following stochastic transi­ tion dynamic relation: Y n = Y n −1 + W n ( X n −1 ) ,

n = 1, 2, . . . ,

(3.31)

where (a) Y0 is a real-valued random variable with a distribution P0(A), (b) W n (x), x ∈ X, n = 1, 2, . . . , is a family of independent random variables taking values δ+ and δ− with probabilities p+ (x) and p− (x) = 1 − p+ (x) which are independent of n = 0, 1, . . . , but are dependent on x ∈ X, (c) X n , n = 0, 1, . . . , is a homoge­ neous Markov chain with the phase space X, an initial distribution P(x) = P{X0 = x}, x ∈ X and one-step transition probabilities P(x , x ) = P{X n = x /X n−1 = x }, 0 , the family of random sequence x , x ∈ X, n = 1, 2, . . ., (d) the random variable Y  W n (x), x ∈ X, n = 1, 2, . . . , and the Markov chain X n , n = 0, 1, . . . , are independent. Also usual assumptions are (e) δ− < δ+ and (f) 0 ≤ p+ (x) ≤ 1, x ∈ X. The Markov modulating index X n is included in the model. Correspondingly, a pay-off function g(n, e y , x) and the corresponding log-reward functions ϕ n (y, x) do depend on the index argument. Let us assume that the initial state of the modulated log-price process Z0 = (Y 0 , X0 ) = z0 = (y0 , x0 ) with probability 1, i.e. the initial distribution P0 (A) = I (y0 ∈ A, x = x0 ) is concentrated at the point z0 ∈ Z = R1 × X. The tree structure is well visible from the following recurrent algorithm that de­ scribe the sequential transitions of the modulated log-price process Z n = (Y n , X n ). If Z n−1 = (y, x ) for some n = 0, 1, . . . , N − 1, then Z n = (y + δ+ , x ) or Z n+1 = (y + δ− , x ), where x ∈ X. The initial point z0 = (y0 , x0 ) determines the unique tree T((y0 ,x0),0) with the root in the node ((y0 , x0 ), 0) in the space–time domain Z × {0, 1, . . . }. According to the algorithm described above the tree T((y0 ,x0),0) may include 2m nodes after the first step, (2m)2 additional nodes after the second steps, etc. The tree may include (2m)n additional nodes after the nth step. If this is the case, i.e. the tree T((y0 ,x0),0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithm based on such a tree would not be effective from the computational point of view. But it is not the case, since the recombining condition holds. This condition means that the modulated random walk Z n = (Y n , X n ), which is in a state (y, x ) at a moment n − 1, and has the first transition in a state (y + δ+ , x ) and then the second transition

3.1 Binomial tree reward algorithms

| 87

to a state (y + δ+ + δ− , x ) ends up at the same point as if it has first a transition in a state (y + δ− , x ) and then a transition in a state (y + δ− + δ+ , x ). This holds for any y ∈ R1 and x , x , x ∈ X. The tree T((y0 ,x0),0) includes one initial node ((y0 , x0 ), 0) and 2m additional nodes ((y0 + δ+ , x1 ), 1), ((y0 + δ− , x1 ), 1), x1 ∈ X after the first step. Due to the recombin­ ing property, the tree includes 3m additional nodes ((y0 + 2δ+ , x2 ), 2), ((y0 + δ+ + δ− , x2 ), 2), ((y0 + 2δ− , x2 ), 2), x2 ∈ X after the second step, etc. The states that can appear in the tree after the nth step depend on the number l of jumps δ+ in n steps for the random walk Y n but does not depend of moments when these jumps occurred. They are given by the following formulas: y n,l = y0 + δ+ l + δ− (n − l) ,

l = 0, 1, . . . , n ,

n = 0, 1, . . . .

(3.32)

Note also that these numbers satisfy the inequalities y n,0 < y n,1 < · · · < y n,n ,

n = 0, 1, . . . .

(3.33)

The nodes appearing in the tree T((y0 ,x0),0) after the n steps are given by the fol­ lowing formula: z n,l = (y n,l , x n ) ,

l = 0, 1, . . . , n ,

n = 0, 1, . . . ,

(3.34)

where y n,l = y0 + δ+ l + δ− (n − l) ∈ R1 , x n ∈ X. It is useful to note that all points z n,l are located on the grid of points G m [y0 , δ+ , δ− ] = {(y0 + δ+ l+ + δ− l− , x), l± = 0, 1, . . . , x ∈ X}. Moreover, in the standard symmetric case, where ±δ± = δ > 0, this grid reduces to the simple grid of points, G m [y0 , δ] = {(y0 + δl, x), l = 0, ±1, . . . , x ∈ X}. The tree T((y0 ,x0),0) includes (n + 1)m additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 + 2m + · · · + (n + 1)m = 1 − m + (n+2)(2n+1)m . This means that the tree possesses a quadratic rate of growth for the number of nodes in the tree after n steps as a function of n. This makes algorithms based on such a tree effective from the computational point of view. According to formulas (3.34), the tree T((y0 ,x0),0) is the following set of nodes: T((y0 ,x0),0) = ((y n,l , x n ), n), l = 0, 1, . . . , n, x n ∈ X, n = 0, 1, . . .  .

(3.35)

The tree T((y0 ,x0),0) includes a particular node ((y, x), n) if y n,l = y0 + δ+ l + δ− (n − l) = y, x n = x for some l = 0, 1, . . . , n. The condition A2 holds since, in this case, the following inequality takes place for any z = (y, x) ∈ Z, n = 0, 1, . . . , N:     Ez,n max g(r, e Y r , X r ) n≤r≤N   g(r, e y r,l , x r ) < ∞ , ≤ max max (3.36) n ≤ r ≤ N m ≤ l ≤ m + r − n, x r∈X

88 | 3 Backward recurrence reward algorithms where the points y r,l are given by formulas (3.32), with y0 and 0 ≤ m ≤ n chosen such that y n,m = y. Theorem 2.3.3 takes in this case the following form. Lemma 3.1.4. Let the log-price process Y n be represented by the binomialMmodulated random walk defined by the stochastic transition dynamic relation (3.31). Then the logreward functions ϕ n (y n,l), for points z n,l = (y n,l , x n ), l = 0, 1, . . . , n, n = 0, 1, . . . , N given by formulas (3.34), are the unique solution for the following finite recurrence system of linear equations: ⎧ y N,l ⎪ ⎪ ⎪ ϕ N (y N,l , x N ) = g(N, e , x N ) , ⎪ ⎪ ⎪ ⎪ ⎪ l = 0, 1, . . . , N , x N ∈ X , ⎪ ⎪ ⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ n (y n,l , x n ) = max ⎝g(n, e y n,l , x n ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

 ⎨ ϕ n+1 (y n+1,l , x n+1)p− (x n )P(x n , x n+1 ) (3.37) ⎪ ⎪ x n+1 ∈X ⎪ ⎪ ⎞ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ ⎪ + ϕ n+1 (y n+1,l+1, x n+1 )p+ (x n )P(x n , x n+1 ) ⎠ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = 0, 1, . . . , n, x n ∈ X , ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 . This system lets one find the values of the log-reward function ϕ n (y, x) at points (y n,l , x n ), l = 0, 1, . . . , n given by formulas (3.34), recursively, for n = N, . . . , 0. The particular value ϕ n (y, x) can be found using this system if the initial point y0 is chosen so that y n,l = y0 + δ+ l + δ− (n − l) = y and x n = x, for some l = 0, 1, . . . , n.

3.2 Trinomial tree reward algorithms In this section, we describe backward tree algorithms for finding log-reward functions and optimal expected rewards for the trinomial models of log-price processes. We con­ sider classical homogeneous models, model inhomogeneous in time and space as well as models with Markov modulation. Trinomial models models can be effectively used to approximate more general log-price processes such as Gaussian random walks in such cases where binomial models do not work well, for example, for multivariate models inhomogeneous in time and space.

3.2 Trinomial tree reward algorithms

|

89

3.2.1 A classical homogeneous trinomial tree model This is the simplest model where the log-price Y n is a univariate random walk given by the following stochastic transition dynamic relation: Y n = Y n −1 + W n ,

n = 1, 2, . . . .

(3.38)

where (a) Y0 is a real-valued random variable with the distribution P0 (A), (b) W n , n = 1, 2, . . . , is a sequence of i.i.d. random variables which can take three values δ+ , δ◦ and δ− with probabilities, respectively, p+ , p◦ , and p− and (c) the random variable Y0 and the random sequence W n , n = 1, 2, . . . , are independent. Also, usual assumptions are (d) δ− < δ◦ < δ+ , and (e) p+ , p◦ , p− ≥ 0, p+ + p◦ + p− = 1. It is obvious that the trinomial model is a generalization of the binomial model. The trinomial model reduces to the binomial model if probability p◦ = 0. The main advantage of the trinomial model is in the presence of additional pa­ rameters that may make fitting of trinomial models to parameters of inhomogeneous in time and space models of Gaussian random walks more effective than for the bino­ mial model. The index X n is not included in the model. Correspondingly, a pay-off function g(n, e y ) and the corresponding log-reward functions ϕ n (y) do not depend on the in­ dex argument. Let us assume that the initial state of the log-price process Y0 = y0 with proba­ bility 1, i.e. the initial distribution P0 (A) = I (y0 ∈ A) is concentrated at the point y0 ∈ R1 . The tree structure of trajectories is well visible from the following recurrent algo­ rithm that describes sequential transitions of the log-price process Y n . If Y n = y, for some n = 0, 1, . . . , then, either Y n+1 = y + δ+ with probability p+ or Y n+1 = y + δ◦ with probability p◦ , or Y n+1 = y + δ− with probability p− . The initial point y0 determines the unique tree T(y0 ,0) with the root in the node (y0 , 0) in the space–time domain R1 × {0, 1, . . . }. According to the algorithm described above the tree T(y0 ,0) includes one initial node, may include three additional nodes after the first step, nine additional nodes after the second step, etc. It seems that the tree may include 3n additional nodes after the nth step. If this is the case, the tree T(y0 ,0) possesses an exponential rate of growth for the number of nodes in the tree as a function of the number of steps, any algorithm based on such a tree would not be effective from the computational point of view. However, this is not the case if some natural recombining condition holds. Note that the pairwise recombining condition analogous to those formulated for binomial random walk obviously holds for any pair of possible jumps δ+ and δ◦ , or δ+ and δ− , or δ◦ and δ− . The random walk Y n , which makes the first jump δ+ and then the second jump δ◦ , or makes the first a jump δ◦ and then the second jump δ+ , will

90 | 3 Backward recurrence reward algorithms end at the same point. The similar recombining condition holds for another two pairs of jumps pointed above. These pairwise recombining conditions hold for any possible position of the random walk. But, in the trinomial model, an additional recombining condition may also hold. This condition means that the random walk Y n which makes the first jump δ+ and then the second jump δ− , or makes the first jump δ− and then the second jump δ+ , or makes both jumps δ◦ , should end up at the same point. This should hold for any possible position of the random walk. This leads to the following recombining condition: δ◦ =

δ− + δ+ . 2

(3.39)

Indeed, if Y n−1 = y for some n = 1, 2, . . . , then Y n+1 = y + δ+ + δ− if W n = δ+ , W n+1 = δ− or Y n+1 = y + δ− + δ+ if W n = δ− , W n+1 = δ+ or Y n+1 = y + 2δ◦ = y + δ− + δ+ if W n = δ◦ , W n+1 = δ◦ . In all cases, Y n+1 takes the same value. The tree T(y0 ,0) obviously includes one initial node (y0 , 0) and three additional nodes (y0 + δ+ , 1), (y0 + δ◦ , 1), and (y0 + δ− , 1) after the first step. Due to the recom­ bining property, the tree includes five additional nodes (y0 + 2δ+ , 2), (y0 + δ+ + δ◦ , 2), (y0 + 2δ◦ , 2), (y0 + δ◦ + δ− , 2), and (y0 + 2δ− , 2) after the second step, etc. If the recombining condition (3.39) holds, then the tree includes 2n + 1 additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 + 3 + · · · + (2n + 1) = (n + 1)2 .

(3.40)

This means that the tree possesses a quadratic rate of growth for the number of nodes in the tree after n steps as a function of n. This makes algorithms based on such a tree effective from the computational point of view. The nodes appearing in the tree after n steps are determined by the numbers l+ and l− of jumps δ+ and δ− in n steps, which also determine the number l◦ = n − l+ − l− of jumps δ◦ in n steps. What is important is that these nodes do not depend of moments when these jumps occurred. The states that can be reached by the trinomial random walk at moments n = 0, 1, . . . , N are given by the following formula: y n,l+ ,l− = y0 + δ+ l+ + δ− l− + δ◦ (n − l+ − l− ) ,

(3.41)

where l+ , l− ≥ 0, l+ + l− ≤ n, n = 0, 1, . . . , N. A disadvantage of the above formula is that it does not count one-to-one possi­ ble states of the random walk. Different pairs of parameters l+ and l− can, in fact, determine the same state. Let us denote δ = δ+ − δ◦ = δ◦ − δ− =

δ+ − δ− . 2

(3.42)

Now, we can rewrite formulas for y n,l+ ,l− in the following form, y n,l+ ,l− = y0 + (δ+ − δ◦ )l+ + (δ− − δ◦ )l− + δ◦ n = y0 + δ(l+ − l− ) + δ◦ n. This form shows that states that can be reached by the trinomial random walk at moments n = 0, 1, . . . , N

3.2 Trinomial tree reward algorithms

| 91

can be indexed by the one parameter l = l+ − l− , which is the difference between the numbers of jump δ+ and δ− in n steps. This parameter takes values −n, . . . , 0, . . . , n and counts one-to-one states that can be reached by the trinomial random walk at moments n = 0, 1, . . . , N. According to the above remarks, the states that can be reached by the trinomial random walk at moments n = 0, 1, . . . , are given by the following formula: y n,l = y0 + δl + δ◦ n ,

l = 0, ±1, . . . , ±n ,

n = 0, 1, . . . .

(3.43)

It is useful to note that, under the recombining condition (3.39), the random jumps W n can be represented in the form W n = W n + δ◦ , n = 1, 2, . . . , where W n , n = 1, 2, . . . , are i.i.d. random variables taking values δ, −δ and 0 with probabilities, respectively, p+ , p− and p◦ = 1 − p+ − p− . Correspondingly, the basic relation (3.38) defining the trinomial random walk Y n can be rewritten in the following form that is consistent with formulas (3.43): Y n = Y n−1 + W n + δ◦ = y0 + W1 + · · · + W n + δ◦ n ,

n = 0, 1, . . . .

(3.44)

It should be mentioned that the procedure for the construction of the tree T(y0 ,0) described above does not take into account values of probabilities for appearance of nodes in the tree generated by trajectories of the random walk Y n . The probability that a node y n,l will be reached after n steps is given by the follow­ ing formula, for l = 0, ±1, . . . , ±n, n = 0, 1, . . . :

p n,l =

l + ,l −≥0,l − + l + ≤ n,l +− l − = l

n! l l p++ p−− p◦n−l+−l− , l+ !, l− !(n − l+ − l− )!

(3.45)

where 00 should be counted as 1. Note also that the numbers y n,l satisfy the inequalities, y n,−n < · · · < y n,0 < · · · < y n,n ,

n = 0, 1, . . . .

(3.46)

It is useful to note that all these points are located on the grid of points G[y0 , δ, δ◦ ] = {y0 + δl + δ◦ n, l = 0, ±1, . . . , n = 0, 1, . . . }. Moreover, in the stan­ dard symmetric case, where δ◦ = 0, this grid reduces to the simple grid of points, G[y0 , δ] = {y0 + δl, l = 0, ±1, . . . }. According to formula (3.43), the tree T(y0 ,0) is the following set of nodes: T(y0 ,0) = (y n,l , n), l = 0, ±1, . . . , ±n, n = 0, 1, . . .  .

(3.47)

The tree T(y0 ,0) includes a particular point (y, n), if y n,l = y0 + δl + δ◦ n = y for some l = 0, ±1, . . . , ±n. The condition A2 holds since, in this case, the following inequality takes place, for y ∈ R1 , n = 0, 1, . . . , N:       g(r, e y r,l ) < ∞ , max (3.48) Ey,n max g(r, e Y r ) ≤ max n≤r≤N

n ≤ r ≤ N m − r + n ≤ l ≤ m +r −n

92 | 3 Backward recurrence reward algorithms where the points y r,l are given by formulas (3.43), with y0 and −n ≤ m ≤ n chosen such that y n,m = y. Theorem 2.3.3 takes in this case the following form. Lemma 3.2.1. Let the log-price process Y n be represented by the trinomial random walk defined by the stochastic transition dynamic relation (3.38). Then the log-reward func­ tions ϕ n (y n,l ), for points y n,l = y0 + δl + δ◦ n, l = 0, ±1, . . . , ±n, n = 0, 1, . . . N given by formulas (3.43), are the unique solution for the following finite recurrence system of linear equations: ⎧ ⎪ ϕ N (y N,l ) = g(N, e y N,l ) , l = 0, ±1, . . . , ±N , ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ ϕ n (y n,l ) = max g(n, e y n,l ), ϕ n+1 (y n+1,l−1)p− + ϕ n+1 (y n+1,l)p◦ (3.49) ⎪ ⎪ +ϕ n+1 (y n+1,l+1)p+ , l = 0, ±1, . . . , ±n , ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 , This system gives the values of the log-reward function ϕ n (y) at points y n,l , l = 0, ±1, . . . , ±n, n = 0, 1, . . . , N given by formulas (3.43). The particular value ϕ n (y) can be calculated using this system if the initial value y0 is chosen so that y n,l = y0 + δ+ l + δ− (n − l) = y for some l = 0, ±1, . . . , ±n. Remarks concerning simultaneous solution of the system (3.49) for the set of ini­ tial states y0 + δl, r− ≤ l ≤ r+ , subtrees with an initial node y n,l and the corresponding subsystems of equations for log-reward functions generated by such subtrees as well as formulas for finding optimal expected rewards for a given initial distribution, anal­ ogous to those made in Subsection 3.1.1 for the binomial tree model, can be translated in an obvious way to the case of the trinomial model.

3.2.2 A multivariate modulated trinomial tree model inhomogeneous in time and space In this subsection, we consider the model of a discrete time k-dimensional log-price process represented by a inhomogeneous trinomial random walk modulated by a Markov chain with a finite phase space X = {1, . . . , m}. We would like to define the model in a dynamic form. In order to do this, we first describe, in which way one can construct, for a  = (W1 , . . . , W k ), another k-dimensional k-dimensional trinary random vector W    trinomial random vector W = (W1 , . . . , W k ), which has the same distribution and  = (U1 , . . . , U k ) uniformly is a nonrandom transformation of a random vector U distributed in the k-dimensional cube Uk = [0, 1] × · · · × [0, 1].  = (W1 , . . . , W k ) takes values δ¯¯ȷ k = (δ1,ȷ1 , . . . , δ k,ȷ k ) with Let a random vector W probability p¯ȷ k for ¯ȷ k = (ȷ1 , . . . , ȷ k ) ∈ Jk , where Jk = {¯ȷ k : ȷ i = +, ◦, −, i = 1, . . . , k }. The following conditions are also assumed: (a) δ i,− < δ i,◦ < δ i,+, i = 1, . . . , k,  (b) p¯ȷ k ≥ 0, ¯ȷ k ∈ Jk and ¯ȷ k ∈Jk p¯ȷ k = 1.

3.2 Trinomial tree reward algorithms

| 93

 is determined by the pair of parameter The distribution of the random vector W families δ¯ =  δ¯¯ȷ k , ¯ȷ k ∈ Jk  and p¯ =  p¯ȷ k , ¯ȷ k ∈ Jk . Let us now define the joint distributions for the first 1 ≤ r ≤ k components of the  random vector W

p¯ȷ r = p ȷ1 ,...,ȷ r = p¯ȷ k , ¯ȷ r = (ȷ1 , . . . , ȷ r ) ∈ Jr . (3.50) ¯ȷ k :ȷ r+1 ,...,ȷ N =+ , ◦ , −

Let us also define intervals I¯ȷ r+1 [p¯ ] for ¯ȷ r+1 = for r = 0, ⎧ ⎪ ⎪[0, p+ ) ⎪ ⎨ I¯ȷ1 [p¯ ] = ⎪[p+ , 1 − p− ) ⎪ ⎪ ⎩[1 − p , 1]

(ȷ1 , . . . , ȷ r+1 ), ȷ1 , . . . , ȷ r+1 = +, ◦ , −,

−

if ȷ1 = + , if ȷ1 = ◦ ,

(3.51)

if ȷ1 = − ,

and, for r = 1, . . . , k − 1, ⎧ ⎪ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎨

p ȷ1 ,...,ȷ r ,+ p ȷ1 ,...,ȷ r



p

p

,ȷ r ,+ r ,− I¯ȷ r+1 [p¯ ] = ⎪ pȷ1ȷ,...,...,ȷ , 1 − pȷ1ȷ,...,ȷ r 1 1 ,...,ȷ r ⎪ ⎪ ⎪ ! ⎪ ⎪ ⎩ 1 − p ȷ1 ,...,ȷ r ,− , 1 p ȷ ,...,ȷ 1



r

if ȷ r+1 = + , if ȷ r+1 = ◦ ,

(3.52)

if ȷ r+1 = − ,

p

r ,± where one should count quotients pȷ1ȷ,...,ȷ , for example, as 31 , if p ȷ1 ,...,ȷ r = 0. 1 ,... ,ȷ r Finally, let us define disjoint cubes, whose union is Uk

¯I¯ȷ k [p¯ ] = I¯ȷ1 [p¯ ] × · · · × I¯ȷ k [p¯ ] ,

¯ȷ k ∈ Jk .

(3.53)

By the definition, the length of the interval I¯ȷ1 [p¯ ] is p ȷ1 and the length I¯ȷ r [p¯ ] is for r = 2, . . . , k. This holds for any ¯ȷ k ∈ Jk . Thus,

p ȷ1 ,...,ȷ r p ȷ1 ,...,ȷ r−1

 ∈ ¯I¯ȷ k [p ¯ ]} = p ȷ1 P{ U

k  p ȷ1 ,...,ȷ r = p ȷ1 ,...,ȷ k , p r =2 ȷ 1 ,...,ȷ r−1

¯ȷ k ∈ Jk .

(3.54)

 has the same distribution It follows from relation (3.54) that a random vector W with the following random vector:

 ∈ ¯I¯ȷ k [p  = ¯ ]) . δ¯¯ȷ k I (U W (3.55) ¯ȷ k ∈Jk

Now we are prepared to define the model of the discrete time k-dimensional log-price process, represented by a trinomial inhomogeneous in space and time ran­ dom walk modulated by an inhomogeneous Markov chain, in a dynamical form. We restrict consideration by the model where values of jumps do not depend on the index argument, while the corresponding jump probabilities do depend on the index argument.

94 | 3 Backward recurrence reward algorithms In this model, the log-price process is given by the following stochastic transition dynamic relation: ⎧    ⎪ ⎪ ⎪ Y n = Y n −1 + W n , ⎪

⎪ ⎨  n −1 ) I ( U  n ∈ ¯I¯ȷ k [p n−1 , X n−1 )]) , ¯ n (Y Wn = δ¯ n,¯ȷ k (Y (3.56) ⎪ ¯ȷ k ∈Jk ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking value in Rk with a distri­ where (a) Y  n = (U n,1 , . . . , U n,k ), n = 1, 2, . . . , is a sequence of i.i.d. random bution P0 (A), (b) U vectors uniformly distributed in the k-dimensional unit cube Uk , (c) X n , n = 0, 1, . . . , is an inhomogeneous in time Markov chain with the phase space X, initial distribution P(x) = P{X0 = x}, x ∈ X and one-step transition probabilities P n (x , x ) = P{X n = 0 , the random se­ x /X n−1 = x }, x , x ∈ X, n = 1, 2, . . ., (d) the random vector Y  n , n = 1, 2, . . . , and the Markov chain X n , n = 0, 1, . . . , are independent, quence U (e) δ¯ n,¯ȷ k ( y) = (δ n,1,ȷ1 ( y), . . . , δ n,k,ȷ k ( y)), ¯ȷ k = (ȷ1 , . . . , ȷ k ) ∈ Jk , n = 1, 2, . . . , where δ n,i,ȷ i ( y), ¯ȷ k ∈ Jk , i = 1, . . . , k, n = 1, 2, . . . , are measurable functions acting from Rk to R1 , and (f) p¯ n ( y , x) =  p n,¯ȷ k ( y , x), ¯ȷ ∈ Jk , where p n,¯ı k ( y , x), ¯ȷ k ∈ Jk , n = 1, 2, . . . , N are measurable functions acting from Z = Rk × X to [0, 1]. Also, usual assumptions are (g) δ n,i, −( y) < δ n,i, ◦( y ) < δ n,i, +( y ),  y ∈ Rk , i = 1, . . . , k, n = 1, . . . , and (h) p n,¯ȷ k ( y , x) ≥ 0, ( y , x) ∈ Z, ¯ȷ k ∈ Jk , n = 1, . . . , and  y , x) = 1, ( y , x) ∈ Z, n = 1, . . . . ¯ȷ k ∈Jk p n,¯ȷ k ( The Markov modulating index X n is included in the model. Correspondingly, the y , x) do pay-off function g(n, ey , x) and the corresponding log-reward functions ϕ n ( depend on the index argument. 0 , X 0 ) Let us assume that the initial state of the modulated log-price process (Y = ( y0 , x0 ) with probability 1, i.e. the initial distribution P0 (A)P(x) = I ( y0 ∈ A)I (x = x0 ) is concentrated at a point ( y0 , x0 ) ∈ Z. The tree structure of trajectories for the modulated log-price process  Z n is analo­ gous to those described above for the simplest trinomial tree model. It is well visible from the recurrent algorithm that describes the sequential transitions of the modu­ n , X n ). If  lated log-price process  Z n = (Y Z n−1 = ( y , x) for some n = 1, 2, . . . , then   ¯  Z n = ( y + δ n,¯ȷ k ( y), x ) for some ¯ȷ k ∈ Jk and x ∈ X. The initial point z0 = ( y0 , x0 ) determines the unique tree T((y0 ,x0),0) with the root in the node (( y0 , x0 ), 0) in the space–time domain Z × {0, 1, . . . }. According to the algorithm described above, the tree T((y0 ,x0),0) includes one ini­ tial node and it may include 3k m additional nodes after the first step, 32k m additional nodes after the second step, etc. So, it seems, that the tree may include 3nk m additional nodes after the nth step. If this is the case, i.e., the tree T((y0 ,x0),0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithm based on such a tree would not be effective from the computational point of view.

3.2 Trinomial tree reward algorithms

| 95

In order it would not be the case, the recombining condition should hold for the  n = (Y n,1 , . . . , Y n,k). This condition means the following. random walk Y  n −1 =  Let us assume that Y y = (y1 , · · · , y k ) for some n = 1, 2, . . . ,. Let us also y) assume that the random walk makes from this position two sequential jumps δ¯ n,¯ȷk (   and then δ¯ n+1,¯ȷk ( y + δ¯ n,¯ık ( y )), where ¯ȷk = (ȷ1 , . . . , ȷk ), ¯ȷ = ( ȷ , . . . , ȷ ) ∈ J . Then, k 1 k k for every j = 1, . . . , k, the resulting position for the ith component of the random walk y) + δ n+1,i,ȷi ( y + δ¯ n,¯ȷk ( y)), should coincide for the after these two jumps, y i + δ n,i,ȷi (      following three cases: (a) ȷ i = +, ȷ i = ◦ or ȷ i = ◦, ȷ i = +, (b) ȷ i = −, ȷ i = ◦ or         ȷ i = ◦, ȷ i = −; and (c) ȷ i = +, ȷ i = − or ȷ i = −, ȷ i = + or ȷ i = ◦, ȷ i = ◦. This y ∈ R1 , and should hold for any values of other components of vectors ¯ȷk , ¯ȷ k , position  n = 1, 2, . . . . In the case of the model with values of jumps homogeneous in time and space, y) = δ i,ȷ , ȷ = +, ◦, − where there exists δ i,− < δ i,◦ < δ i,+, i = 1, . . . , k such that δ n,i,ȷ( for any i = 1, . . . , k,  y ∈ Rk , n = 1, 2 . . . , the recombining condition take the form of the following relation: δ i,+ + δ i,− (3.57) , i = 1, . . . , k . 2 In the case of the model with values of jumps inhomogeneous in time but homo­ geneous in space, where there exists δ n,i, − < δ n,i, ◦ < δ n,i, +, i = 1, . . . , k, n = 1, 2, . . . , y) = δ n,i,ȷ, ȷ = +, ◦, − for i = 1, . . . , k,  y ∈ Rk , n = 1, 2 . . . , the such that δ n,i,ȷ( recombining condition holds if the following equalities take place for i = 1, . . . , k, n = 1, 2, . . . : ⎧ ⎪ ⎪ ⎪ δ n,i, + + δ n+1,i, ◦ = δ n,i, ◦ + δ n+1,i, + , ⎨ δ n,i, − + δ n+1,i,◦ = δ n,i, ◦ + δ n+1,i,− , (3.58) ⎪ ⎪ ⎪ ⎩δ +δ =δ +δ =δ +δ . δ i,◦ =

n,i, +

n +1,i, −

n,i, −

n +1,i, +

n,i, ◦

n +1,i, ◦

It follows from the third equality in (3.58) that there exists δ i > 0, j = 1, . . . , k such that δ n,i, + − δ n,i, − = 2δ i , n = 1, 2, . . . , i = 1, . . . , k. Thus, δ n,i, + = δ i + δ n,i , δ n,i, − = δ +δ −δ i + δ n,i , n = 1, 2, . . . , i = 1, . . . , k, where δ n,i = n,i,+ 2 n,i,− , n = 1, 2, . . . , i = 1, . . . , k. By substituting the above δ n,i, ± in the expression on the right-hand side of the third equality in (3.58), we get the equalities δ n,i + δ n+1,i = δ n,i, ◦ + δ n+1,i,◦, n = 1, 2, . . . , i = 1, . . . , k. Also, by adding and devising by 2 the first two equalities in (3.58), we get equalities δ n,i − δ n+1,i = δ n,i, ◦ − δ n+1,i,◦, n = 1, 2, . . . , i = 1, . . . , k. The above two series of equalities imply in an obvious way that δ n,i = δ n,i, ◦, n = 1, 2, . . . , i = 1, . . . , k. Summarizing the above calculations, we can reformulate the recombining condi­ tion (3.58) in the equivalent form as the condition that require the existence of δ i > 0, i = 1, . . . , k such that the following relation hold for n = 1, 2, . . . , i = 1, . . . , k: δ n,i, + = δ i + δ n,i, ◦ ,

δ n,i, − = −δ i + δ n,i, ◦ .

(3.59)

As an example of the model with values of jumps inhomogeneous in time and space, let us consider the model with values of jumps δ n,i,ȷ( y) = δ n,i,ȷ y i , ȷ = +, ◦, −

96 | 3 Backward recurrence reward algorithms are, for every i = 1, . . . , k, linear functions of y i for n = 1, 2, . . . , such that δ n,i, − < δ n,i, ◦ < δ n,i, +, i = 1, . . . , k, n = 1, 2, . . . ,. In this case, the recombining condition takes the form of the following relations that should hold for any n = 0, 1, . . . , i = 1, . . . , k: ⎧ ⎪ δ n,i, + + δ n+1,i,◦ + δ n,i, +δ n+1,i,◦ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = δ n,i, ◦ + δ n+1,i, + + δ n,i, ◦δ n+1,i, + , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δ n,i, − + δ n+1,i, ◦ + δ n,i, −δ n+1,i, ◦ ⎪ ⎪ ⎨ = δ n,i, ◦ + δ n+1,i, − + δ n,i, ◦δ n+1,i, − , ⎪ ⎪ ⎪ ⎪ ⎪ δ n,i, + + δ n+1,i,− + δ n,i, +δ n+1,i,− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = δ n,i, − + δ n+1,i, + + δ n,i, −δ n+1,i, + ⎪ ⎪ ⎩ = δ n,i, ◦ + δ n+1,i, ◦ + δ n,i, ◦δ n+1,i, ◦.

(3.60)

For example, if the coefficients δ n,i,ȷ = δ i,ȷ , ȷ = +, ◦, −, i = 1, . . . , k do not depend on n, then the first three equalities in (3.60) automatically hold, and, thus, relations (3.60) reduce to the following relations that should hold for i = 1, . . . , k: δ i,+ + δ i,− + δ i,+ δ i,− = 2δ i,◦ + δ2i,◦ .

(3.61)

Let us return back to the general case where the recombining condition holds. The tree T((y0 ,x0),0) includes one initial node (( y0 , x0 ), 0). It includes 3k m additional  ¯ y0 + δ1,¯ȷk ( y0 ), x1 ), 1) , ¯ȷ k ∈ Jk , x1 ∈ X after the first step. If the recombining nodes ( y0 + δ¯ 1,¯ȷk ( y0 ) + δ¯ 2,¯ȷk (y0 + condition holds, the tree contains 5k m additional nodes (   ¯δ1,¯ȷ ( y0 )), x2 ), 2), ¯ȷ k , ¯ȷ k ∈ Jk , x2 ∈ X after the second step. Indeed, the ith component k y0 ) + δ¯ 2,¯ȷk (y0 + δ¯ 1,¯ȷk ( y0 )) can take five different values, δ1,i,+( y0 ) + of the vector δ¯ 1,¯ȷk ( δ2,i,+( y0 + δ¯ 1,¯ȷk ( y0 )) if ith components of both vectors ¯ȷk and ¯ȷ equals + ; δ ( 1,i, + y 0 ) + k ¯ ¯ δ2,i,◦( y0 + δ1,¯ȷk ( y0 )) = δ1,i,◦( y0 ) + δ2,i,+( y0 + δ1,¯ȷk ( y0 )) if the ith components of the equal, respectively + and ◦ , or vice versa; δ1,i,+( y0 ) + δ2,i,−( y0 + vectors ¯ȷk and ¯ȷ k ¯δ1,¯ȷ ( ¯ ¯   y0 )) = δ1,i,−( y0 ) + δ2,i,+( y0 + δ1,¯ȷ k ( y0 )) = δ1,i,◦( y0 ) + δ2,i,◦( y0 + δ1,¯ȷ k ( y0 )) if the k equal, respectively, + and − , or − and +, or ith components of the vectors ¯ȷk and ¯ȷ k y0 ) + δ2,i,◦( y0 + δ¯ 1,¯ȷk ( y0 )) = δ1,i,◦( y0 ) + δ2,i,−( y0 + δ¯ 1,¯ȷk ( y0 )) if both equal ◦; δ1,i,−(   the ith components of the vectors ¯ȷ k and ¯ȷ k equal, respectively − and ◦, or vice versa; y0 ) + δ2,i,−( y0 + δ¯ 1,¯ȷk ( y0 )) if the ith components of both vectors ¯ȷk and, finally, δ1,i,−(  and ¯ȷ k equals −. Subsequently, the tree includes 7k m additional nodes after the third step, etc. The tree T(y0 ,x0),0) includes (2n + 1)k m additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 + 3k m + · · · + (2n + 1)k m.

3.2 Trinomial tree reward algorithms

| 97

Obviously, (2n + 1)k =1−m+m 1−m+m 2(k + 1)

n +1

(2x − 1)k dx ≤ L n 1 2

n +2

(2x − 1)k dx = 1 − m + m

≤1−m+m 1

(2n + 3)k − 1 . (3.62) 2(k + 1)

Therefore, the tree possesses the polynomial of order k + 1 rate of growth for the number of nodes in the tree after n steps as a function of n. This makes algorithms based on such a tree effective from the computational point of view. Due to the recombining condition, states that can be reached by the ith com­ ponent of the random walk after the nth step are determined by the number l n,i , which is the difference between the numbers of jumps δ· ,i,+(·) and δ· ,i,−(·) made by the ith component of the random walk in n steps. By the definition, the pa­ rameters l n,i , i = 1, . . . , k can take values −n, . . . , 0, . . . , n. Correspondingly, states which can be reached by the random walk after n steps are determined by the vector ¯l n = (l n,1 , . . . , l n,k ), which take values in the set Ln = {¯l n : |l n,i | ≤ n, i = 1, . . . , k }. The states  y n,¯l n = (y n,1,l n,1 , . . . , y n,k,l n,k ), ¯l n ∈ Ln , which can be reached by the n = (Y 1 , . . . , Y n ) with the initial value Y 0 =  log-price process Y y0 = (y0,1 , . . . , y0,k ), at moments n = 0, 1, . . . , are given by the following recurrence relation, for −n ≤ l n,i ≤ n, i = 1, . . . , k, n = 0, 1, . . . : ⎧ ⎪ y n,¯l n ) if l n+1,i = l n,i + 1 , ⎪ ⎪ δ n+1,i, +( ⎨ (3.63) y n+1,i,l n+1,i = y n,i,l n,i + ⎪ δ n+1,i,◦( y n,¯l n ) if l n+1,i = l n,i , ⎪ ⎪ ⎩δ ( y ¯ ) if l = l −1, n +1,i, −

n, l n

n +1,i

n,i

where y0,i,0 = y0,i , i = 1, . . . , k. Note that calculations in (3.63) should be made for l i = −n, . . . , n, i = 1, . . . , k sequentially for n = 1, 2, . . . . This procedure let one find vectors  y n,¯l n , ¯l n ∈ Ln sequen­ ¯ tially for n = 1, 2, . . . , for any given initial vector  y0,¯l0 =  y0 , l0 = (0, . . . , 0). Note that numbers y n,i,l n,i satisfy the inequalities y n,i, −n < · · · < y n,i,n ,

i = 1, . . . , k ,

n = 0, 1, . . . .

(3.64)

According to the above remarks, the tree T((y0 ,x0),0) is the following set of nodes: T((y0 ,x0),0) = (( y n,¯l n , x n ), n), ¯l n ∈ Ln , x n ∈ X, n = 0, 1, . . .  .

(3.65)

y , x), n), if ( y n,¯l n , x n ) = ( y , x) for The tree T((y0 ,x0),0) includes a particular point (( some ¯l n ∈ Ln .

98 | 3 Backward recurrence reward algorithms The condition A2 holds since, in this case, the following inequality take place for any  z = ( y , x) ∈ Z = Rk × X, n = 0, 1, . . . , N:      Ez ,n max g(r, e Y r , X r ) n≤r≤N     g(r, ey r,¯l r , x r ) < ∞ , ≤ max max (3.66) n ≤ r ≤ N m i − r + n ≤ l i≤ m i + r − n,i =1,...,k,x r∈X

¯ = ( m 1 , . . . , m k ), where the points  y r,¯l are given by formulas (3.63), with  y0 and m −n ≤ m i ≤ n, i = 1, . . . , k chosen such that  y n, m¯ =  y. Theorem 2.2.3 takes in this case the following form. Lemma 3.2.2. Let the log-price process Y n be represented by the trinomial random walk defined by the stochastic transition dynamic relation (3.56). Then, the log-reward func­ y n,¯l n , x n ), for points ( y n,¯l n , x n ), ¯l n ∈ Ln , x n ∈ X, n = 0, 1, . . . , N given by tions ϕ n ( formulas (3.63), are the unique solution for the following finite recurrence system of lin­ ear equations: ⎧ ⎪ y N,¯l N , x N ) = g(N, ey N,¯l N , x N ) , ¯l N ∈ LN , x N ∈ X , ⎪ ⎪ ϕ N ( ⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ n ( y n,¯l n , x n ) = max ⎝g(n, e y n,¯l n , x n ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎨ ϕ n+1 ( y n+1,¯l n+e¯¯ȷ k , x n+1 ) ⎪ ¯ȷ k ∈Jk ,x n+1 ∈X ⎪ ⎪ ⎪ ⎞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y n,¯l n , x n )P n+1(x n , x n+1 )⎠ , ¯l n ∈ Ln , x n ∈ X , × p n+1,¯ȷ k ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 , (3.67) where the points  y n,¯l n are given by formulas (3.63) and vector e¯¯ȷ k = (e ȷ1 , . . . , e ȷ k ), where e ȷ i = 1 if ȷ i = +, or e ȷ i = 0 if ȷ i = ◦, or e ȷ i = −1 if ȷ i = −, for i = 1, . . . , k. y , x) at points This system lets one compute the values of the log-reward function ϕ n ( ¯ ( y n,¯l , x n ), l ∈ Ln , x n ∈ X given by formulas (3.63), recursively, for n = N, . . . , 0. The particular value ϕ n ( y , x) can be calculated using this system if the initial y0 is chosen so that y n,¯l n =  y, for some ¯l n ∈ Ln , and x n = x. value  The extension and reduction of the system of backward recurrence relations (3.67) can be realized in the same way as as described in Subsection 3.1.1, for the system of backward recurrence relations (3.8). Also, the remarks concerning finding of optimal expected reward analogous to those made in Subsection 3.1.1 can be made.

3.2 Trinomial tree reward algorithms

| 99

3.2.3 Binomial and trinomial random walk models In this subsection a recurrence algorithm for computing the transition probabilities for multivariate binomial and trinomial random walks is presented. Let us introduce the sets Lr = {¯l r = (l r,1 , . . . , l r,k ), l r,1 , . . . , l r,k = −r, . . . , r} ,

r = 1, 2, . . . .

(3.68)

 r = (W r,1 , . . . , W r,k ), r = 1, 2, . . . , which Consider the trinomial random vectors W can be represented in the following form: 1 + · · · + U r , r = U W

(3.69)

 n = (U n,1 , . . . , U n,k ), n = 1, 2, . . . , are i.i.d. trinary random vectors taking where (a) U  1 = ¯l1 } = p(1, ¯l1 ) and (b) p(1, ¯l1 ) ≥ 0, ¯l1 ∈ L1 and values ¯l1 ∈ L1 with probabilities P{U  ¯ ¯l 1 ∈L1 p (1, l 1 ) = 1.  r can be referred to as a trinomial random vector. In the par­ The random vector W ticular case, where probabilities p(1, ¯l1 ) = 0 for any ¯l1 = (l1,1 , . . . , l1,k ) such that "k  i =1 l 1,i = 0, the random vector W r can also be referred to as a binomial random vector.  r takes values ¯l r ∈ Lr , for every r = 1, 2, . . . . Let us denote The random vector W the corresponding probabilities by  r = ¯l r } = p(r, ¯l r ) , P{ W

¯l r ∈ Lr , r = 1, 2, . . . . (3.70)  By the definition, p(r, ¯l r ) ≥ 0, ¯l r ∈ Lr and ¯l r ∈Lr p(r, ¯l r ) = 1. The probabilities p(r, ¯l r ), ¯l r ∈ Lr satisfy the following convolution recurrence relations, which can be used for their sequential computing:

p(r + 1, ¯l r+1 ) = p(r, ¯l r+1 − ¯l1 )p(1, ¯l1 ) , r = 1, 2, . . . . (3.71) ¯l 1 ∈L1 ,¯l r+1 −¯l 1 ∈Lr

It can be easily calculated that computing the probabilities p(r, ¯l r ), ¯l r ∈ Lr us­  ing these recurrence relations requires N r = rm−=11 3(2m + 1)k multiplication and 32 N r summation operations. Note that N r =

3 ((2r − 1)k+1 − 1) = 3 2(k + 1)

r

≤ N r ≤ 3 (2x + 1)k dx = 1

r −1

(2x + 1)k dx 0

3 ((2r + 1)k+1 − 3k+1 ) = N r . 2(k + 1)

(3.72)

One can, however, improve the above recurrence algorithm. The probabilities p(r, ¯l r ), ¯l r ∈ Lr also satisfy the following convolution recurrence relations, which can be used for their sequential computing:

p(2r, ¯l2r ) = p(r, ¯l2r − ¯l r )p(r, ¯l r ) , r = 2m , m = 0, 1, . . . . (3.73) ¯l r ∈Lr ,¯l 2r −¯l r ∈Lr

100 | 3 Backward recurrence reward algorithms Any r = 1, 2, . . . , can be uniquely decomposed in the binary sum r = i1 +· · ·+ i n r , where 1 ≤ i1 = 2m r,1 · · · < i n r = 2m r,nr and 0 ≤ m r,1 < · · · < m r,n r are integer numbers. Note that m r,n r = [ln2 r], m r,n r−1 = [ln2 (r − i n r )], . . . , m r,1 = [ln2 (r − i n r − · · · − i2 )]. Let us also denote j1 = i1 , j2 = i1 + i2 , . . . , j n r = i1 + · · · + i n r = r. The probabilities p(r, ¯l r ), ¯l r ∈ Lr can be computed using the following recurrence relations:

p(j m , ¯l j m+1 − ¯l i m ) p(j m+1 , ¯l j m+1 ) = ¯l i m ∈Li m ,¯l j ¯ m +1 − l i m ∈Lj m

× p(i m , ¯l i m ) ,

m = 1, 2, . . . , n r − 1 .

(3.74)

The algorithm described above can be in obvious way translated to the general recombining case, where random variables U n , n = 1, 2, . . . , take values δ + δ◦ , δ◦ , and −δ + δ◦ for some δ > 0, δ◦ ∈ R1 , instead of values 1, 0 and −1.

3.3 Random walk reward algorithms In this section, we describe backward tree algorithms for finding log-reward functions and optimal expected rewards for models, where log-price processes are represented by discrete random walks with bounded jumps. We consider classical homogeneous models, model inhomogeneous in time as well as models with Markov modulation. Random walk models can be effectively used for approximation of more general price processes represented by random walks with arbitrary distributions of jumps.

3.3.1 A classical homogeneous in time random walk model This is the simplest model, where the log-price Y n is a univariate random walk, Y n = Y n −1 + W n ,

n = 1, 2, . . . .

(3.75)

where (a) Y0 is a real-valued random variable with the distribution P0 (A), (b) W n , n = 1, 2, . . . , is a sequence of i.i.d. random variables which take values lδ, l = m− , . . . , m+ with probabilities, respectively, p(l), l = m− , . . . , m+ , and (c) the random variable Y0 and the random sequence W n , n = 1, 2, . . . , are independent. Also, usual assumptions are (d) δ > 0, (e) m− < m+ are integers, and (f) 0 ≤ p(l) ≤ 1, l = m− , . . . , m+ , and p(m− ) + · · · + p(m+ ) = 1. In this case, the stochastic modulating index X n is not included in the model. Cor­ respondingly, a pay-off function g(n, e y ) and the corresponding log-reward functions ϕ n (y) do not depend on the index argument. Let us assume that the initial state of the log-price process Y0 = y0 with probabil­ ity 1, i.e. the initial distribution P0 (A) = I (y0 ∈ A) is concentrated in a point y0 ∈ R1 .

3.3 Random walk reward algorithms

| 101

The tree structure of trajectories is well visible from the recurrent algorithm, which describes sequential transitions for the log-price process Y n . If Y n−1 = y for some n = 1, 2, . . . , then Y n = y + lδ, where l takes one of the values m− , . . . , m+ . The initial point y0 determines the unique tree T(y0 ,0) with the root in the node (y0 , 0), in the space–time domain R1 × {0, 1, . . . }. According to the algorithm described above the tree T(y0 ,0) has one initial node (y0 , 0). It may include m+ − m− + 1 additional nodes after the first step, then (m+ − m− + 1)2 additional nodes after the second step, etc. The tree may include (m+ − m− + 1)n additional nodes after the nth step. If this is the case, i.e. the tree T(y0 ,0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithms based on such a tree would not be effective from the computational point of view. However, this is not the case. In fact, the tree includes only m+ − m− + 1 additional nodes after the first step, and n(m+ − m− ) + 1 additional nodes after the nth step, for every n ≥ 2. This is because the tree T(y0 ,0) satisfies the recombining condition. In this model, the recombining condition means that the random walk Y n , which is in a position Y n−1 = y, for some n = 1, 2, . . . , and makes from this position the first jump l δ and then the second jump l δ should end up at the same point as if it would make the first jump l δ and then the second jump l δ. This should hold for any l and l such that l + l = l, for any l = 2m− , . . . , 2m+ , y ∈ R1 , and n = 1, 2, . . . . Indeed, if Y n−1 = y for some n = 1, 2, . . . , then Y n+1 = y + l δ + l δ = lδ if W n = l δ, W n+1 = l δ or Y n+1 = y + l δ + l δ = lδ if W n = l δ, W n+1 = l δ, for the case where l + l = l. In all such cases, Y n+1 take the same value. The tree T(y0 ,0) obviously includes one initial node (y0 , 0) and m+ − m− + 1 additional nodes (y0 + m− δ, 1), . . . , (y0 + m+ δ, 1) after the first step. Due to the recombining property, the tree includes 2(m+ − m− ) + 1 additional nodes (y0 + 2m− δ, 2), . . . , (y0 + 2m+ δ, 2) after the second step, etc. The tree T(y0 ,0) includes n(m+ − m− ) + 1 additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 + (m+ − m− ) + 1 + · · · + n(m+ − m− ) + 1 = n + 1 + (m+ − m− ) n(n2+1) . This means that the tree possesses a quadratic rate of growth for the number of nodes in the tree after n steps. This makes algorithms based on such a tree effective from the computational point of view. The nodes appearing in the tree after n steps are determined by the number l = l1 + · · · + l n , where l1 δ, . . . , l n δ are jumps at moments 1, . . . , n. The states that can be reached by the random walk at moments n = 0, 1, . . . , are given by the following formula: (3.76) y n,l = y0 + lδ , l = nm− , . . . , nm+ n = 0, 1, . . . . The procedure for the construction of the tree T (y0 ,0) described above does not take into account actual values of probabilities for nodes to appear in the tree generated by the random walk Y n .

102 | 3 Backward recurrence reward algorithms The probability that a node y n,l will be reached after n steps is given by the con­ volution recurrence relation, for nm− ≤ l n ≤ nm+ , n = 1, 2, . . . , ⎧ ⎨p( l 1 ) if l1 = m− , . . . , m+ , n = 1 , (3.77) p n,l n = ⎩m+ p p(l) if l = nm , . . . , nm , n > 1 , l=m−

n −1,l n − l

n

−

+

where one should count p n,l = 0 if l < nm− or l > nm+ , for n = 1, 2, . . . . Note also that numbers y n,l satisfy the inequalities, y n,nm− < y n,nm−+1 < · · · < y n,nm+ ,

n = 0, 1, . . . .

(3.78)

It is useful to note that these points are located on the grid of points G[y0 , δ] = {y0 + lδ, l = 0, ±1, ±2, . . . }. According to formulas (3.3), the tree T(y0 ,0) is the following set of nodes: T(y0 ,0) = (y n,l , n), l = nm− , . . . , nm+ , n = 0, 1, . . .  .

(3.79)

The tree T(y0 ,0) includes a particular point (y, n), if y n,l = y0 + lδ = y, for some nm− ≤ l ≤ nm+ . The condition A2 holds since, in this case, the following inequality takes place for any y ∈ R1 , n = 0, 1, . . . , N:     Ey,n max g(r, e Y r ) n≤r≤N   g(r, e y r,l ) < ∞ , ≤ max max (3.80) n ≤ r ≤ N m +( r − n ) m −≤ l ≤ m +( r − n ) m +

where the points y r,l are given by formulas (3.76), with y0 and nm− ≤ m ≤ nm+ chosen such that y n,m = y. Theorem 2.2.3 takes in this case the following form. Lemma 3.3.1. Let the log-price process Y n be represented by the homogeneous random walk defined in the stochastic transition dynamic relation (3.75). Then the log-reward functions ϕ n (y n,l ), for points y n,l = y0 + lδ, l = nm− , . . . , nm+ , n = 0, 1, . . . , N given by formulas (3.76), are the unique solution for the following finite recurrence system of linear equations: ⎧ ⎪ ϕ N (y N,l ) = g(N, e y N,l ) , l = Nm− , . . . , Nm+ , ⎪ ⎪ ⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ n (y n,l ) = max ⎝g(n, e y n,l ) , ⎪ ⎪ ⎨ ⎞ (3.81) ⎪ m+ ⎪

⎪ ⎪ ⎠ ⎪ ϕ n+1 (y n+1,l+r)p(r) , l = nm− , . . . , nm+ , ⎪ ⎪ ⎪ ⎪ r=m− ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 . This system lets one find the values of the log-reward function ϕ n (y) at points y n,l , given by formulas (3.76), recursively, for n = N, . . . , 0. The particular value ϕ n (y) can be calculated using this system if the initial value y0 is chosen so that y n,l = y0 + lδ = y, for some nm− ≤ l ≤ nm+ .

3.3 Random walk reward algorithms

| 103

3.3.2 An inhomogeneous in time random walk model This is the model, where the log-price Y n is a univariate random walk, Y n = Y n −1 + W n ,

n = 1, 2, . . . .

(3.82)

where (a) Y0 is a real-valued random variable with the distribution P0 (A), (b) W n , n = 1, 2, . . . , is a sequence of independent random variables which takes values lδ + δ n,◦ , l = m n,− , . . . , m n,+ with probabilities, respectively, p n (l), l = m n,− , . . . , m n,+, and (c) the random variable Y0 and the random sequence W n , n = 1, 2, . . . , are inde­ pendent. Also usual assumptions are (d) δ > 0; (e) δ n,◦ ∈ R1 , n = 1, . . . ; (f) m n,− < m n,+, n = 1, 2, . . . , are integers; (g) 0 ≤ p n (l) ≤ 1, l = m n,− , . . . , m n,+, n = 1, 2, . . . ; (h) p n (m n,−) + · · · + p n (m n,+) = 1, n = 1, 2, . . . . Note that usually it is also assumed that (i) δm n,− + δ n,◦ < 0 < δm n,+ + δ n,◦, n = 1, 2, . . . . In this case, the stochastic modulating index variable X n is not included in the model. Correspondingly, a pay-off function g(n, e y ) and the corresponding log-reward functions ϕ n (y) do not depend on the index argument. Let us assume that the initial state of the log-price process Y0 = y0 with probabil­ ity 1, i.e. the initial distribution P0 (A) = I (y0 ∈ A) is concentrated in a point y0 ∈ R1 . The tree structure of trajectories is well visible from the following recurrent algo­ rithm, which describes sequential transitions of the log-price process Y n . If Y n−1 = y for some n = 1, 2, . . . , then Y n = y + lδ + δ n,◦, where l takes one of the values m n,− , . . . , m n,+. The initial point y0 determines the unique tree T(y0 ,0) with the root in the node (y0 , 0), in the space–time domain R1 × {0, 1, . . . }. According to the algorithm described above, the tree T(y0 ,0) has one initial node (y0 , 0). It may include m1, + − m1, − + 1 additional nodes after the first step, then (m1, + − m1,− + 1)(m2,+ − m2,− + 1) additional nodes after the second step, etc. The tree may " include nk=1 (m k,+ − m k,− + 1) additional nodes after the nth step. If this is the case, i.e. the tree T(y0 ,0) possesses an exponential rate of growth for the number of nodes as a function of the number of steps n, any algorithms based on such a tree would not be effective from the computational point of view. However, this is not the case. In fact, the tree includes only m1,+ − m1,− + 1 addi­ tional nodes after the first step, m1,+ − m1,− + m2,+ − m2,− + 1 additional nodes after the second step, etc. This is because the tree T(y0 ,0) satisfies the recombining condition. In this model, the recombining condition means that the random walk Y n , which is in a position Y n−1 = y, for some n = 1, 2, . . . , and makes from this position the first jump l δ + δ n,◦ and then the second jump l δ + δ n+1,◦, ends up at the same point as if it would make the first jump l δ + δ n,◦ and then the second jump l δ + δ n+1,◦. This should hold for

104 | 3 Backward recurrence reward algorithms any l and l such that l + l = l. In this case, the end point is lδ + δ n,◦ + δ n+1,◦. This should hold for any l = m n,− + m n+1,− , . . . , m n,+ + m n+1,+, y ∈ R1 , and n = 1, 2, . . . . The tree T(y0 ,0) obviously includes one initial node (y0 , 0) and m1,+ − m1,− + 1 additional nodes (y0 + m1,− δ + δ1,◦ , 1), . . . , (y0 + m1,+ δ + δ1,◦ , 1) after the first step. Due to the recombining property, the tree includes m1,+ − m1,− + m2,+ − m2,− + 1 additional nodes (y0 + (m1,− + m2,−)δ + δ1,◦ + δ2,◦ , 2), . . . , (y0 + (m1,+ + m2,+ )δ + δ1,◦ + δ2,◦ , 2) after the second step, etc.  The tree T(y0 ,0) includes nk=1 (m k,+ − m k,−)+ 1 additional nodes after the nth step. The total number of nodes in the tree after n steps is L n = 1 +(m1,+ − m1,− ) + 1 +· · · + n k =1 ( m k, + − m k, − ) + 1 = n + 1 + n( m 1, + − m 1, − ) + · · · + ( m n, + − m n, − ). This means that, in the most important case, where m n,+−m n,− ≤ L, n = 1, . . . , N, the tree possesses not more than a quadratic rate of growth for the number of nodes in the tree after n steps. Indeed, in this case, L n ≤ n + 1 + L n(n2+1) , n = 1, . . . . This makes algorithms based on such a tree effective from the computational point of view. Nodes appeared in the tree after n steps are determined by the number l = l1 + · · · + l n , where l1 δ + δ1, ◦ , . . . , l n δ + δ n, ◦ are jumps at moments 1, . . . , n. The states that can be reached by the random walk at moments n = 0, 1, . . . , are given by the formulas y n,l = y0 + lδ + δ1,◦ + · · · + δ n,◦ ,

l = M n,− , . . . , M n,+ n = 0, 1, . . . .

(3.83)

 where M n,± = nk=1 m k,±, n = 1, 2, . . . . It should be mentioned that the procedure for the construction of the tree T(y0 ,0) described above does not take into account the values of appearance probabilities for nodes in the tree generated by trajectories of the random walk Y n . The probability that a node y n,l will be reached after n steps is given by the con­ volution recurrence relation, for M n,− ≤ l n ≤ M n,+, n = 0, 1, . . . , ⎧ ⎨p1 ( l 1 ) if M1,− ≤ l1 ≤ M1,+ , n = 1 , (3.84) p n,l n = ⎩m n,+ p n−1,l −l p n (l) if M n,− ≤ l n ≤ M n,− , n > 1 , l = m n,−

n

where one should count p n,l = 0 if l < M n,− or l > M n,+, for n = 1, 2, . . . . Note also that the states y n,l satisfy the inequalities, y n,M n,− < y n,M n,− +1 < · · · < y n,M n,+ ,

n = 1, . . . .

(3.85)

According to formula (3.3), the tree T(y0 ,0) is the following set of nodes: T(y0 ,0) = (y n,l , n), l = M n,− , . . . , M n,+ , n = 0, 1, . . .  .

(3.86)

The tree T(y0 ,0) includes a particular point (y, n), if y n,l = y0 + lδ + δ1,◦ + · · · + δ n,◦ = y for some M n,− ≤ l ≤ M n,+.

3.3 Random walk reward algorithms

| 105

The condition A2 holds since, in this case, the following inequality takes place for any y ∈ R1 , n = 0, 1, . . . , N:     Ey,n max g(r, e Y r ) n≤r≤N   g(r, e y r,l ) < ∞ , ≤ max max (3.87) n ≤ r ≤ N m + M r,− − M n,− ≤ l ≤ m + M r,+− M n,+

where the points y r,l are given by formulas (3.83), with y0 and M n,− ≤ m ≤ M n,+ chosen such that y n,m = y. Theorem 2.3.3 takes in this case the following form. Lemma 3.3.2. Let the log-price process Y n be represented by the inhomogeneous in time random walk defined by the stochastic transition dynamic relation (3.82). Then the logreward functions ϕ n (y n,l ), for points y n,l , l = M n,− , . . . , M n,+, n = 0, 1, . . . , N given by formulas (3.83), are the unique solution for the following finite recurrence system of linear equations: ⎧ ⎪ ϕ N (y N,l ) = g(N, e y N,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = l = M N,− , . . . , M N,+ , ⎪ ⎪ ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ m n+1,+ ⎨

(3.88) ϕ n (y n,l ) = max ⎝g(n, e y n,l ) , ϕ n+1 (y n+1,l+r)p n+1 (r)⎠ , ⎪ ⎪ r = m ⎪ n + 1, − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = M n,− , . . . , M n,+ , ⎪ ⎪ ⎪ ⎪ ⎩ n = N − 1, . . . , 0 . The system of equations (3.88) let one find values of the log-reward function ϕ n (y) at points y n,l , l = 0, 1, . . . , n given by formulas (3.83), recursively, for n = N, . . . , 0. The particular value ϕ n (y) can be calculated using this system if the initial value y0 is chosen so that y n,l = y0 + lδ + δ1,◦ + · · · + δ n,◦ = y, for some M n,− ≤ l ≤ M n,+. Moreover, if y n,l y = y for some 0 ≤ n ≤ N, y ∈ R1 , then the value ϕ n (y) can be found by solving the reduced version of the system of equations (3.88), ⎧ y N,l ⎪ ⎪ ⎪ ϕ N (y N,l ) = g(N, e ) , ⎪ ⎪ ⎪ ⎪ l = l y + M N,− − M n,− , . . . , l y + M N,+ − M n,+ , ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ m s+1,+ ⎨

y ϕ s (y s,l ) = max ⎝g(s, e s,l ), ϕ s +1 (y s +1,l+r)p s +1 (r)⎠ , (3.89) ⎪ ⎪ r = m s+1,− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = l y + M s,− − M n,− , . . . , l y + M s,+ − M n,+ , ⎪ ⎪ ⎪ ⎪ ⎩ s = N − 1, . . . , n , where y s,l = y + lδ + δ n+1,◦ +· · ·+ δ s,◦, l = l y + M s,− − M n,− , . . . , l y + M s,+ − M n,+ , s = n, . . . , N. In the above model, it is also natural to assume that the initial distribution P0 (A) is concentrated at a finite set of points y0,l = δl + δ0,◦ , l = m0,− , . . . , m0,+ , with the cor­ responding probabilities p0,l , l = m0,− , . . . , m0,+ . In this case, the optimal expected

106 | 3 Backward recurrence reward algorithms reward Φ(Mmax,N ) can be found using the following summation formula:

m 0,+

Φ(Mmax,N ) =

p0,l ϕ0 (y0,l ) .

(3.90)

l = m 0,−

3.4 Markov chain reward algorithms In this section, we describe backward tree algorithms for finding log-reward functions and optimal expected rewards for models where modulated log-price processes are represented by Markov chains, with transition probabilities (distributions), which are concentrated at finite sets of points. Such log-price processes can be effectively used for approximation of more general modulated Markov-type log-price processes. Such so-called space-skeleton approximations are studied in Chapters 7 and 8.

3.4.1 An atomic Markov chain model n , X n ) is an inhomogeneous We assume that the modulated log-price process  Z n = (Y in time Markov chain with a phase space Z = Rk × X, an initial distribution P0 (A) = 0 ∈ A} and one-step transition probabilities P n ( P{ Z z , A ) = P{  Z n ∈ A/ Z n −1 =  z}. In this case, the stochastic modulating index X n is included in the model. Corre­ spondingly, a pay-off function g(n, e y , x) and the corresponding log-reward functions ϕ n ( y , x) do depend on the index argument. We assume that one-step transition probabilities P n ( z , A) are concentrated at fi­ z , n), l = m− ( z , n), . . . , m+ ( z , n)} ∈ BZ , with the correspond­ nite sets Fz ,n = {f l ( ing one-point transition probabilities p l ( z , n ) = P{  Z n = f l ( z , n) /  Z n −1 =  z}, l = − + m ( z , n), . . . , m ( z , n), i.e. for every  z ∈ Z and n = 1, 2, . . . ,

P n ( z , A) = p l ( z , n) , A ∈ BZ . (3.91) f l ( z ,n )∈ A

In this case, it is natural to assume that the initial distribution P0 (A) also is con­ + z0,l , l = m− centrated at a finite set F0 = { 0 , . . . , m 0 } ∈ BZ with the corresponding + 0 =  one-point probabilities p0,l = P{Z z0,l }, l = m− 0 , . . . , m 0 , i.e.

P0 (A) = p0,l , A ∈ BZ . (3.92)  z 0,l ∈ A

In the case, where the above conditions hold,  Z n can be referred to as an atomic Markov chain. The above assumption (3.91) implies that the r-step transition probabilities P(n,  z , n + r, A) = P{ Z n+r ∈ A/ Zn =  z } are also concentrated on finite sets Fz ,n,n+r =

3.4 Markov chain reward algorithms

| 107

{f l ( z , n, n + r), l = m− ( z , n, n + r), . . . , m+ ( z , n, n + r)}, with the corresponding transition probabilities p l ( z , n, n + r) = P{ Z n+r = f l ( z , n, n + r)/ Zn =  z }, l = − + m ( z , n, n + r), . . . , m ( z, n, n + r), i.e. for every  z ∈ Z and n = 0, 1, . . . , and r = 1, 2, . . . ,

P(n, z , n + r, A) = p l ( z , n, n + r) , A ∈ BZ . (3.93) f l ( z ,n,n + r )∈ A

z , n, n + r) and the The following recurrent relations link transition functions f l ( corresponding transition probabilities p l ( z , n, n + r) for r = 1, 2, . . . : z , n, n + r) f l ( ⎧ ⎪ z , n + 1) ⎪ ⎪ f l ( ⎪ ⎪ ⎪ − ⎪ ⎪ m ( z , n + 1) ≤ l ≤ m+ ( z , n + 1), ⎪ ⎨ = f j (f i ( z , n, n + r − 1), n + r) ⎪ ⎪ ⎪ ⎪ ⎪ (i, j) ∈ Fz ,n,n+r,l , ⎪ ⎪ ⎪ ⎪ − ⎩ m ( z , n, n + r) ≤ l ≤ m+ ( z , n, n + r).

if r = 1 , if r > 1 ,

(3.94)

and p l ( z , n, n + r) ⎧ ⎪ p l ( z , n + 1) ⎪ ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ m ( z , n + 1) ≤ l ≤ m+ ( z , n + 1), ⎪ ⎨ = z , n, n + r − 1) ( i,j )∈Fz,n,n+r,l p i ( ⎪ ⎪ ⎪ ⎪ ⎪ ×p j (f i ( z , n, n + r − 1), n + r) ⎪ ⎪ ⎪ ⎪ ⎩ m− ( z , n, n + r) ≤ l ≤ m+ ( z , n, n + r) ,

if r = 1 , (3.95) if r > 1 ,

where  Fz ,n,n+r,l = (i, j) : f j (f i ( z , n, n + r − 1), n + r) = f l ( z , n, n + r)

m− ( z , n, n + r − 1) ≤ i ≤ m+ ( z , n, n + r − 1),

(3.96)  m (f i ( z , n, n + r − 1), n + r) ≤ j ≤ m (f i ( z , n, n + r − 1), n + r) . −

+

Any node ( z , n) generates the tree of nodes T(z ,n). This tree includes one ini­ z , n), additional nodes (f l ( z , n, n + 1), n + 1), l = m− ( z , n, n + 1), . . . , tial node ( + m ( z , n, n + 1), appearing in the tree after the first transition, additional nodes (f l ( z , n, n + 2), n + 2), l = m− ( z , n, n + 2), . . . , m+ ( z , n, n + 2), appearing in the tree after the second transition, etc. According to the above remarks, the tree T(z ,n) is the following set of nodes: T(z ,n) = (f l ( z , n, n + r), n + r), z , n, n + r), . . . , l = m− ( m+ ( z , n, n + r), r = 0, 1, . . .  .

(3.97)

108 | 3 Backward recurrence reward algorithms where f0 ( z , n, n) =  z , m± ( z , n, n) = 0. The total number of nodes in the tree after r transitions is L r ( z , n) = 1 + (m+ ( z , n, n + 1) − m− ( z , n, n + 1) + 1) z , n, n + r) − m− ( z , n, n + r) + 1) . + · · · + (m+ (

(3.98)

We are specially interested in trees, where the number L r ( z , n) ≤ Lr h , r = 1, 2, . . . , where L, h > 0, i.e. models where the trees have not more than polynomial rate of z , n) as a function of n. growth for L r ( Examples of such binomial, trinomial and general discrete random walk trees have been presented in Sections 3.1–3.3. Below, we extend this list by presenting socalled skeleton-type models. We should introduce additional notations explicitly separating log-price and in­ z , n, n + r) ∈ Z, by presenting these points in the dex components for points f l ( form z , n, n + r) = (f l ( z , n, n + r), f l ( z , n, n + r)) , f l (

(3.99)

where f l ( z , n, n + r) ∈ Rk and f l ( z , n, n + r) ∈ X. The condition A2 holds since, in this case, the following inequality takes place for z = ( y , x) ∈ Z, n = 0, 1, . . . , N: any       Ez ,n max g(r, e Y n+r , X n+r ) 0≤ r ≤ N − n

≤

max

max

0≤ r ≤ N − n m − ( z ,n,n + r )≤ l≤ m+( z ,n,n + r )

     g(n + r, e f l (z ,n,n+r) , f l ( z , n, n + r))

< ∞.

(3.100)

Theorem 2.3.3 takes in this case the following form. Z n be an atomic Markov chain with transition Lemma 3.4.1. Let the log-price process  probabilities and initial distribution defined, respectively, by relations (3.91) and (3.92). Then the log-reward functions ϕ n ( z) and ϕ n+r (f l ( z , n, n + r)), for points f l ( z , n, n + r), z , n, n + r), . . . , m+ ( z , n, n + r), r = 1, . . . N − n given by formulas (3.94), are, l = m− ( for every  z ∈ Z, n = 0, . . . , N, the unique solution for the following finite recurrence

3.4 Markov chain reward algorithms

system of linear equations: ⎧  ⎪ z , n, N )) = g(N, e f l (z ,n,N ), f l ( z , n, N )) , ⎪ ⎪ ϕ N (f l ( ⎪ ⎪ ⎪ − ⎪ l = m ( z , n, N ), . . . , m+ ( z , n, N ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ n+r (f l ( z , n, n + r)) ⎪ ⎪ ⎪ ⎛ ⎪ ⎪ ⎪  # ⎪ ⎪ ⎪ ⎝g n + r, e f l (z ,n,n+r), f  ( ⎪ max z , n, n + r ) , = ⎪ l ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ m + ( z ,n,n

+r+1) ⎪ ⎪ ϕ n+r+1(f l (f l ( z , n, n + r), n + r + 1)) ⎪ ⎪ ⎪ ⎪ l  = m − ( z ,n,n + r +1) ⎪ ⎪ ⎪ ⎞ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z , n, n + r), n + r + 1)⎠ , × p l (f l ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = m− ( z , n, n + r), . . . , m+ ( z , n, n + r) , ⎪ ⎪ ⎪ ⎪ ⎩ r = N − n − 1, . . . , 0 ,

|

109

(3.101)

z , n, n) =  z, m± ( z , n, n) = 0. where f0 ( As far as the optimal expected reward Φ(Mmax,N ) is concerned, it can be found using the following summation formula: +

Φ(Mmax,N ) =

m0

p0,l ϕ0 ( z0,l ) .

(3.102)

l=m− 0

3.4.2 An atomic Markov chain model with a fixed skeleton structure This model is a particular case of the model introduced in Subsection 3.4.1. + Let us assume that the sets of points Fn = { z n,l , l = m− n , . . . , m n } ∈ BZ are − defined for n = 0, 1, . . . , and that points f l ( z , n) =  z n,l , l = m ( z , n) = m − n,..., z , n) = m + , n = 1, 2, . . . , and, respectively, sets F = F , n = 1, 2, . . . , do m+ ( n  z ,n n   not depend on  z ∈ Z, while the probabilities p l ( z , n ) = P{ Z n =  z n,l /Z n−1 =  z }, + l = m− z ∈ Z. n , . . . , m n , n = 1, 2, . . . , do depend on  z , A) have the following form, for  z ∈ Z and n = In this case, measures P n ( 1, 2, . . . :

P n ( z , A) = p l ( z , n) , A ∈ BZ . (3.103)  z n,l ∈ A

The log-price process  Z n can be referred to as a space-skeleton Markov chain. The tree of nodes T(z ,n) for the skeleton Markov chain  Z n includes one initial node + − + ( z , n), m n+1 − m n+1 + 1, additional nodes ( z n+1,l , n + 1), l = m− n +1 , . . . , m n +1 , appearing

110 | 3 Backward recurrence reward algorithms − in the tree after the first transition, m+ z n+2,l , n + 2), n +2 − m n +2 + 1 additional nodes ( − + l = m n+2 , . . . , m n+2 , appearing in the tree after the second transition, etc. According to the above remarks, the tree T(z ,n) is the following set of nodes: + z , n), ( z n+r,l , n + r), l = m− T(z ,n) = ( n + r , . . . , m n + r , r = 0, 1, . . .  .

(3.104)

The total number of nodes in the tree after r transitions is given by the following formula: − + − L r ( n) = 1 + ( m + n +1 − m n +1 + 1) + · · · + ( m n + r − m n + r + 1) .

(3.105)

Theorem 2.3.3 takes in this case the following form. Lemma 3.4.2. Let the log-price process  Z n be a skeleton Markov chain with transi­ tion probabilities defined by relation (3.103). Then the log-reward functions ϕ n ( z) and + ϕ n+r ( z n+r,l), for points  z n+r,l, l = m− , . . . , m , r = 1, . . . N − n are, for every  z ∈ Z, n+r n+r n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equations: ⎧ + ⎪ ϕ N ( z N,l ) = g(N, ey N,l , x N,l ) , l = m− ⎪ N , . . . , mN , ⎪ ⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z n+r,l) = max ⎝g(n + r, ey n+r,l , x n+r,l) , ⎪ ⎪ ϕ n+r ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎞ ⎪ ⎪ m+ ⎪ n + r +1 ⎨

ϕ n+r+1( z n+r+1,l )p l ( z n+r,l , n + r + 1)⎠ , (3.106) ⎪ ⎪ =m− ⎪ l n + r +1 ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ r = N − n − 1, . . . , 1 , l = m− ⎪ n+r , . . . , m n+r , ⎪ ⎪ ⎪ ⎞ ⎛ ⎪ ⎪ m+ n +1 ⎪

⎪ ⎪  y ⎪ ⎝ ⎪ ϕ n ( z ) = max g(n, e , x) , ϕ n+1 ( z n+1,l )p l ( z , n + 1)⎠ . ⎪ ⎩ −  l = m n +1

3.4.3 A space-skeleton Markov chain model This model is a particular case of the model introduced in Subsection 3.4.2. + Let us choose some δ n,i > 0, i = 1, . . . , k, n = 0, 1, . . . , integer m− n,i ≤ m n,i , i = 1, . . . , k, n = 0, 1, . . . , and λ n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . . + Let us now define points y n,i,l = lδ n,i + λ n,i , l = m− n,i , . . . , m n,i , i = 1, . . . , k, + n = 0, 1, . . . , and also choose points x n,l ∈ X, l = m− n,0 , . . . , m n,0 , n = 0, 1, . . . . + ¯ Now, let us define skeleton points for l = (l0 , l1 , . . . , l k ), l j = m− n,j , . . . , m n,j , j = 0, . . . , k, n = 0, 1, . . . ,  z n,¯l = ( y n,¯l , x n,¯l ) = ((y n,1,l1 , . . . , y n,k,l k ), x n,l0 ) .

(3.107)

The difference in notations (for  z n,l and  z n,¯l and between p l ( z , n) and p¯l ( z , n)) can − be removed by a natural renumeration of indices, namely (m− , m , . . . , m− n,1 n,0 n,k ) ↔

3.4 Markov chain reward algorithms

|

111

− − + + + + − + m− n , . . . , ( m n,0 + 1, m n,1 , . . . , m n,k ) ↔ m n + 1, . . . , ( m n,0 , m n,1 , . . . , m n,k ) ↔ m n . Ob­ viously, the following relation holds, for n = 0, 1, . . . :

− m+ n − mn + 1 =

k  − (m+ n,j − m n,j + 1) .

(3.108)

j =0

The simplest variant is to choose integers ±m± n ≥ 0, n = 0, 1, . . . . − The tree of nodes T(z ,n) includes one initial node ( z , n ), m + n +1 − m n +1 + 1 ad­ − + ¯ z n+1,¯l , l = (l0 , l1 , . . . , l k ), l j = m n+1,j , . . . , m n+1,j, j = 0, . . . , k af­ ditional nodes  − ter the first jump, m+ z n+2,¯l , ¯l = (l0 , l1 , . . . , l k ), n +2 − m n +2 + 1 additional nodes  − + l j = m n+2,j , . . . , m n+1,2, j = 0, . . . , k after the second jump, etc. According to the above remarks, the tree T(z ,n) is the following set of nodes: T(z ,n) = ( z , n), ( z n+r,¯l , n + r), lj =

m− n + r,j ,

...,

m+ n + r,j ,

¯l = (l0 , l1 , . . . , l k ), j = 0, . . . , k, r = 1, . . .  .

(3.109)

The total number of nodes in the tree after r transitions is given by the following formula: − + − L r ( n) = 1 + ( m + n +1 − m n +1 + 1) + · · · + ( m n + r − m n + r + 1) .

(3.110)

Theorem 2.2.3 takes in this case the following form. Z n is a skeleton Markov chain with skeleton Lemma 3.4.3. Let the log-price process  points defined in relation (3.107). Then the log-reward functions ϕ n ( z ) and ϕ n+r ( z n+r,¯l), for points z n+r,¯l = ( y n+r,¯l, x n+r,¯l) = ((y n+r,1,l1 , . . . , y n+r,k,l k ), x n+r,l0 ), ¯l = (l0 , l1 , . . . , l k ), + z ∈ Z, n = 0, . . . , N, l j = m− n + r,j , . . . , m n + r,j, j = 0, . . . , k, r = 1, . . . , N −n, are, for every  the unique solution for the following recurrence finite system of linear equations: ⎧ ⎪ ϕ N ( z N,¯l ) = g(N, ey N,¯l , x N,¯l ) , ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ j = 0, . . . , k , ⎪ N,j , . . . , m N,j , ⎪ ⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ n+r ( z n+r,¯l) = max ⎝g(n + r, ey n+r,¯l , x n+r,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎞ ⎪ ⎪ ⎪ ⎪

⎨ ⎟ ϕ n+r+1( z n+r+1,¯l  )p¯l  ( z n+r,¯l , n + r + 1)⎠ , (3.111) ⎪ ⎪  z n+r+1,¯l  ∈Fn+r+1 ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ j = 0, . . . , k , ⎪ n + r,j , . . . , m n + r,j , ⎪ ⎪ ⎪ ⎪ ⎪ r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ ⎪ ⎪

⎪ ⎜ ⎟ ⎪ ⎪ ⎪ ϕ n ( z) = max ⎝g(n, ey , x) , ϕ n+1 ( z n+1,¯l  )p¯l  ( z , n + 1)⎠ . ⎪ ⎪ ⎩  z n+1,¯l  ∈Fn+1

112 | 3 Backward recurrence reward algorithms

3.4.4 A space-skeleton Markov random walk model n , X n ), n = 0, 1, . . . , can always be represented A modulated Markov chain  Z n = (Y n , X n ) = (Y  n −1 + W  n , X n ), Z n = (Y in the form of the modulated Markov random walk  0 , X0 ) and random jumps W n − Y  n −1 , n = Y n = 1, 2, . . . , with an initial state  Z0 = (Y n = 1, 2, . . . . In this case, it is convenient to operate with the transition jump probabilities ˜ n (  n , X n ) ∈ A/Y  n −1 =  y , x, A) = P{(W y , X n−1 = x}, which are connected with P the transition probabilities of the Markov chain  Z n by the following relations, for  z = ( y , x) ∈ Z, A ∈ BZ , n = 0, 1, . . . :   ˜ n (  n , X n ) ∈ A/Y  n −1 =  P y , x, A) = P (W y , X n −1 = x    n −1 =  n −  = P (Y y , X n ) ∈ A/Y y , X n −1 = x #  = Pn  z , A[y] , (3.112)

z  = ( y  , x) : ( y − y , x) ∈ A}. where A[y] = { Let us consider particular cases of the model introduced above in Subsection 3.4.1, which can also be interpreted as a space-skeleton-type Markov random walk. In this case, one can use the standard space-skeleton construction applied to jumps for the random walk. + Let us choose some δ i > 0, i = 1, . . . , k, integers m− n,j ≤ m n,j , j = 0, . . . , k, n = 0, 1, . . . , and λ n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . . + Let us now define points y n,i,l i = l i δ i + λ n,i , l i = m− n,i . . . , m n,i , i = 1, . . . , k, n = − + 0, 1, . . . , and, also, choose points x n,l0 ∈ X, l0 = m n,0 , . . . , m n,0, n = 0, 1, . . . . z n,¯l = ( y n,¯l , x n,¯l ), ¯l = (l0 , l1 , . . . , l k ), l j = Finally let us define skeleton points  − + m n,j , . . . , m n,j, j = 0, . . . , k, n = 0, 1, . . . ,  z n,¯l = ( y n,¯l , x n,¯l ) = ((y n,1,l1 , . . . , y n,k,l k ), x n,l0 ) .

(3.113)

z , n) = ( y + y n,¯l , x n,¯l ) : ( y n,¯l , x n,¯l ) ∈ Let us now introduce skeleton sets Fz ,n = {f¯l ( + z n,¯l : ¯l = (l0 , l1 , . . . , l k ), l j = m− , . . . , m , j = 0, . . . , k } for Fn }, where Fn = { n,j n,j  z = ( y , x) ∈ Z and n = 0, 1, . . . . In this case, skeleton points f¯l ( z , n) and sets Fz ,n depend on  z ∈ Z. Note that paranetrs δ i , i = 1, . . . , k, do not depend on n. This implies that the corresponding recombining condition holds.  n , Xn) = ˜¯l ( y , x, n) = P{(W The corresponding one-point transition probabilities p ¯    y n,¯l , x n,¯l )/Z n−1 =  z } = p¯l ( z , n) = P{Z n = f¯l ( z , n)/Z n−1 =  z}, l = (l0 , l1 , . . . , l k ), ( + z = ( y , x) ∈ Z. l j = m− n,j , . . . , m n,j , j = 0, . . . , k also depend on 

3.4 Markov chain reward algorithms

| 113

˜ n ( z , A) have the following form, for  z = In this case, the probability measures P y , x) ∈ Z and n = 0, 1, . . . : (

˜ n ( ˜¯l ( P p y , x, A) = y , x, n), ( y n,¯l ,x n,¯l )∈ A

= P n ( z , A[y] )

=

p¯l ( z , n) ,

A ∈ BZ .

(3.114)

( y+ y n,¯l ,x n,¯l )∈ A [y]

The difference in notations (for f l ( z , n) and f¯l ( z , n) and between p l ( z , n) and z , n)) can be removed, as above, by a natural renumeration of indices, namely p¯l ( − − − − + − − − (m− n,0 , m n,1 , . . . , m n,k ) ↔ m n , . . . , ( m n,0 + 1, m n,1 , . . . , m n,k ) ↔ m n + 1, . . . , ( m n, + , + + + m n,1, . . . , m n,k ) ↔ m n , where − m+ n − mn + 1 =

k 

− (m+ n,j − m n,j + 1) ,

n = 0, 1, . . . .

(3.115)

j =0

The simplest variant is to choose integers ±m± n ≥ 0, n = 0, 1, . . . . Let us denote  y n,n+r,¯l = (δ1 l1 + λ n,n+r,1, . . . , δ k l k + λ n,n+r,k), x n,n+r,¯l = x n+r,l0 , ¯l = + ± ± (l0 , . . . , l k ), l j = m− n,n + r,j , . . . , m n,n + r,j, j = 0, . . . , k, where m n,n + r,0 = m n + r,0 and n+r n+r ± ± m n,n+r,j = l=n+1 m l,j , λ n,n+r,j = l=n+1 λ l,j , j = 1, . . . , k, for 0 ≤ n < n + r < ∞. n , X n ) = ( If (Y y , x) for some ( y , x) ∈ Z and n = 0, 1, . . . , then the possible n+r , X n+r ) for r = 1, 2, . . . , are ( states for (Y y+ y n,n+r,¯l, x n,n+r,¯l), ¯l = (l0 , . . . , l k ), l j = − + m n,n+r,j, . . . , m n,n+r,j, j = 0, . . . , k. + Let us also define integer numbers m− n,n + r ≤ m n,n + r , 0 ≤ n < n + r < ∞ such that the following relation holds for 0 ≤ n < n + r < ∞: − m+ n,n + r − m n,n + r + 1 =

k 

− (m+ n,n + r,j − m n,n + r,j + 1) .

(3.116)

j =0

The simplest variant is to choose integers ±m± n,n + r ≥ 0, 0 ≤ n < n + r < ∞. − y , x ) , n ), m + The tree of nodes T((y ,x),n) includes one initial node (( n,n +1 − m n,n +1 + 1 additional nodes (( y+ y n,n+1,¯l,  x n,n+1,¯l), n + 1), ¯l = (l0 , l1 , . . . , l k ), l j = m− n,n +1,j, + + − . . . , m n,n+1,j, j = 0, . . . , k after the first jump, m n,n+2 − m n,n+2 + 1 additional nodes + (( y + y n,n+2,¯l , x n+2,¯l), n + 2), ¯l = (l0 , l1 , . . . , l k ), l j = m− n,n +2,j , . . . , m n,n +2,j, j = 0, . . . , k after the second jump, etc. According to the above remarks, the skeleton Markov random walk tree T((y ,x),n) is the following set of nodes: T((y ,x),n) = (( y , x), n), (( y + y n,n+r,¯l, x n+r,¯l), n + r), + ¯l = (l0 , l1 , . . . , l k ), l j = m− n,n + r,j, . . . , m n,n + r,j , j = 0, . . . , k, r = 1, . . .  .

(3.117)

114 | 3 Backward recurrence reward algorithms The total number of nodes in the tree after r jumps is given by the following for­ mula: − + − L r ( n) = 1 + m + n,n +1 − m n,n +1 + 1 + · · · + m n,n + r − m n,n + r + 1 .

(3.118)

Theorem 2.3.3 takes in this case the following form. Lemma 3.4.4. Let the log-price process  Z n be a skeleton Markov random walk with skeleton points and transition probabilities defined, respectively, by relations (3.113) y , x) and ϕ n+r ( y+ y n,n+r,¯l, x n+r,¯l), for and (3.114). Then the log-reward functions ϕ n ( + , . . . , m , j = 0, . . . , k, r = 1, . . . N − n are, points ¯l = (l0 , l1 , . . . , l k ), l j = m− n,n + r,j n,n + r,j z ∈ Z, n = 0, . . . , N, the unique solution for the following finite recurrence for every  system of linear equations: ⎧ ϕ N ( y+ y n,N,¯l , x N,¯l ) = g(N, ey n,N,¯l , x n,N,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ j = 0, . . . , k , ⎪ n,N,j , . . . , m n,N,j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ n+r ( y+ y n,n+r,¯l,x n+r,¯l) ⎪ ⎪ ⎪ ⎪ ⎪ ⎛ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = max ⎝g(n + r, ey+y n,n+r,¯l , x n+r,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ # 

⎪ ⎪ ⎪ ⎪ ϕ y + y y ¯l +  ¯l  , x n + r +1,¯l  n + r +1  n,n + r, n + r + 1, ⎪ ⎪ ⎪ ⎪ ( y n+r+1,¯l ,x n+r+1,¯l )∈Fn+r+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎞ ⎪ ⎪ ⎪ #  ⎪ ⎪ ⎪ ⎪ × p¯l  y+ y n,n+r,¯l, x n+r,¯l , n + r + 1 ⎠ , ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

¯l = (l0 , l1 , . . . , l k ) ,

+ l j = m− n + r,j , . . . , m n + r,j ,

j = 0, . . . , k ,

r = N − n − 1, . . . , 1 , ⎛ y , x) = max ⎝g(n, ey , x) , ϕ n (

ϕ n+1 ( y+ y n+1,¯l , x n+1,¯l )

( y n+1,¯l ,x n+1,¯l )∈Fn+1

⎞ × p¯l ( y , x, n + 1)⎠ .

(3.119)

4 Upper bounds for option rewards In this chapter, we give upper bounds for log-reward functions and optimal expected rewards for American-type options for multivariate modulated Markov log-price pro­ cesses. We consider models with pay-off functions, which do not have more than polyno­ mial rate of growth in price arguments and multivariate modulated Markov log-price processes with bounded and unbounded characteristics. In Section 4.1, we give upper bounds for log-reward functions and optimal ex­ pected rewards for American-type options for univariate and multivariate modulated Markov log-price processes with bounded exponential moments of increments. In Section 4.2, we give upper bounds for log-reward functions and optimal ex­ pected rewards for American-type options for univariate and multivariate modulated log-price processes represented by standard and modulated random walks. In Section 4.3, we present upper bounds for log-reward functions and optimal ex­ pected rewards for American-type options for univariate and multivariate modulated Markov log-price processes with unbounded exponential moments of increments. In Section 4.4, we investigate upper bounds for log-reward functions and optimal expected rewards for American-type options for univariate Markov Gaussian log-price processes with bounded and unbounded drift and volatility coefficients. In Section 4.5, we give upper bounds for log-reward functions and optimal ex­ pected rewards for American-type options for multivariate modulated Markov Gaus­ sian log-price processes with bounded and unbounded drift and volatility coefficients. The main results for multivariate modulated Markov log-price processes with bounded exponential moments of increments and for log-price processes represented by random walks are, respectively, given in Theorems 4.1.1–4.1.4 and 4.2.1–4.2.4. These results generalize results obtained in papers Silvestrov, Jönsson, and Sten­ berg (2006, 2008, 2009), for univariate modulated Markov log-price processes and by Lundgren and Silvestrov (2009, 2011), and Silvetsrov and Lundgren (2011) for multi­ variate Markov log-price processes. The results for multivariate modulated Markov log-price processes with un­ bounded exponential moments of increments given in Theorems 4.3.1–4.3.8, as well as results for univariate and multivariate modulated Markov Gaussian log-price pro­ cesses given in Theorems 4.4.3–4.4.4 and 4.5.3–4.5.4, are new.

4.1 Markov LPP with bounded characteristics In this section, we give explicit upper bounds for log-reward functions and opti­ mal expected rewards for American-type options for discrete time-modulated Markov

116 | 4 Upper bounds for option rewards log-price processes, with conditional exponential moments of increments for log-price processes bounded as functions of log-price and index arguments.

4.1.1 Upper bounds for exponential moments of maxima for Markov log-price processes with bounded characteristics n , X n ) is an inhomogeneous We assume that a modulated log-price process  Z n = (Y Markov chain with a phase space Z = Rk ×X, an initial distribution P0 (A) = P{ Z 0 ∈ A }, and one-step transition probabilities P n ( z , A ) = P{  Z n ∈ A/ Z n −1 =  z }. Let us define the first-type modulus of exponential moment compactness for com­ n = (Y n,1 . . . , Y n,k), for β ≥ 0, i = 1, . . . , k, ponents of the log-price process Y

Δ β (Y· ,i , N ) = max sup Ez ,n e β| Y n+1,i −Y n,i | . 0≤ n ≤ N − 1  z∈Z

(4.1)

Remind that we use the notations Pz ,n and Ez ,n for conditional probabilities and Zn =  z. expectations under the condition that  We use the following first-type condition of exponential moment compactness for n , assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) the log-price process Y with nonnegative components: C1 [β¯ ]: Δ β i (Y· ,i , N ) < K1,i , i = 1, . . . , k, for some 1 < K1,i < ∞, i = 1, . . . , k . It is useful to note that the condition C1 [β¯ ] implies that the condition C1 [β¯  ] holds with the same constants K1,i , i = 1, . . . , k, for any vector parameter β¯  = (β 1 , . . . , β k ) such that 0 ≤ β i ≤ β i , i = 1, . . . , k. The following lemma gives explicit upper bounds for conditional exponential mo­ n . ments of the maximal absolute values of components for the log-price process Y Lemma 4.1.1. Let the condition C1 [β¯ ] holds. Then there exist constants 0 ≤ M1,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N, i = 1, . . . , k: $ % Ez ,n exp β i max |Y r,i | ≤ M1,i e β i | y i | . (4.2) n≤r≤N

Proof. Note that Δ0 (Y i,· , N ) ≡ 1 and, thus, the relation penetrating condition C1 [β¯ ] holds automatically for i such that β i = 0. Also, the expectation on the left-hand side of inequality (4.2) equals 1, while the expression on the right-hand side of this inequal­ ity equals M1,i . Thus, this inequality holds, in this case, with the constant M1,i = 1. Therefore, we should prove (4.2) only for i such that β i > 0.

4.1 Markov LPP with bounded characteristics

|

117

Let us use the following inequality: % $ exp β i max |Y r,i| n≤r≤N ⎫ ⎧ r −1 ⎬ ⎨

= max exp β i |Y n,i + (Y l+1,i − Y l,i )| ⎭ ⎩ n≤r≤N

l=n

⎧ ⎨

≤ exp β i |Y n,i | + β i ⎩

N −1

r=n

⎫ ⎬ |Y r+1,i − Y r,i| . ⎭

(4.3)

n and then the recurrent condi­ Using Markov property of the log-price process Y tion C1 [β¯ ], we get the following inequalities, for  z ∈ Z, 0 ≤ n ≤ N: ⎫ ⎧ −1 ⎬ ⎨ N

β i |Y r+1,i − Y r,i|⎭ Ez ,n exp ⎩ r=n ⎫ ⎧ ⎧ ⎫ −1 ⎬ ⎨ ⎨ N

⎬ = Ez ,n E exp β i |Y r+1,i − Y r,i|⎭ /FN −1⎭ ⎩ ⎩ r=n

⎫ ⎧ ⎧ ⎫ −2 ⎬ ⎨ ⎨ N

 ⎬ = Ez ,n exp β i |Y r+1,i − Y r,i | · E exp β i |Y N,i − Y N −1,i |/ Z N −1 ⎭ ⎭ ⎩ ⎩ r =1 ⎫ ⎧ −2 ⎬ ⎨ N

N −n ≤ Ez ,n exp β i |Y r,i| · K1,i ≤ · · · ≤ K1,i . ⎭ ⎩

(4.4)

r=n

Finally, using relations (4.3) and (4.4) we get the following inequality, for  z ∈ Z, 0 ≤ n ≤ N: $ % Ez ,n exp β i max |Y r,i | n≤r≤N ⎫ ⎧ N −1 ⎬ ⎨

≤ Ez ,n exp β i |Y n,i | + β i |Y r +1,i − Y r,i | ⎭ ⎩ r=n ⎫ ⎧ −1 ⎬ ⎨ N

N −n = e β i | y i | Ez ,n exp ⎩ β i |Y r+1,i − Y r,i|⎭ ≤ e β i | y i | K1,i . (4.5) r=n

The remarks made at the beginning of the proof and inequality (4.5) imply that N . inequality (4.2) holds with the constant K1,i Remark 4.1.1. The constants M1,i , i = 1, . . . , k are given by the following formulas: N M1,i = K1,i .

(4.6)

Moreover, inequalities (4.5) give upper bounds for exponential moments of the max­ ima for the log-price processes Y n,i over discrete time intervals [n, N ], with the con­ N −n stants K1,i depending on n.

118 | 4 Upper bounds for option rewards The first-type modulus of exponential moment compactness Δ β (Y· ,i , N ) involves the supremum of conditional exponential moments for increments of the correspond­ n over the whole phase space Z. In some ing component for the log-price processes Y cases, for example, in applications to space-skeleton approximation models, this can be too strong restriction. One can improve the definition of the above modulus in the following way. Let Z = Z0 , . . . , ZN  be a sequence of measurable sets and Δ β,Z (Y· ,i , N ) be a modified first-type modulus of exponential moment compactness defined for β ≥ 0 by the following formula: Δ β,Z (Y· ,i , N ) = max sup Ez ,n e β| Y n+1,i −Y n,i | . 0≤ n ≤ N − 1  z∈Z

(4.7)

n

Obviously, if Z = Z, . . . , Z, then modulus Δ β,Z (Y· ,i , N ) = Δ β (Y· ,i , N ), i = 1, . . . , k, β ≥ 0. A sequence of measurable subsets Z = Z0 , . . . , ZN  is called a complete se­ quence of phase sets for the multivariate modulated Markov log-price process  Z n with z , A), if Pn ( z , Zn ) = 1,  z∈ an initial distribution P0(A) and transition probabilities P n ( Zn−1 , n = 1, . . . , N, and an ultimately complete sequence of phase sets for this process, if also P(Z0 ) = 1. The following property of complete sequence of phase sets and ultimately com­ plete sequence of phase sets follows in an obvious way from their definitions. Lemma 4.1.2. If Z = Z0 , . . . , ZN  is a complete sequence of phase sets for the mul­ r ∈ Zr } = 1,  tivariate modulated Markov log-price process  Z n , then Pz ,n {Z z ∈ Zn , 0 ≤ n ≤ r ≤ N. If Z is an ultimately complete sequence of sets for this process, then r ∈ Zr } = 1, 0 ≤ r ≤ N. also P{Z Let us formulate a condition of exponential moment compactness for the log-price ¯ ]. It should be assumed to hold for  n , which generalizes the condition C1 [β process Y some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: C1 [β¯ ]: There exists a sequence of measurable sets Z = Z0 , . . . , ZN  such that   (a) Δ β i ,Z (Y· ,i , N ) < K1,i , i = 1, . . . , k, for some 1 < K1,i < ∞, i = 1, . . . , k; z , Zn ) = 1,  z ∈ Z n −1 , n = (b) Z is a complete sequence of phase sets, i.e., Pn ( 1, . . . , N . The following lemma gives alternative upper bounds for conditional exponential mo­ n . ments of the maximal absolute values of components for the log-price process Y  < Lemma 4.1.3. Let the condition C1 [β¯ ] holds. Then there exist constants 0 ≤ M1,i z = ( y , x) = ∞, i = 1, . . . , k such that the following inequalities take place for  ((y1 , . . . , y k ), x) ∈ Zn , n = 0, . . . , N, i = 1, . . . , k: $ %  e βi |yi | . (4.8) Ez ,n exp β i max Y r,i ≤ M1,i

n≤r≤N

4.1 Markov LPP with bounded characteristics

| 119

Proof. It repeats the proof of Lemma 4.1.1. Again, the case where β i = 0 is trivial. As far as the case β i > 0 is concerned, the only remark should be made that probability r ∈ Zr } = 1, for  Pz ,n {Z z ∈ Zn , 0 ≤ n ≤ r ≤ N, by Lemma 4.1.2, and expectation  , for  z ∈ Zr , 0 ≤ r ≤ N, by condition C1 [β¯ ]. Ez ,r exp{β i |Y r+1,i − Y r,i| ≤ K1,i Using these remarks one can write down a relations analogous to (4.4) and (4.5),  but for  z ∈ Zn , 0 ≤ n ≤ N, and with the constants K1,i , i = 1, . . . , k used instead of K1,i , z ∈ Zn , 0 ≤ n ≤ N, i = 1, . . . , k. These relations imply that inequality (4.8) holds for  i = 1, . . . , k.  Remark 4.1.2. The constants M1,i , i = 1, . . . , k are given by the following formulas:   N M1,i = (K1,i ) .

(4.9)

Moreover, inequalities (4.5) give, in this case, upper bounds for exponential moments of the maxima for the log-price processes Y n,i over discrete time intervals [n, N ], with  N −n the constants (K1,i ) depending on n.

4.1.2 Upper bounds for rewards of Markov log-price processes with bounded characteristics In this book, we study the American-type options with pay-off functions g(n, s , x) that have not more than polynomial rate of growth in argument  s = (s1 , . . . , s k ) ∈ R+ k and are bounded in arguments 0 ≤ n ≤ N and x ∈ X. This condition can also be reformulated as the assumption that pay-off functions g(n, ey , x) have not more than exponential rate of growth in argument  y = (y1 , . . . , y k ) ∈ Rk and are bounded in arguments 0 ≤ n ≤ N and x ∈ X. The following condition is assumed to hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components: B1 [𝛾¯]: max0≤n≤N supz=(y ,x)∈Z L2,1 , . . . , L2,k < ∞ .

| g ( n, ey ,x )| k 1+ i=1 L 2,i e 𝛾i |y i |

< L1 , for some 0 < L1 < ∞ and 0 ≤

The following lemma gives sufficient conditions for the condition A2 to hold and ex­ plicit upper bounds for conditional expectations of the maximal absolute values for the pay-off process g(n, e Yn , X n ). Lemma 4.1.4. Let the conditions B1 [𝛾¯] and C1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exist constants 0 ≤ M2 , M3,i < ∞, i = 1, . . . , k such that the following inequalities take place for 0 ≤ n ≤ N,  z = ((y1 , . . . , y k ), x) ∈ Z: 

Ez ,n max |g(r, e Y r , X r )| ≤ M2 + n≤r≤N

k

i =1

M3,i e𝛾i | y i | .

(4.10)

120 | 4 Upper bounds for option rewards Proof. Using the condition B1 [𝛾¯] we get the following inequality for 0 ≤ n ≤ N: ⎛ ⎞ k  

  r Y 𝛾 | Y | r,i ⎠ L1 L2,i e i max g(r, e , X r ) ≤ max ⎝L1 + n≤r≤N

n≤r≤N

≤ L1 +

i =1

k

L1 L2,i max e𝛾i | Y r,i | n≤r≤N

i =1

= L1 +

k

L1 L2,i e𝛾i maxn≤r≤N | Y r,i | .

(4.11)

i =1

Now, using well-known properties for moments of random variables and z ∈ Z, 0 ≤ n ≤ N: Lemma 4.1.2, we get the following inequality for   #     Ez ,n max g r, e Y r , X r  n≤r≤N

≤ L1 +

k

L1 L2,i Ez ,n e𝛾i maxn≤r≤N | Y r,i |

i =1



≤ L1 +

L1 L2,i +

i:𝛾i =0



= L1 +

i:𝛾i >0

L1 L2,i +

i:𝛾i =0



≤ L1 +



i:𝛾i >0

L1 L2,i +

i:𝛾i =0

= L1 +





&

#

L1 L2,i Ez ,n e𝛾i maxn≤r≤N | Y r,i |

 βi 𝛾i

' 𝛾i

βi

#  𝛾i β L1 L2,i Ez ,n e β i maxn≤r≤N | Y r,i | i #  𝛾i β L1 L2,i e β i | y i | M1,i i

i:𝛾i >0

L1 L2,i +

i:𝛾i =0



𝛾i β

L1 L2,i M1,ii e𝛾i | y i | .

(4.12)

i:𝛾i >0

Thus, inequality (4.10) holds with the constants M2 and M3,i , i = 1, . . . , k, which are given by relation (4.12). This completes the proof. Remark 4.1.3. The explicit formulas for the constants M2 and M3,i , i = 1, . . . , k follow from relations (4.6) and (4.12), 𝛾

M2 = L1 ,

N βi

M3,i = L1 L2,i I (𝛾i = 0) + L1 L2,i K1,i i I (𝛾i > 0) .

(4.13)

Lemma 4.1.5. Let the conditions B1 [𝛾¯] and C1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i =  1, . . . , k. Then, there exist constants 0 ≤ M2 , M3,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ((y1 , . . . , y k ), x) ∈ Zn , 0 ≤ n ≤ N: k  

    M3,i e𝛾i | y i | . Ez ,n max g(r, e Y r , X r ) ≤ M2 + n≤r≤N

(4.14)

i =1

Proof. It repeats the proof of Lemma 4.1.4. The only difference is that the condition C1 [β¯ ] is assumed to hold instead the condition C1 [β¯ ]. In this case, inequality (4.12)

4.1 Markov LPP with bounded characteristics

| 121

z ∈ Zn , 0 ≤ n ≤ N with the same constants L1 , L2,i , i = 1, . . . , k and takes place for   the constants K1,i , i = 1, . . . , k used instead of the constants K1,i , i = 1, . . . , k.  Remark 4.1.4. The explicit formulas for the constants M2 and M3,i , i = 1, . . . , k follow from relations (4.9) and (4.12), and the remarks made in the above proof,

M2 = L1 ,

  M3,i = L1 L2,i I (𝛾i = 0) + L1 L2,i (K1,i )

𝛾

N βi

i

I (𝛾i > 0) .

(4.15)

According to Lemma 4.1.4, the conditions B1 [𝛾¯] and C1 [β¯ ], which are assumed to hold with parameters 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k, imply that the condition A2 holds, and, therefore, the log-reward functions ϕ n ( y , x) are well defined by the following relation, z = ( y , x) ∈ Z, n = 0, . . . , N: for any  y , x) = ϕ n (

sup τ n ∈Mmax,n,N



Ez ,n g(τ n , e Y τn , X τ n ) .

(4.16)

Moreover, for Lemma 4.1.4 let us obtain the following upper bounds for the logZn . reward functions of American-type options for the modulated log-price process  Theorem 4.1.1. Let the conditions B1 [𝛾¯] and C1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the log-reward functions ϕ n ( y , x) satisfy the following inequalities for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N: k

  ϕ n ( y , x) ≤ M2 + M3,i e𝛾i | y i | .

(4.17)

i =1

Proof. Using Lemma 4.1.4 and the definition of the log-reward functions, we get the z = ( y , x) ∈ Z, 0 ≤ n ≤ N: following inequalities for          τn Y ϕ n (   y , x) =  sup E g ( τ , e , X ) n τ  z ,n n  τ ∈M n max,n,N k  

   ≤ Ez ,n max g(r, e Y r , X r ) ≤ M2 + M3,i e𝛾i | y i | , n≤r≤N

(4.18)

i =1

which proves the theorem. Let us now assume that the conditions B1 [𝛾¯] and C1 [β¯ ] hold with parameters 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Lemma 4.1.5 guarantees in this case that the log-reward functions ϕ n ( y , x) are defined by relation (4.16) only for  z = ( y , x) ∈ Zn , n = 0, . . . , N. Moreover, for Lemma 4.1.5 let us get the following upper bounds for the log-reward Zn. functions of American-type options for the modulated log-price process  Theorem 4.1.2. Let the conditions B1 [𝛾¯] and C1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the log-reward functions ϕ n ( y , x) satisfy the following inequalities for  z = ((y1 , . . . , y k ), x) ∈ Zn , 0 ≤ n ≤ N: k

   ϕ n ( y , x) ≤ M2 + M3,i e𝛾i | y i | . i =1

(4.19)

122 | 4 Upper bounds for option rewards Proof. It repeats the proof of Theorem 4.1.1. The only change is that Lemma 4.1.5 should be used instead of Lemma 4.1.4. This lets one write down inequality (4.18), for   z = ( y , x) ∈ Zn , 0 ≤ n ≤ N and with the constants M2 , M3,i , i = 1, . . . , k used instead of the constants M2 , M3,i , i = 1, . . . , k. As far as points  z = ( y , x) ∈ Zn , n = 0, . . . , N are concerned, there is no guarantee that expectations on the right-hand side in formulas (4.16) do exist. Moreover, if these expectations are finite, there is no guarantee that suprema penetrating these formulas are finite. Thus, the log-reward functions ϕ n ( y , x) may not be defined for such points. There exist three alternatives. First, some additional conditions should be imposed on transition probabilities P n ( z , A), in order to guarantee that formula (4.16) would well define the log-reward functions ϕ n ( y , x) for points  z = ( y , x) ∈ Zn , n = 0, 1, . . . , N. For example, in the important case of a space-skeleton approximation model, z , ·) are concentrated on finite sets of points Z∗ z) ⊆ Zn , for  z = distributions P n ( n ( y , x) ∈ Z, n = 1, . . . , N. In this case, the log-reward functions ϕ n ( y , x) are automati­ ( z = ( y , x) ∈ Z, n = 0, . . . , N. As cally finite and well defined by formula (4.16), for any  far as upper bounds for log-reward functions are concerned, they do require assuming conditions penetrating Theorems 4.1.1 or 4.1.2. Second, one can define and operate with the log-reward functions only for points  z = ( y , x) ∈ Zn , n = 0, . . . , N. Third, one can use the assumption that Z = Z0 , . . . , ZN  is a complete sequence y , x), for points  z = of phase sets such that values of the log-reward functions ϕ n ( y , x) ∈ Zn , n = 0, . . . , N, do not depend on transition probabilities P n ( z , A) for (  z = ( y , x) ∈ Zn−1 , n = 1, . . . , N. z , A) for This assumption makes it possible to change transition probabilities P n (  z = ( y , x) ∈ Zn−1, n = 1, . . . , N that do not change the values of the log-reward functions ϕ n ( y , x) for  z = ( y , x) ∈ Zn , n = 0, . . . , N. For example, one can define z , A) = I ( z ∈ A) for  z = ( y , x) ∈ Zn−1, n = 1, . . . , N. new transition probabilities P n (  In this case, obviously, Pz {Z r =  z , n ≤ r ≤ N } = 1 for  z = ( y , x) ∈ Zn , n = 0, . . . , N and, therefore, ϕ n ( y , x) = maxn≤r≤N g(n, ey , x) for  z = ( y , x) ∈ Zn , n = 0, . . . , N. Let us now give upper bounds for optimal expected rewards using the above upper bounds for reward functions. To do this, we should impose the following first-type condition of exponential mo­ Z n . This con­ ment boundedness on the initial distribution of the log-price process  dition should be assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: D1 [β¯ ]: Ee β i | Y0,i | < K2,i , i = 1, . . . , k, for some 1 < K2,i < ∞, i = 1, . . . , k . It is appropriate to comment here, why conditions of type C are referred to as moment compactness conditions, while conditions of type D are referred to as moment bound­ edness conditions. As a matter of fact, conditions of type C are moment boundedness

4.1 Markov LPP with bounded characteristics

|

123

conditions imposed on increments of log-price processes, while conditions of type D are moment boundedness conditions imposed on values of log-price processes them­ selves. The following lemma gives sufficient conditions for the condition A1 to hold and explicit upper bounds for the expectation of maximum for the pay-off process g(n, e Yn , X n ). Lemma 4.1.6. Let the conditions B1 [𝛾¯], C1 [β¯ ], and D1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exists a constant 0 ≤ M4 < ∞ such that the following inequality takes place:      (4.20) E max g(n, e Y n , X n ) ≤ M4 . 0≤ n ≤ N

Proof. Lemma 4.1.4, we get $ %   #         E max g(n, e Y n , X n ) = E E max g n, e Y n , X n  / Z0 0≤ n ≤ N 0≤ n ≤ N ⎛ ⎞ k

≤ E ⎝M2 + M3,i e𝛾i | Y0,i | ⎠ i =1

≤ M2 +



M3,i +

i:𝛾i =0

≤ M2 +



#  𝛾i β M3,i Ee β i | Y0,i | i

i:𝛾i >0

M3,i +

i:𝛾i =0



𝛾i β

M3,i K2,ii .

(4.21)

i:𝛾i >0

Relation (4.21) proves that inequality (4.20) holds with the constant M4 given in formula (4.21). Remark 4.1.5. The explicit formula for the constant M4 follows from the formulas given in Remark 4.1.3 and the last inequality in (4.21), M4 = L1 +

i:𝛾i =0

L1 L2,i +



𝛾

N βi

𝛾i β

L1 L2,i K1,i i K2,ii .

(4.22)

i:𝛾i >0

In the case, where the condition C1 [β¯ ] is used instead of the condition C1 [β¯ ], the fol­ lowing modified first-type condition of exponential moment boundedness should be assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative com­ ponents:   , i = 1, . . . , k, for some 1 ≤ K2,i < ∞, i = 1, . . . , k; D1 [β¯ ]: (a) Ee β i | Y0,i | < K2,i  (b) P{Z0 ∈ Z0 } = 1, where Z0 is the set penetrating condition C1 [β¯ ] . Lemma 4.1.7. Let the conditions B1 [𝛾¯], C1 [β¯ ], and D1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exists a constant 0 ≤ M4 < ∞ such that the following inequality takes place:      E max g(n, e Y n , X n ) ≤ M4 . (4.23) 0≤ n ≤ N

124 | 4 Upper bounds for option rewards Proof. It repeats the proof of Lemma 4.1.6. The only difference is that Lemma 4.1.5 should be used instead of Lemma 4.1.4. The relation analogous to (4.21) can be writ­   ten down, with the same constants L1 , L2,i , i = 1, . . . , k and the constants K1,i , K2,i , i = 1, . . . , k used instead of the constants K1,i , K2,i , i = 1, . . . , k. Remark 4.1.6. The explicit formula for the constant M4 follows from the formulas given in Remark 4.1.4, the last inequality in (4.21) and the remarks made in the above proof, 𝛾i 𝛾i



 N βi  L1 L2,i + L1 L2,i (K1,i ) (K2,i ) βi . (4.24) M4 = L1 + i:𝛾i =0

i:𝛾i >0

Let us assume that either conditions of Lemma 4.1.6 or Lemma 4.1.7 hold. In this case, Z n can be defined by the optimal expected reward Φ = Φ(Mmax,N ) for the log-price  the following relation: Φ=

sup τ 0 ∈Mmax,N



Eg(τ0 , e Y τ0 , X τ0 ) .

(4.25)

The following two theorems give upper bound for the optimal expected reward Φ. Theorem 4.1.3. Let the conditions B1 [𝛾¯], C1 [β¯ ], and D1 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the following inequality takes place: |Φ| ≤ M 4 .

(4.26)

Proof. It follows from the definition of the functional Φ and Lemma 4.1.6 that           |Φ| ≤ sup E g(τ 0 , e Y τ0 , X τ0 ) ≤ E max g(n, e Y n , X n ) ≤ M4 . (4.27) 0≤ n ≤ N

τ 0 ∈Mmax,N

This inequality proves the theorem. Theorem 4.1.4. Let the conditions B1 [𝛾¯], C1 [β¯ ], and D1 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the following inequality takes place: |Φ| ≤ M4 .

(4.28)

Proof. It repeats the proof of Theorem 4.1.3. The only difference is that Lemma 4.1.7 should be used instead of Lemma 4.1.6. The relation analogous to (4.27) can be written down, and the constant M4 used instead of the constant M4 .

4.1.3 Modified conditions of exponential moment compactness for Markov log-price processes with bounded characteristics The condition C1 [β¯ ], used in Lemmas 4.1.1 and 4.1.2, is based on exponential moments for absolute values for increments of log-price process. It is useful for some models, for

4.1 Markov LPP with bounded characteristics

|

125

example, for log-price processes represented by random walks, to replace this condi­ tion by an alternative condition based on moment-generating functions for log-price processes. Let us define the second-type modulus of exponential moment compactness for  n = (Y n,1 . . . , Y n,k ), for β ≥ 0, i = 1, . . . , k, components of the log-price process Y ± β( Y n+1,i − Y n,i ) . Ξ± z ,n e β ( Y · ,i , N ) = max sup E

0≤ n ≤ N − 1  z∈Z

(4.29)

The following second-type condition of exponential moment compactness based n , is assumed on moment-generating functions for increments of log-price processes Y ¯ to hold for some vector parameter β = (β 1 , . . . , β k ) with nonnegative components: E1 [β¯ ]: Ξ± β i ( Y · ,i , N ) < K 3,i , i = 1, . . . , k, for some 1 < K 3,i < ∞, i = 1, . . . , k . Lemma 4.1.8. The condition E1 [β¯ ] implies condition C1 [β¯ ] to hold. z ∈ Z, i = Proof. The following inequality takes place, for every 0 ≤ n ≤ N − 1,  1, . . . , k, Ez ,n e β i | Y n+1,i −Y n,i | = Ez ,n e β i (Y n+1,i −Y n,i ) I (Y n+1,i ≥ Y n,i ) + Ez ,n e−β i (Y n+1,i −Y n,i ) I (Y n+1,i < Y n,i ) ≤ Ez ,n e β i (Y n+1,i −Y n,i ) + Ez ,n e−β i (Y n+1,i −Y n,i ) .

(4.30)

Relation (4.30) obviously implies that the following inequality holds, for i = 1, . . . , k: − Δ β i (Y· ,i , N ) ≤ Ξ+ (4.31) β i ( Y · ,i , N ) + Ξ β i ( Y · ,i , N ) . Thus, the condition E1 [β¯ ] implies that condition C1 [β¯ ] holds. Remark 4.1.7. As follows from inequality (4.31), if condition E1 [β¯ ] holds with the con­ stants K3,i , i = 1, . . . , k, then the condition C1 [β¯ ] holds with the constants K1,i , i = 1, . . . , k given by the following formulas: K1,i = 2K3,i , i = 1, . . . , k .

(4.32)

Let Z = Z0 , . . . , ZN  be a sequence of measurable sets, and Ξ± β, Z ( Y · ,i , N ) be a modi­ fied second-type exponential moment modulus of compactness defined for β ≥ 0 by the following formula: ± β( Y n+1,i − Y n,i ) . Ξ± z ,n e β, Z ( Y · ,i , N ) = max sup E

0≤ n ≤ N − 1  z∈Z

n

(4.33)

126 | 4 Upper bounds for option rewards The following modified second-type condition of exponential moment compact­ ness is based on moment-generating functions of increments for the log-price pro­ ¯ ] and is assumed to hold for some vector  n . It generalizes the condition E1 [β cesses Y ¯ parameter β = (β 1 , . . . , β k ) with nonnegative components: E1 [β¯ ]: There exists a sequence of measurable sets Z = Z0 , . . . , ZN  such that   (a) Ξ± β i , Z ( Y · ,i , N ) < K 3,i , i = 1, . . . , k, for some 1 < K 3,i < ∞, i = 1, . . . , k; z , Zn ) = 1,  z ∈ Z n −1 , n = (b) Z is a complete sequence of phase sets, i.e., Pn ( 1, . . . , N . Lemma 4.1.9. The condition E1 [β¯ ] implies the condition C1 [β¯ ] to hold. Proof. Relation (4.30) implies analogously to (4.31) the following inequality, for i = 1, . . . , k: − Δ β i ,Z (Y· ,i , N ) ≤ Ξ+ (4.34) β i , Z ( Y · ,i , N ) + Ξ β i , Z ( Y · ,i , N ) . Thus, the condition E1 [β¯ ] implies that condition C1 [β¯ ] holds. Remark 4.1.8. As follows from inequality (4.31), if condition E1 [β¯ ] holds with the con­   stants K3,i , i = 1, . . . , k, then the condition C1 [β¯ ] holds with the constants K1,i ,i = 1, . . . , k given by the following formulas:   = 2K3,i , i = 1, . . . , k . K1,i

(4.35)

One also can replace the first-type condition of exponential moment boundedness D1 [β¯ ] by the following second-type condition of exponential moment boundedness 0 , based on moment-generating functions for the initial value of log-price processes Y ¯ assumed to hold for some vector parameter β = (β 1 , . . . , β k ) with nonnegative com­ ponents: F1 [β¯ ]: Ee±β i Y0,i < K4,i , i = 1, . . . , k, for some 1 < K4,i < ∞, i = 1, . . . , k . Lemma 4.1.10. The condition F1 [β¯ ] implies condition D1 [β¯ ] to hold. Proof. The following inequality holds for any β ≥ 0: Ee β| Y0,i | ≤ Ee+βY0,i + Ee−βY0,i .

(4.36)

The proposition of Lemma 4.1.10 obviously follows from the above inequality. Remark 4.1.9. As follows from inequality (4.36), if condition F1 [β¯ ] holds with the con­ stants K4,i , i = 1, . . . , k, then the condition D1 [β¯ ] holds with the constants K2,i , i = 1, . . . , k given by the following formulas: K2,i = 2K4,i ,

i = 1, . . . , k .

(4.37)

4.2 LPP represented by random walks

| 127

Analogously, the condition D1 [β¯ ] can be replaced by the following second-type condi­ tion of exponential moment boundedness assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: F1 [β¯ ]: (a) Ee±β i Y0,i < K4,i , i = 1, . . . , k, for some 1 < K4,i < ∞, i = 1, . . . , k; ¯] . 0 ∈ Z0 } = 1, where Z0 is the set penetrating condition D1 [β (b) P{Z Lemma 4.1.11. The condition F1 [β¯ ] implies condition D1 [β¯ ] to hold. Remark 4.1.10. As follows from inequality (5.26), if condition F1 [β¯ ] holds with the   , i = 1, . . . , k, then the condition D1 [β¯ ] holds with the constants K2,i , constants K4,i i = 1, . . . , k given by the following formulas:   = 2K4,i , i = 1, . . . , k . K2,i

(4.38)

4.2 LPP represented by random walks In this section, we get explicit upper bounds for log-reward functions and optimal expected rewards for American-type options for log-price processes represented by random walks.

4.2.1 Upper bounds for option rewards for log-price processes represented by standard random walks n = Let us consider the simplest model, where the multivariate Markov log-price Y (Y n,1 . . . , Y n,k ) is a standard random walk,  n −1 + W n , n = Y Y

n = 1, 2, . . . .

(4.39)

0 = (Y 0,1 , . . . , Y 0,k ) is a k-dimensional random vector, (b) W  n = (W n,1 , where (a) Y . . . , W n,k ), n = 1, 2, . . . , is a sequence of k-dimensional independent random vectors, 0 and the random sequence W  n , n = 1, 2, . . . , are inde­ and (c) the random vector Y pendent. In this case, the model without index component is considered. In fact, one can always to define a “virtual” constant index component X n which takes the constant value x0 for every n = 0, 1, . . . . In this model, a pay-off function g(n, ey , x) = g(n, ey ) does not depend on the index argument x. The first-type modulus of exponential moment compactness Δ β (Y· ,i , N ), ex­  n , takes the following form, for β ≥ 0, pressed in terms of components of jumps W

128 | 4 Upper bounds for option rewards i = 1, . . . , k:

Δβ (Y· ,i , N ) = max Ee β| W n+1,i | . 0≤ n ≤ N − 1

(4.40)

The condition C1 [β¯ ], which is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, takes the following simpler form: C2 [β¯ ]: Δβ i (Y· ,i , N ) < K5,i , i = 1, . . . , k, for some 1 < K5,i < ∞, i = 1, . . . , k . The condition B1 [𝛾¯], which is assumed to hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components, takes the following form: B2 [𝛾¯]: max0≤n≤N supy∈Rk L4,1 , . . . , L4,k < ∞ .

1+

| g ( n, ey)| k 𝛾i |y i | i =1 L 4,i e

< L3 , for some 0 < L3 < ∞ and 0 ≤

The condition D1 [β¯ ] does not change. The second-type modulus of exponential moment compactness for the compo­ n , takes the following form for β ≥ 0, i = 1, . . . , k: nents of the log-price process Y ± βW n+1,i Ξ± . β ( Y · ,i , N ) = max Ee

0≤ n ≤ N − 1

(4.41)

The condition E1 [β¯ ], which is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, takes the following simpler form: E2 [β¯ ]: Ξ± β i ( Y · ,i , N ) < K 6,i , i = 1, . . . , k, for some 1 < K 6,i < ∞, i = 1, . . . , k . Note that, according to Lemma 4.1.8, the condition E2 [β¯ ] implies the condition C2 [β¯ ] to hold. The conditions F1 [β¯ ] do not change. Lemmas 4.1.1, 4.1.3–4.1.11 and Theorems 4.1.1 and 4.1.3 can be reformulated with the replacement of the conditions B1 [𝛾¯], C1 [β¯ ], and E1 [β¯ ], respectively, by the condi­ tions B2 [𝛾¯], C2 [β¯ ] and E2 [β¯ ] and the constants L1 , L2,i , i = 1, . . . , k and K1,i , K3,i , i = 1, . . . , k, respectively by the constants L3 , L4,i , i = 1, . . . , k and K5,i , K6,i , i = 1, . . . , k. As was mentioned above the conditions D1 [β¯ ] and F1 [β¯ ] do not change, and, thus, the same constants K2,i , K4,i , i = 1, . . . , k should be used in the above listed lemmas and theorems. Since the pay-off function g(n, ey , x) does not depend on the index argument x, the log-reward functions ϕ n ( y , x) = ϕ n ( y ) also do not depend on the index argu­ ment x. Theorem 4.1.1 takes, according to the above remarks, the following form. Theorem 4.2.1. Let the conditions B2 [𝛾¯] and C2 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the log-reward functions ϕ n ( y) satisfy the following inequalities for  y= (y1 , . . . , y k ) ∈ Rk , 0 ≤ n ≤ N: |ϕ n ( y)| ≤ M5 +

k

i =1

M6,i e𝛾i | y i | .

(4.42)

4.2 LPP represented by random walks |

129

Remark 4.2.1. The explicit formulas for the constants M5 and M6,i , i = 1, . . . , k follow from formula (4.13) given in Remark 4.1.3, 𝛾

M5 = L3 ,

N βi

M6,i = L3 L4,i I (𝛾i = 0) + L3 L4,i K5,i i I (𝛾i > 0) .

(4.43)

Theorem 4.1.3 takes, according to the above remarks, the following form. Theorem 4.2.2. Let the conditions B2 [𝛾¯], C2 [β¯ ], and D1 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the following inequality takes place: |Φ| ≤ M 7 .

(4.44)

Remark 4.2.2. The explicit formula for the constant M7 follows from formula (4.22) given in Remark 4.1.5, M7 = L3 +

i:𝛾i =0

L3 L4,i +



𝛾

N βi

𝛾i

L3 L4,i K5,i i (K2,i ) βi .

(4.45)

i:𝛾i >0

4.2.2 Upper bounds for option rewards for price processes represented by modulated random walks n , X n ) can be introduced via the n = (Y In this model, a modulated price process Z following stochastic transition dynamic relation: ⎧ ⎨Y ˜ n ( X n −1 , U n ) , n = Y  n −1 + W  n, n = B where W (4.46) ⎩ X n = C n ( X n −1 , U n ) , n = 1, 2, . . . , 0 , X0 ) = ((Y 0,1 , . . . , Y 0,k ), X0 ) is a random vector taking values in space where (a) (Y Z, (b) U n , n = 1, 2, . . . , is a sequence of independent random variables taking val­ ues in some measurable space U with a σ-algebra of measurable subsets BU , (c) the 0 , X0 ) and the random sequence U n , n = 1, 2, . . . , are independent, random vector (Y ˜ n (x, u ) = (B ˜ n,1(x, u ), . . . , B ˜ n,k (x, u )), n = 1, 2, . . . , are vector functions, which (d) B components are measurable functions acting from the space X × U to R1 , (e) C n (x, u ), n = 1, 2, . . . , are measurable functions acting from the space X × U to the space X.  n = (W n,1 , . . . , W n,k ), n = 1, 2, . . . , where In this model, (f) the sequence of jumps W ˜ n,i (X n−1 , U n ), i = 1, . . . , k, n = 1, 2, . . . . W n,i = B In this case, a pay-off function g(n, ey , x) depends on the index argument x. The first-type modulus of exponential moment compactness Δ β ( Y · ,i , N ), ex­  n , takes the following form, for β ≥ 0, pressed in terms of components of jumps W

i = 1, . . . , k:

β| W n+1,i | Δ , β ( Y · ,i , N ) = max sup Ex,n e 0≤ n ≤ N −1 x ∈X

where Ex,n is the conditional expectation under the condition X n = x.

(4.47)

130 | 4 Upper bounds for option rewards The condition C1 [β¯ ], which is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, takes the following simpler form: C3 [β¯ ]: Δ β i ( Y · ,i , N ) < K 7,i , i = 1, . . . , k for some 1 < K 7,i < ∞, i = 1, . . . , k . The conditions B1 [𝛾¯] and D1 [β¯ ] do not change. The second-type modulus of exponential moment compactness for components n , takes the following form for β ≥ 0, i = 1, . . . , k: of the log-price process Y ± βW n+1,i . Ξ± β ( Y · ,i , N ) = max sup Ex,n e

0≤ n ≤ N −1 x ∈X

(4.48)

The condition E1 [β¯ ], which is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, takes the following form: E3 [β¯ ]: Ξ± β i ( Y · ,i , N ) < K 8,i , i = 1, . . . , k, for some 1 < K 8,i < ∞, i = 1, . . . , k . The following lemma is a particular case of Lemma 4.1.8. Lemma 4.2.1. The condition E3 [β¯ ] implies condition C3 [β¯ ] to hold. The condition F1 [β¯ ] does not change. Lemmas 4.1.1, 4.1.3–4.1.11 and Theorems 4.1.1 and 4.1.3 can be reformulated with the replacement of the conditions C1 [β¯ ], and E1 [β¯ ], respectively, by the conditions C3 [β¯ ] and E3 [β¯ ], and the constants K1,i , K3,i , i = 1. . . . , k, respectively, by the con­ stants K7,i , K8,i , i = 1. . . . , k. As was mentioned above the conditions B1 [𝛾¯], D1 [β¯ ] and F1 [β¯ ] do not change, and, thus, the same constants L1 , L2,i , i = 1, . . . , k and K2,i , K4,i , i = 1, . . . , k should be used in the above listed lemmas and theorems. Theorem 4.1.1 takes, according to the above remarks, the following form. Theorem 4.2.3. Let the conditions B1 [𝛾¯] and C3 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the log-reward functions ϕ n ( y , x) satisfy the following inequalities for  z = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N: k

  ϕ n ( y , x) ≤ M8 + M9,i e𝛾i | y i | .

(4.49)

i =1

Remark 4.2.3. The explicit formulas for the constants M8 and M9,i , i = 1, . . . , k follow from formula (4.13) given in Remark 1.4.1.3, 𝛾

N βi

M8 = L1 , M9,i = L1 L2,i I (𝛾i = 0) + L1 L2,i K7,i i I (𝛾i > 0) .

(4.50)

Theorem 4.1.3 takes, according to the above remarks, the following form. Theorem 4.2.4. Let the conditions B1 [𝛾¯], C3 [β¯ ], and D1 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the following inequality takes place: |Φ| ≤ M10 .

(4.51)

4.2 LPP represented by random walks | 131

Remark 4.2.4. The explicit formula for the constant M10 follows from formula (4.22) given in Remark 4.1.5, M10 = L1 +

i:𝛾i =0

L1 L2,i +



𝛾

N βi

𝛾i β

L1 L2,i K7,i i K2,ii .

(4.52)

i:𝛾i >0

4.2.3 Upper bounds for option rewards for price processes represented by Lévy random walks Models of log-price processes represented by random walks often appear as the result of time discretization procedures applied to the continuous time log-price processes with independent increments.  n, In this case, a natural assumption can be such that the random variables W n = 1, 2, . . . , are infinitely divisible. We can restrict consideration by univariate random variables and processes since the corresponding moment conditions used in this section are formulated in terms of one-dimensional components of jumps for the corresponding multivariate random walks. As is known, the characteristic function for an infinitely divisible random vari­ able W is given by Lévy–Khintchine representation formula, ψ W (s) = E exp{isW } ⎧ ⎪

# ⎨  1 2 2 = exp ⎪iμs − σ s + eisy − 1 − isy Π (dy) 2 ⎩ | y | 0. Using condition B3 [𝛾], we get the following inequality for 0 ≤ n ≤ N:     max g(r, e Y r , X r ) ≤ max (L5 + L5 L6 e𝛾| Y r | ) n≤r≤N

n≤r≤N

≤ L5 + L5 L6 max e𝛾| Y r | = L5 + L5 L6 e𝛾maxn≤r≤N | Y r | . n≤r≤N

(4.70)

Now, using well-known properties of moments for random variables and Lemma 4.3.1, we get the following inequality for z = (y, x) ∈ Z, 0 ≤ n ≤ N:     Ez,n max g(r, e Y r , X r ) n≤r≤N

≤ L5 + L5 L6 Ez,n e𝛾maxn≤r≤N | Y r | & '𝛾 # β β 𝛾 max n≤r≤N | Y r | 𝛾 ≤ L5 + L5 L6 Ez,n e # 𝛾 β = L5 + L5 L6 Ez,n e β maxn≤r≤N | Y r |    𝛾 ≤ L5 + L5 L6 exp A N −n (β )|y| M5 β ( ) 𝛾 A N − n ( β )𝛾 β = L5 + L5 L6 (M5 ) exp |y| . β

(4.71)

Thus, inequality (4.69) holds with the constants M12 and M13 , which are given by relation (4.71). This completes the proof. Remark 4.3.3. The constants M12 and M13 are given by the following formulas: 𝛾

Nβ

M12 = L5 , M13 = L5 L6 I (𝛾 = 0) + L5 L6 K9 I (𝛾 > 0) .

(4.72)

138 | 4 Upper bounds for option rewards Lemma 4.3.4. Let the conditions B3 [𝛾] and C4 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, there   exist the constants 0 ≤ M12 , M13 < ∞ such that the following inequalities take place for z = (y, x) ∈ Zn , 0 ≤ n ≤ N: Ez,n max |g(r, e Y r , X r )| n≤r≤N

≤

 M12

+

 M13 I (𝛾 =

(

0) +

 M13 exp

) A N − n ( β )𝛾 |y| I (𝛾 > 0) . β

(4.73)

Proof. It repeats the proof of Lemma 4.3.3. The only difference is that, in the case, where the condition C4 [β ] is assumed to hold instead of the condition C4 [β ], inequal­ ity (4.71) takes place for z ∈ Zn , 0 ≤ n ≤ N, with the same constants L5 , L6 and con­ stant K9 replacing the constant K9 .   Remark 4.3.4. The constants M12 and M13 are given by the following formulas: 𝛾

N   M12 = L5 , M13 = L5 L6 I (𝛾 = 0) + L5 L6 (K9 ) β I (𝛾 > 0) .

(4.74)

Lemma 4.3.3 implies that under conditions of this lemma, the log-reward functions ϕ n (y, x) are well defined by the following relation, for z = (y, x) ∈ Z, n = 0, . . . , N: ϕ n (y, x) =

sup τ n ∈Mmax,n,N

Ez ,n g(τ n , e Y τn , X τ n ) .

(4.75)

Moreover, for Lemma 4.3.3 let us get the following upper bounds for the log-reward functions of American-type options for the log-price process Z n . Theorem 4.3.1. Let the conditions B3 [𝛾] and C4 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, the log-reward functions ϕ n (y, x) satisfy the following inequalities for z = (y, x) ∈ Z, 0 ≤ n ≤ N: |ϕ n (y, x)| ≤ M12 + M13 I (𝛾 = 0) ) ( A N − n ( β )𝛾 |y| I (𝛾 > 0) . + M13 exp β

(4.76)

Proof. Using Lemma 4.3.3 and the definition of the log-reward functions, we get the following inequalities for z = (y, x) ∈ Z, 0 ≤ n ≤ N:       Y τn  |ϕ n (y, x)| =  sup E g ( τ , e , X ) z,n n τ n  τ ∈M n max,n,N     ≤ Ez,n max g(r, e Y r , X r ) n≤r≤N ) ( A N − n ( β )𝛾 ≤ M12 + M13 I (𝛾 = 0) + M13 exp |y| I (𝛾 > 0) , (4.77) β which proves the theorem.

4.3 Markov LPP with unbounded characteristics

| 139

Lemma 4.3.4 implies that under conditions of this lemma, the log-reward functions ϕ n (y, x) are well defined by the relation (4.75), only for z = (y, x) ∈ Zn , n = 0, . . . , N. Also, Lemma 4.3.4 let us get the following upper bounds for the log-reward func­ tions of American-type options for the modulated log-price process Z n . Theorem 4.3.2. Let the conditions B3 [𝛾] and C4 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, the log-reward functions ϕ n (y, x) satisfy the following inequalities for z = (y, x) ∈ Zn , 0 ≤ n ≤ N:   ϕ n (y, x) ≤ M  + M  I (𝛾 = 0) 12 13 ( ) A N − n ( β )𝛾  |y| I (𝛾 > 0) . (4.78) + M13 exp β Proof. It repeats the proof of Theorem 4.3.1. The only change is that Lemma 4.3.4 should be used instead of Lemma 4.3.3. This let one get inequalities analogous to   (4.77), for z = (y, x) ∈ Zn , 0 ≤ n ≤ N, with the constants M12 , M13 replacing the constants M12 , M13 . We can now get upper bounds for optimal expected rewards. To do this, we should impose the following first-type condition of exponential mo­ ment boundedness (on the initial distribution of the log-price process), assumed to hold for some β ≥ 0: D2 [β ]: E exp{A N (β )|Y0 |} < K10 , for some 1 < K10 < ∞ . The following lemma gives sufficient conditions for the condition A1 to hold and explicit upper bound for the expectation of the maximum for the pay-off process g(n, e Y n , X n ). Lemma 4.3.5. Let the conditions B3 [𝛾], C4 [β ], and D2 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, there exists a constant M14 < ∞ such that for the following inequality takes place:     E max g(n, e Y n , X n ) ≤ M14 . (4.79) 0≤ n ≤ N

Proof. If 𝛾 = 0 then inequality (4.79) holds with the constant M14 = M12 + M13 . Let 𝛾 > 0. Using Lemma 4.3.3, we get     E max g(n, e Y n , X n ) 0≤ n ≤ N $ %     = E E max g(n, e Y n , X n ) /Z0 0≤ n ≤ N )' & ( A N ( β )𝛾 |Y 0 | ≤ E M12 + M13 exp β  𝛾   ≤ M12 + M13 E exp A N (β )|Y 0 | β 𝛾 β

≤ M12 + M13 K10 .

Relation (4.80) proves that inequality (4.79) holds.

(4.80)

140 | 4 Upper bounds for option rewards Remark 4.3.5. The following formulas gives the explicit expression for constant M14 : 𝛾

Nβ

𝛾 β

M14 = L5 + L5 L6 I (𝛾 = 0) + L5 L6 K9 K10 I (𝛾 > 0) .

(4.81)

In the case, where the condition C4 [β ] is used instead the condition C4 [β ], the follow­ ing condition should replace the condition D2 [β ] and be assumed to hold for some β ≥ 0:   , for some 1 < K10 < ∞; D2 [β ]: (a) E exp{A N (β )|Y0 |} < K10 (b) P{Y0 ∈ Z0 } = 1, where Z0 is the set penetrating condition C4 [β ] . Lemma 4.3.6. Let conditions the conditions B3 [𝛾], C4 [β ], and D2 [β ] hold and 0 ≤ 𝛾 ≤  β < ∞. Then, there exists a constant M14 < ∞ such that for the following inequality takes place:  . (4.82) E max |g(n, e Y n , X n )| ≤ M14 0≤ n ≤ N

Proof. It repeats the proof of Lemma 4.3.5. The only difference is that Lemma 4.3.4 should be use instead of Lemma 4.3.3. The relation analogous to (4.80) can be written  down, with the same constants L5 , L6 and the constants K9 , K10 used instead of the constants L5 , L6 and the constants K9 , K10 .  Remark 4.3.6. The following formula gives the explicit expression for constant M14 : 𝛾

𝛾

N   β M14 = L5 + L5 L6 I (𝛾 = 0) + L5 L6 (K9 ) β (K10 ) I (𝛾 > 0) .

(4.83)

Let us assume that either conditions of Lemma 4.3.5 or Lemma 4.3.6 hold. In this case, the optimal expected reward Φ = Φ(Mmax,N ) for the log-price Z n can be defined by the following relation: Φ=

sup τ 0 ∈Mmax,N

Eg(τ0 , e Y τ0 , X τ0 ) .

(4.84)

The following two theorems give explicit upper bounds for the optimal expected reward Φ. Theorem 4.3.3. Let the conditions B3 [𝛾], C4 [β ], and D2 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, the following inequality takes place: |Φ| ≤ M14 .

(4.85)

Proof. The following inequality follows directly from the definition of the functional Φ and Lemma 4.3.5:         |Φ| ≤ sup E g(τ 0 , e Y τ0 , X τ0 ) ≤ E max g(n, e Y n , X n ) ≤ M14 . (4.86) τ 0 ∈Mmax,N

This inequality proves the theorem.

0≤ n ≤ N

4.3 Markov LPP with unbounded characteristics

| 141

Theorem 4.3.4. Let the conditions B3 [𝛾], C4 [β ], and D2 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, the following inequality takes place:  |Φ| ≤ M14 .

(4.87)

Proof. It repeats the proof of Theorem 4.3.3. The only difference is that Lemma 4.3.6 should be used instead of Lemma 4.3.5. The relation analogous to (4.86) can be written  down, with and the constant M14 used instead of the constant M14 .

4.3.3 Modified conditions of exponential moment compactness for univariate modulated Markov log-price processes with unbounded characteristics The condition C4 [β ] used in Lemmas 4.3.1 and 4.3.2 is based on exponential moments for absolute values of increments for the log-price process Z n . In some case, it is use­ ful to replace this condition by an alternative condition based on moment-generating functions. Let us define a second-type A-modulus of exponential moment compactness for the log-price process Y n , for β ≥ 0, Ez,n e±A N −n−1 (β)(Y n+1 −Y n ) . 0≤ n ≤ N −1 z =( y,x )∈Z e A(A N −n−1(β))| y|

Ξ± β,A ( Y · , N ) = max

sup

(4.88)

The following second-type condition of exponential moment compactness based on moment-generating functions of increments for the log-price process Y n , is as­ sumed to hold for some β ≥ 0: E4 [ β ] : Ξ ± β,A ( Y · , N ) < K 11 , for some 1 < K 11 < ∞ . Lemma 4.3.7. The condition E4 [β ] implies condition C4 [β ] to hold. Proof. The following inequality takes place, for every 0 ≤ n ≤ N − 1, z ∈ Z, i = 1, . . . , k, and β ≥ 0: Ez,n e A N −n−1 (β)| Y n+1 −Y n | = Ez,n e A N −n−1 (β)(Y n+1 −Y n ) I (Y n+1 ≥ Y n ) + Ez,n e−A N −n−1 (β)(Y n+1 −Y n ) I (Y n+1 < Y n ) ≤ Ez,n e A N −n−1 (β)(Y n+1 −Y n ) + Ez,n e−A N −n−1 (β)(Y n+1 −Y n ) .

(4.89)

Relation (4.89) obviously implies the following inequality: − Δ β,A (Y· , N ) ≤ Ξ+ β,A ( Y · , N ) + Ξ β,A ( Y · , N ) .

Thus the condition E4 [β ] implies that condition C4 [β ] holds.

(4.90)

142 | 4 Upper bounds for option rewards Remark 4.3.7. As follows from inequality (4.31), if condition E4 [β ] holds with the con­ stant K11 , then condition C4 [β ] holds with the constant K9 given by the following for­ mula: (4.91) K9 = 2K11 . Let Z = Z0 , . . . , ZN  be a sequence of measurable sets and Ξ± β,A, Z( Y · , N ) be the mod­ ified second-type A-modulus of exponential moment compactness, defined by the fol­ lowing formula for β ≥ 0: Ez,n e±A N −n−1 (β)(Y n+1 −Y n ) . 0≤ n ≤ N −1 z =( y,x )∈Zn e A(A N −n−1(β))| y|

Ξ± β,A, Z( Y · , N ) = max

sup

(4.92)

The following modified second-type condition of exponential moment compactness is based on moment-generating functions of increments for the log-price process Y n . It generalizes the condition E4 [β ] and is assumed to hold for some β ≥ 0: E4 [β ]: There exists a sequence of sets Z = Z0 , . . . , ZN  such that   (a) Ξ± β,A, Z( Y · , N ) < K 11 , for some 1 < K 11 < ∞; (b) Z is a complete sequence of phase sets, i.e., Pn (z, Zn ) = 1, z ∈ Zn−1, n = 1, . . . , N . Lemma 4.3.8. The condition E4 [β ] implies condition C4 [β ] to hold. Proof. Relation (4.89) implies analogously to (4.90) the following inequality: − Δ β,A,Z(Y· , N ) ≤ Ξ+ β,A, Z( Y · , N ) + Ξ β,A, Z( Y · , N ) .

(4.93)

Thus, the condition E4 [β ] implies that condition C4 [β ] holds. Remark 4.3.8. As follows from inequality (4.31), if condition E4 [β ] holds with the  constant K11 , then condition C4 [β ] holds with the constant K9 given by the following formula:  . (4.94) K9 = 2K11 Similarly, one can replace condition D2 [β ] by the following second-type condition of exponential moment boundedness based on moment-generating functions for the ini­ 0 and assumed to hold for some β ≥ 0: tial value of log-price processes Y ± A N (β)Y 0 F2 [β ]: Ee < K12 , for some 1 < K12 < ∞ . Lemma 4.3.9. The condition F2 [β ] implies condition D2 [β ] to hold. Remark 4.3.9. If the condition F2 [β ] holds with the constant K12 , then the condition D2 [β ] holds with the constant K10 given by the following formula: K10 = 2K12 .

(4.95)

4.3 Markov LPP with unbounded characteristics

| 143

Analogously, the condition D2 [β ] can be replaced by the following condition assumed to hold for some β ≥ 0:   F2 [β ]: (a) Ee±A N (β)Y0 < K12 , for some 1 < K12 < ∞; (b) P{Y0 ∈ Z0 } = 1, where Z0 is the set penetrating condition D2 [β ] . Lemma 4.3.10. The condition F2 [β ] implies condition D2 [β ] to hold.  , then the condition Remark 4.3.10. If the condition F2 [β ] holds with the constant K12   D2 [β ] holds with the constant K10 given by the following formula:   = 2K12 . K10

(4.96)

4.3.4 Upper bounds for exponential moments of maxima for multivariate modulated Markov log-price processes with unbounded characteristics n , X n ) with the phase space We consider a modulated Markov log-price process  Z n = (Y ¯ Z = Rk × X. Our goal is to generalise the condition C1 [β ].  (β¯ ) = (A1 (β¯ ), Let us denote by Ak , the class of measurable, vector functions A . . . , A k (β¯ )) defined for β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k , whose components A i (β¯ ), i = 1, . . . , k are real-valued functions, nondecreasing in every argument β i , ¯ ) = 0, i = 1, . . . , k. i = 1, . . . , k and such that A i (0  (β¯ ) ∈ Ak generates a sequence of functions A  n (β¯ ) = (A n,1 (β¯ ), Any function A . . . , A n,k (β¯ )), n = 0, 1, . . . from the class Ak that is defined by the following recur­ rence relation, for any β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k: ⎧ ⎨ β¯ for n = 0 ,  n (β¯ ) = A (4.97) ⎩A  (A  n−1 (β¯ )) for n = 1, 2, . . . .  n−1 (β¯ ) + A

Note that, by the definition, A0,j (β¯ ) ≤ A1,j (β¯ ) ≤ · · · for every β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k and j = 1, . . . , k.  -modulus of exponential moment compactness for Let us define the first-type A ¯  the log-price process Y n , for β = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k,  Δ β, ¯ A  (Y· , N ) =

max

sup

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Z

Ez ,n e

k

e

j =1

k

j =1

A N −n−1,j ( β¯ )| Y n+1,j − Y n,j |  N −n−1 ( β¯ ))| y j | A j (A

.

(4.98)

Remind that we use the notations Pz ,n and Ez ,n for conditional probabilities and expectations under the condition  Zn =  z.

144 | 4 Upper bounds for option rewards Let us introduce vectors β¯ i = (0, . . . , 0, β i , 0, . . . , 0), β i ≥ 0, i = 1, . . . , k (the vector β¯ i has the ith component equals to β i and other components equal to 0). If to denote β i,j = β j I (j = i), j = 1, . . . , k, then the vectors β¯ i = (0, . . . , 0, β i , 0, . . . , 0) = (β i,1 , . . . , β i,k ), i = 1, . . . , k. We use the following first-type condition of exponential moment compactness for  n , which is assumed to hold for some vectors β¯ = (β 1 , . . . , β k ) the log-price process Y with nonnegative components: · , N ) < K13,i , i = 1, . . . , k, for some 1 < K13,i < ∞, i = 1, . . . , k . C5 [β¯ ]: Δ β¯ i , A (Y The following lemma gives explicit upper bounds for conditional moments of the max­ n . imum of absolute values for the log-price process Y Lemma 4.3.11. Let the condition C5 [β¯ ] hold. Then there exist the constants 0 ≤ M15,i < ∞, i = 1, . . . , k such that the following inequality takes place for  z = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N and i = 1, . . . , k: ⎫ ⎧ % k ⎬ ⎨

 Ez ,n exp β i max |Y r,i| ≤ M15,i exp ⎩ A N −n,j β¯ i |y j |⎭ . $

n≤r≤N

(4.99)

j =1

Proof. Note that Δ0¯ , A (Y· , N ) ≡ 1 and, thus, the relation penetrating condition C5 [β¯ ] for a given i = 1, . . . , k holds automatically if β i = 0. Also, the expectation on the left hand side of inequality (4.99) equal 1, while the expression on the right-hand side of this inequality equal M15,i , if β i = 0. Thus, inequality (4.99) holds with the constant M15,i = 1 for i such that β i = 0. Therefore, we should prove (4.99) only for i such that β i > 0. Let us use the following inequality, for β i ≥ 0, i = 1, . . . , k: ⎧ ⎛ ⎞⎫ % N −1 ⎬ ⎨

|Y r+1,i − Y r,i |⎠ . exp β i max |Y r,i | ≤ exp ⎩β i ⎝|Y n,i | + ⎭ n≤r≤N $

(4.100)

r=n

Z n and then recurrently condition Using Markov property of the log-price process  C5 [β¯ ], we get the following inequalities, for  z ∈ Z, 0 ≤ n ≤ N, i = 1, . . . , k:

4.3 Markov LPP with unbounded characteristics

⎧ ⎨

⎛

Ez ,n exp ⎩ β i ⎝|Y n,i | +

= Ez ,n exp

⎧ k ⎨

⎩

N −1

r=n

| 145

⎞⎫ ⎬ |Y r+1,i − Y r,i |⎠ ⎭

⎛

β i,j ⎝|Y n,j | +

N −1

r=n

j =1

⎞⎫ ⎬ |Y r+1,j − Y r,j|⎠ ⎭

⎧ ⎧ ⎫ ⎞⎫ ⎛ k N −1 ⎬* ⎨ ⎨

⎬

⎝ = Ez ,n E exp β |Y r+1,j − Y r,j|⎠ F N −1 i,j |Y n,j | + ⎭ ⎩ ⎩ ⎭ r=n

j =1

⎛ = Ez ,n ⎝exp

⎧ k ⎨

⎩

≤ Ez ,n ⎝exp

× exp

≤ Ez ,n exp

≤ Ez ,n exp

≤ Ez ,n exp

⎩

j =1

⎧ k ⎨

⎩

β i,j ⎝|Y n,j | +

j =1

N

−2 r=n

⎞⎫ ⎬ |Y r+1,j − Y r,j|⎠ ⎭

(4.101)

⎫⎞ ⎬  ¯ A j β i |Y N −1,j |⎭⎠ K13,i 

⎛

β i,j + A j (β¯ i ) ⎝|Y n,j | +

N

−2 r=n

j =1

⎧ k ⎨

⎩

⎛

⎧ k ⎨

⎧ k # ⎨

⎩

j =1

⎞⎫ ⎬ |Y r+1,j − Y r,j|⎠ ⎭

⎫⎞ ⎬ β i,j |Y N,j − Y N −1,j|/ Z N −1 ⎭ ⎠

j =1

⎩

N

−2 r =1

⎧ k ⎨

⎧ k ⎨

⎩

β i,j ⎝|Y n,j | +

j =1

× E exp ⎛

⎛

⎞⎫ ⎬ |Y r+1,j − Y r,j|⎠ K13,i ⎭

⎛ ⎞⎫ N

−2 ⎬  A1,j β¯ i ⎝|Y n,j | + |Y r+1,j − Y r,j|⎠ K13,i ⎭ r=n

⎛

A2,j (β¯ i ) ⎝|Y n,j| +

N

−3 r=n

j =1

⎞⎫ ⎬ 2 |Y r+1,j − Y r,j|⎠ K13,i ⎭

⎫ ⎧ k ⎬ ⎨

N −n ¯ ≤ · · · ≤ Ez ,n exp A N − n,j ( β i )|Y n,j | K 13,i ⎭ ⎩ j =1

= exp

⎧ k ⎨

⎩

j =1

⎫ ⎬ N −n A N −n,j(β¯ i )|y j |⎭ K13,i .

146 | 4 Upper bounds for option rewards Finally, using relations (4.100) and (4.101) we get the following inequalities, for  z ∈ Z, 0 ≤ n ≤ N, i = 1, . . . , k: $ % Ez ,n exp β i max |Y r,i | n≤r≤N ⎧ ⎧ ⎛ ⎞⎫ N −1 ⎨ ⎨ ⎬

⎠ ≤ Ez ,n exp β i ⎝|Y n,i | + | Y − Y | r + 1,i r,i ⎩ ⎩ ⎭ r=n

⎫ ⎧ k ⎬ ⎨

N −n ≤ exp A N −n,j(β¯ i )|y j ||⎭ K13,i . ⎩

(4.102)

j =1

The remarks made at the beginning of the proof and inequality (4.102) implies that N . inequality (4.99) holds with the constant K13,i Remark 4.3.11. The constants M15,i , i = 1, . . . , k are given by the following formulas: N M15,i = K13,i .

(4.103)

Moreover, inequalities (4.102) give upper bounds for exponential moments of the max­ ima for the log-price processes Y n,i over discrete time intervals [n, N ], with the con­ N −n stants K13,i depending on n. Let Z = Z0 , . . . , ZN  be a sequence of measurable sets. Let us define the first-­ · , N ) by the  -modulus of exponential moment compactness Δ ¯ A , Z (Y type modified A β, ¯ following formula, for β = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k:  Δ β, ¯ A  ,Z (Y· , N ) =

max

sup

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Zn

Ez ,n e

k

e

j =1

k

j =1

A N −n−1,j ( β¯ )| Y n+1,j − Y n,j |  N −n−1 ( β¯ ))| y j | A j (A

.

(4.104)

The following modified first-type condition of exponential moment compactness ¯ ], and should be assumed  n generalizes the condition C5 [β for the log-price process Y to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: C5 [β¯ ]: There exists a sequence of measurable sets Z = Z0 , . . . , ZN  such that   · , N ) < K13,i (a) Δ β¯ i , A ,Z (Y , i = 1, . . . , k, for some 1 < K13,i < ∞, i = 1, . . . , k; z , Zn ) = 1,  z ∈ Z n −1 , n = (b) Z is a complete sequence of phase sets, i.e., Pn ( 1, . . . , N . Lemma 4.3.12. Let the condition C5 [β¯ ] holds. Then there exist the constants 0 ≤  M15,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ((y1 , . . . , y k ), x) ∈ Zn , 0 ≤ n ≤ N and i = 1, . . . , k: ⎫ ⎧ $ % k ⎬ ⎨

  (4.105) Ez ,n exp β i max |Y r,i | ≤ M15,i exp ⎩ A N −n,j β¯ i |y j |⎭ . n≤r≤N j =1

4.3 Markov LPP with unbounded characteristics

| 147

Proof. Analogously to the proof of Lemma 4.3.10. Lemma 4.1.2 implies that probability z ∈ Zn , 0 ≤ n ≤ r ≤ N − 1. The condition C5 [β¯ ] implies that ex­ Pz ,n {Z r ∈ Zr } = 1, for  k   pectation Ez ,r exp{ j=1 A N −r−1(β¯ i )|Y r+1,j−Y r,j |} ≤ K13,i exp{ kj=1 A j (A N −r−1(β¯ i ))|y j |}, z = ((y − 1, . . . , y k ), x) ∈ Zr , r = 0, . . . , N − 1. For these remarks let for i = 1, . . . , k,  one write down relations analogous to (4.101) and (4.102), for  z ∈ Zn , 0 ≤ n ≤ N and  with the constants K13,i , i = 1, . . . , k replacing the constants K13,i , i = 1, . . . , k. This completes the proof.  Remark 4.3.12. The constants M15,i , i = 1, . . . , k are given by the following formulas:   = (K13,i )N . M15,i

(4.106)

Moreover, inequality (4.102), modified according to remarks made in the above proof, give upper bounds for exponential moments of the maxima for the log-price processes  Y n,i over discrete time intervals [n, N ], with the constants (K13,i )N −n depending on n.

4.3.5 Upper bounds for rewards of multivariate modulated Markov log-price processes with unbounded characteristics The following lemma gives sufficient conditions for the condition A2 to hold and ex­ plicit upper bounds for conditional expectations of the maximal absolute values for the pay-off process g(n, e Yn , X n ). Lemma 4.3.13. Let the conditions B1 [𝛾¯] and C5 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exist the constants 0 ≤ M16 , M17,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N:  

   M17,i Ez ,n max g(r, e Y r , X r ) ≤ M16 + n≤r≤N

i:𝛾i =0

⎧⎛ ⎞ ⎫ k ⎨

 𝛾⎬ + M17,i exp ⎩⎝ A N −n,j β¯ i |y j |⎠ i ⎭ . βi i:𝛾 >0 j =1

(4.107)

i

Proof. Using the condition B1 [𝛾¯] we get for 0 ≤ n ≤ N, ⎛ ⎞ k  

  r Y 𝛾i | Y r,i | ⎠ ⎝ max g(r, e , X r ) ≤ max L1 + L1 L2,i e n≤r≤N

n≤r≤N

≤ L1 +

k

i =1

= L1 +

k

i =1

i =1

L1 L2,i max e𝛾i | Y r,i | n≤r≤N

L1 L2,i e𝛾i maxn≤r≤N | Y r,i | .

(4.108)

148 | 4 Upper bounds for option rewards Using relation (4.108) and well-known properties of moments for random vari­ ables and Lemma 4.3.11, we get the following inequality, for  z ∈ Z, 0 ≤ n ≤ N:      Ez ,n max g(r, e Y r , X r ) n≤r≤N

≤ L1 +

k

L1 L2,i Ez ,n e𝛾i maxn≤r≤N Y r,i

i =1



≤ L1 +

L1 L2,i +

i:𝛾i =0



= L1 +

i:𝛾i >0

≤ L1 +

i:𝛾i =0

+

i:𝛾i >0

&

#

L1 L2,i Ez ,n e

𝛾i max n≤r≤N Y r,i

 βi

' 𝛾i

𝛾i

βi

#  𝛾i β L1 L2,i Ez ,n e β i maxn≤r≤N Y r,i i

i:𝛾i >0

L1 L2,i

⎫⎞ 𝛾i ⎧ k ⎬ βi ⎨

L1 L2,i ⎝M15,i exp ⎩ A N −n,j(β¯ i )|y j |⎭⎠



= L1 +



L1 L2,i +

i:𝛾i =0





⎛

j =1

L1 L2,i

i:𝛾i =0

⎧⎛ ⎞ ⎫ k ⎨

𝛾⎬ + L1 L2,i M15,i exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ . βi i:𝛾 >0 j =1

𝛾i βi

(4.109)

i

Thus, inequality (4.107) holds with the constants M16 and M17,i , i = 1, . . . , k, which are given by relation (4.109). This completes the proof. Remark 4.3.13. The explicit formulas for the constants M16 and M17,i , i = 1, . . . , k follow from formulas (4.6) and (4.109), 𝛾

M16 = L1 ,

N βi

M17,i = L1 L2,i I (𝛾i = 0) + L1 L2,i K13,ii I (𝛾i > 0) .

(4.110)

Lemma 4.3.14. Let the conditions B1 [𝛾¯] and C5 [β¯ ] hold   1, . . . , k. Then, there exist the constants 0 ≤ M16 , M17,i
0

 M17,i

⎧⎛ ⎞ ⎫ k ⎨

𝛾⎬ exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ . βi j =1

(4.111)

Proof. It repeats the proof of Lemma 4.3.13. The only difference is that, in the case, where the condition C5 [β¯ ] is assumed to hold instead of the condition C5 [β¯ ], inequal­ z ∈ Zn , 0 ≤ n ≤ N, with the same constants L1 , L2,i , ity (4.109) takes place for   i = 1, . . . , k and the constants K13,i , i = 1, . . . , k replacing the constants K13,i , i = 1, . . . , k.

4.3 Markov LPP with unbounded characteristics

| 149

  and M17,i , i = 1, . . . , k Remark 4.3.14. The explicit formulas for the constants M16 follow from formulas (4.110) and the remarks made in the above proof,  M16 = L1 ,

  M17,i = L1 L2,i I (𝛾i = 0) + L1 L2,i (K13,i )

𝛾

N βi

i

I (𝛾i > 0) .

(4.112)

Lemma 4.3.12 implies that under conditions of this lemma, the log-reward functions y , x) are well defined by the following relation, for  z = ( y , x) ∈ Z, n = 0, . . . , N: ϕ n ( ϕ n ( y , x) =

sup τ n ∈Mmax,n,N



Ez ,n g(τ n , e Y τn , X τ n ) .

(4.113)

Moreover, for Lemma 4.3.13 let us obtain the following upper bounds for the logZn. reward functions of American-type options for the log-price process  Theorem 4.3.5. Let the conditions B1 [𝛾¯] and C5 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the log-reward functions ϕ n ( y , x) satisfy the following inequalities for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N:

  ϕ n ( y , x) ≤ M16 + M17,i i: 𝛾i =0

⎧⎛ ⎞ ⎫ k ⎨

𝛾⎬ + M17,i exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ . βi i: 𝛾 >0 j =1

(4.114)

i

Proof. Using Lemma 4.3.13 and the definition of the log-reward functions, we get the z = ( y , x) ∈ Z, 0 ≤ n ≤ N: following inequalities for    #        Y ϕ n ( y , x ) =  sup Ez ,n g τ n , e τn , X τ n  τ ∈M  n max,n,N     ≤ Ez ,n max g(r, e Y r , X r ) n≤r≤N ⎧⎛ ⎞ ⎫ k ⎨



𝛾⎬ ≤ M16 + M17,i + M17,i exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ , (4.115) βi i: 𝛾 =0 i: 𝛾 >0 j =1 i

i

which proves the theorem. Lemma 4.3.14 implies that, under conditions of this lemma, the log-reward functions ϕ n ( y , x) are well defined by relation (4.113), only for  z = ( y , x) ∈ Zn , n = 0, . . . , N. Also, for Lemma 4.3.14 let us get upper bounds for the log-reward functions of Zn. American-type options for the modulated log-price  Theorem 4.3.6. Let the conditions B1 [𝛾¯] and C5 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the log-reward functions ϕ n ( y , x) satisfy the following inequalities for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Zn , 0 ≤ n ≤ N:

    ϕ n ( y , x) ≤ M16 + M17,i i: 𝛾i =0

+

i: 𝛾i >0

⎧⎛ k ⎨

 M17,i exp ⎩⎝

⎫ ⎬ 𝛾 i A N −n,j(β¯ i )|y j |⎠ ⎭ . βi j =1 ⎞

(4.116)

150 | 4 Upper bounds for option rewards Proof. It repeats the proof of Theorem 4.3.5. The only difference is that Lemma 4.3.14 should be used instead of Lemma 4.3.13. This let one write down the relation analogous   to (4.115), for  z = ( y , x) ∈ Zn , 0 ≤ n ≤ N and with the constants M16 , M17,i ,i =   1, . . . , k replacing the constants M16 , M17,i , i = 1, . . . , k. We can now get upper bounds for optimal expected rewards of American-type options for the modulated log-price process  Zn . To do this, we should impose the following first-type condition of exponential mo­ Z n , for some vec­ ment boundedness on the initial distribution of the log-price process  ¯ tor β = (β 1 , . . . , β k ) with nonnegative components: k D3 [β¯ ]: E exp{ j=1 A N,j (β¯ i )|Y0,j |} < K14,i , i = 1, . . . , k, for some 1 < K14,i < ∞, i = 1, . . . , k . The following lemma gives sufficient conditions for the condition A1 to hold and explicit upper bounds for the expectation of the maximum for the pay-off process g(n, e Yn , X n ). Lemma 4.3.15. Let the conditions B1 [𝛾¯], C5 [β¯ ], and D3 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exists a constant 0 ≤ M18 < ∞ such that the following inequality takes place:      (4.117) E max g(n, e Y n , X n ) ≤ M18 . 0≤ n ≤ N

Proof. Using Lemma 4.3.13, we get      E max g(n, e Y n , X n ) 0≤ n ≤ N $ %  #     = E E max g n, e Y n , X n  / Z0 0≤ n ≤ N

⎧⎛ ⎞ k ⎨

 𝛾 M17,i + M17,i E exp ⎩⎝ A N −n,j β¯ i |Y0,j |⎠ i } ≤ M16 + β i i:𝛾 =0 i: 𝛾 >0 j =1



i

≤ M16 + ≤ M16 +



i



M17,i +

i:𝛾i =0

i:𝛾i >0





M17,i +

i:𝛾i =0

⎛

⎧ k ⎨

M17,i ⎝E exp ⎩

j =1

⎫⎞ 𝛾i ⎬ βi A N −n,j(β¯ i )|Y0,j |⎭⎠

𝛾i βi

M17,i K14,i .

(4.118)

i:𝛾i >0

Relation (4.118) proves that inequality (4.117) holds with the constant M18 given in (4.118). Remark 4.3.15. The explicit formula for the constant M18 follows from the last in­ equality in (4.118), M18 = L1 +

i:𝛾i =0

L1 L2,i +

i:𝛾i >0

𝛾

N βi

𝛾i β

i L1 L2,i K13,ii K14,i .

(4.119)

4.3 Markov LPP with unbounded characteristics

| 151

In the case, where the conditions C5 [β¯ ] and D3 [β¯ ] are used instead of the conditions C5 [β¯ ] and D3 [β¯ ], the following condition should be assumed to hold for some vector β¯ = (β 1 , . . . , β k ) with nonnegative components: k   , i = 1, . . . , k, for some 1 < K14,i < ∞, D3 [β¯ ]: (a) E exp{ j=1 A N,j (β¯ i )|Y0,j |} < K14,i i = 1, . . . , k; 0 ∈ Z0 } = 1, where Z0 is the set penetrating condition C5 [β ] . (b) P{Z Lemma 4.3.16. Let the conditions B1 [𝛾¯], C5 [β¯ ], and D3 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i <  ∞, i = 1, . . . , k. Then, there exists a constant 0 ≤ M18 < ∞ such that the following inequality takes place:       E max g(n, e Y n , X n ) ≤ M18 . (4.120) 0≤ n ≤ N

Proof. It repeats the proof of Lemma 4.3.15. The only difference is that Lemma 4.3.14 should be used instead of Lemma 4.3.13. This let one write down the relation analogous   to (4.118), for  z = ( y , x) ∈ Zn , 0 ≤ n ≤ N and with the constants M16 , M17,i ,i =   1, . . . , k replacing the constants M16 , M17,i , i = 1, . . . , k.  Remark 4.3.16. The explicit formula for the constant M18 follows from inequality (4.119) given in Remark 4.3.11 and the remarks made in the above proof, 𝛾 𝛾i



N i    = L1 + L1 L2,i + L1 L2,i (K13,i ) βi (K14,i ) βi . (4.121) M18

i:𝛾i =0

i:𝛾i >0

Let us assume that either conditions of Lemma 4.3.15 or Lemma 4.3.16 hold. In this Z n can be defined case, the optimal expected reward Φ = Φ(Mmax,N ) for the log-price  by the following relation: Φ=

sup τ 0 ∈Mmax,N



Eg(τ0 , e Y τ0 , X τ0 ) .

(4.122)

The following two theorems give upper bounds for the optimal expected reward Φ = Φ(Mmax,N ). Theorem 4.3.7. Let the conditions B1 [𝛾¯], C5 [β¯ ], and D3 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the following inequality takes place: |Φ| ≤ M18 .

Proof. As follows from the definition of the functional Φ and Lemma 4.3.15,           |Φ| ≤ sup E g(τ 0 , e Y τ0 , X τ0 ) ≤ E max g(n, e Y n , X n ) ≤ M18 . τ 0 ∈Mmax,N

0≤ n ≤ N

(4.123)

(4.124)

This inequality proves the theorem. Theorem 4.3.8. Let the conditions B1 [𝛾¯], C5 [β¯ ], and D3 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, the following inequality takes place:  |Φ| ≤ M18 .

(4.125)

152 | 4 Upper bounds for option rewards Proof. It repeats the proof of Theorem 4.1.7. The only difference is that Lemma 4.3.16 is used instead of Lemma 4.3.15. The relation analogous to (4.124) can be written down,  with the constant M18 replacing the constant M18 .

4.3.6 Modified conditions of exponential moment compactness for multivariate Markov log-price processes with unbounded characteristics  -modulus of exponential moment compactness the Let us define a second-type A  n , for vectors β¯ = (β 1 , . . . , β k ) with nonnegative components and log-price process Y vectors ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , where Ik = {¯ı k : ı j = +, −, j = 1, . . . , k }. k  Ξ β, ¯ A  (Y· , N )

¯ı

=

max

sup

Ez ,n e

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Z

k

e

¯

j =1 ı j A N −n−1,j ( β)( Y n+1,j − Y n,j )

k

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(4.126)

The following second-type condition of exponential moment boundedness based on moment-generating functions of increments for the multivariate log-price process n is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative Y components: ¯ı k · , N ) < K15,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k, for some 1 < K15,i < ∞, E5 [β¯ ]: Ξ β¯ , A (Y i i = 1, . . . , k . Lemma 4.3.17. The condition E5 [β¯ ] implies condition C5 [β¯ ] to hold. Proof. Let us introduce random indicator variables I (ı j ), j = 1, . . . , k, n = 0, . . . , N − 1 in the following way: ⎧ ⎨I (Y n+1,j − Y n,j ≥ 0) if ı j = +, (4.127) I (ı j ) = ⎩ I (Y − Y < 0) if ı = −. n +1,j

n,j

j

The following inequality takes place, for every 0 ≤ n ≤ N − 1,  z ∈ Z, i = 1, . . . , k: Ez ,n e =

k

j =1

A N −n−1,j ( β¯ i )| Y n+1,j − Y n,j |



Ez ,n e

¯ı k ∈Ik

≤



Ez ,n e

k

¯

j =1 ı j A N −n−1,j ( β i )( Y n+1,j − Y n,j )

k 

I (ı j )

j =1

k

¯

j =1 ı j A N −n−1,j ( β i )( Y n+1,j − Y n,j )

.

(4.128)

¯ı k ∈Ik

Relation (4.128) obviously implies the following inequalities, for i = 1, . . . , k:

¯ı · , N ) ≤ · , N ) . Ξ β¯k , A (Y (4.129) Δ β¯ i , A (Y ¯ı k ∈Ik

i

4.3 Markov LPP with unbounded characteristics

| 153

Thus, the condition E5 [β¯ ] implies that condition C5 [β¯ ] holds. Remark 4.3.17. As follows from inequality (4.129), if condition E5 [β¯ ] holds with the constants K15,i , i = 1, . . . , k, then the condition C5 [β¯ ] holds with the constants K13,i , i = 1, . . . , k given by the following formulas: K13,i = 2k K15,i ,

i = 1, . . . , k .

(4.130)

Let Z = Z0 , . . . , ZN  be a sequence of measurable sets. Let us also introduce a sec­  -modulus of exponential moment compactness, defined for vec­ ond-type modified A ¯ tor β = (β 1 , . . . , β k ) with nonnegative components and vectors ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , where Ik = {¯ı k : ı j = +, −, j = 1, . . . , k }. ¯ı k  Ξ β, ¯ A  ,Z (Y· , N )

=

max

Ez ,n e

sup

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Zn

k

¯

j =1 ı j A N −n−1,j ( β )( Y n+1,j − Y n,j )

e

k

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(4.131)

The following modified second-type condition of exponential moment compact­ ness is based on moment-generating functions of increments of log-price processes Y n . This condition generalizes the condition E5 [β¯ ]. It is assumed to hold for some vec­ tor parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: E5 [β¯ ]: There exists a sequence of sets Z = Z0 , . . . , ZN  such that ¯ı   (a) Ξ β¯k , A ,Z (Y· , N ) < K15,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k, for some 1 < K15,i < i ∞, i = 1, . . . , k; (b) Z is a complete sequence of phase sets, i.e., Pn (z, Zn ) = 1, z ∈ Zn−1 , n = 1, . . . , N . Lemma 4.3.18. The condition E5 [β¯ ] implies condition C5 [β¯ ] to hold. Proof. Relation (4.128) implies analogously to (4.129), the following inequality, for i = 1, . . . , k:

¯ı · , N ) ≤ · , N ) . Δ β¯ i , A ,Z (Y Ξ β¯k , A ,Z (Y (4.132) ¯ı k ∈Ik

i

Thus, the condition E5 [β¯ ] implies that condition C5 [β¯ ] holds. Remark 4.3.18. As follows from inequality (4.129), if condition E5 [β¯ ] holds with the   constants K15,i , i = 1, . . . , k, then the condition C5 [β¯ ] holds with the constants K13,i , i = 1, . . . , k given by the following formulas:   K13,i = 2k K15,i ,

i = 1, . . . , k .

(4.133)

Similarly, one can replace condition D3 [β¯ ] by the following second-type condition of exponential moment boundedness based on moment-generating functions for the ini­ tial value of log-price process. This condition should be assumed to hold for some vec­ tor parameter β¯ = (β 1 , . . . , β k ) with nonnegative components:

154 | 4 Upper bounds for option rewards  F3 [β¯ ]: E exp{ kj=1 ı j A N,j (β¯ i )Y0,j } < K16,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k for some 1 < K16,i < ∞, i = 1, . . . , k .

Lemma 4.3.19. The condition F3 [β¯ ] implies condition D3 [β¯ ] to hold. Proof. It follows from the following inequality: k k

¯ ¯ Ee j=1 ı j A N,j (β i )Y0,j . Ee j=1 A N,j (β i )| Y0,j | ≤

(4.134)

¯ı k ∈Ik

Remark 4.3.19. If the condition F3 [β¯ ] holds with constants K16,i , i = 1, . . . , k, then condition D3 [β¯ ] holds with the constants K14,i , i = 1, . . . , k given by the following formulas: (4.135) K14,i = 2k K16,i , i = 1, . . . , k . Analogously, the condition D3 [β¯ ] can be replaced by the following condition assumed to hold for some some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative compo­ nents:   F3 [β¯ ]: (a) E exp{ kj=1 ı j A N,j (β¯ i )Y0,j } < K16,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k for  some 1 < K16,i < ∞, i = 1, . . . , k; ¯] . 0 ∈ Z0 } = 1, where Z0 is the set penetrating condition D3 [β (b) P{Z Lemma 4.3.20. The condition F3 [β¯ ] implies condition D3 [β¯ ] to hold.  , i = 1, . . . , k, then, Remark 4.3.20. If the condition F3 [β¯ ] holds with constants K16,i  ¯  respectively, condition D3 [β] holds with the constants K14,i , i = 1, . . . , k given by the following formulas:   = 2k K16,i , i = 1, . . . , k . (4.136) K14,i

4.4 Univariate Markov Gaussian LPP In this section, we consider Markov Gaussian log-price processes. This model is one of the natural discrete time analogs for diffusion log-price processes. We get upper bounds for option rewards for two classes of such log-price processes. The first one includes processes with bounded drift and volatility coefficients. The second one in­ cludes models with drift and volatility coefficients with not more than linear rate of growth as a function of space arguments. The latter class includes, for example, meanreverse modulated Markov Gaussian log-price processes.

4.4 Univariate Markov Gaussian LPP

| 155

4.4.1 Upper bounds for option rewards of univariate Markov Gaussian log-price processes with bounded drift and volatility coefficients Let us consider the simplest univariate Markov Gaussian log-price process without modulation. In this case, the log-price process Y n , n = 0, 1, . . . , is given by the fol­ lowing stochastic transition dynamic relation: Y n = Y n −1 + μ n ( Y n −1 ) + σ n ( Y n −1 ) W n ,

n = 1, 2, . . . ,

(4.137)

where (a) Y0 is a real-valued random variable, (b) W n , n = 1, 2, . . . , is a sequence of standard i.i.d. normal random variables with mean 0 and variance 1, (c) random variable Y0 and sequence of random variables W n , n = 1, 2, . . . , are independent, (d) μ n (y) are real-valued Borel functions defined on R1 , for n = 1, 2, . . . , and (e) σ n (y) are real-valued Borel functions defined on R1 , for n = 1, 2, . . . . This is a model without an index component, and, therefore, a pay-off function g(n, e y ) is also assumed to be a function of the argument y. In the sequel, the logreward functions ϕ n (y) also depend only on argument y. In this subsection, we give upper bounds for option rewards for the model, where the following condition holds: G1 : max0≤n≤N −1 supy∈R1 (|μ n+1 (y)| + σ 2n+1 (y)) < K17 , for some 0 < K17 < ∞ . In this case, there exist the moment-generating functions Ey,n e β(Y n+1 −Y n ) , y ∈ R1 , n = 0, 1, . . . , N − 1, defined for β ∈ R1 , 1

Ey,n e β(Y n+1 −Y n ) = Ee β(μ n+1 (y) +σ n+1 (y) W n+1 ) = e βμ n+1 (y) + 2 β

2 2 σ n +1 ( y )

.

(4.138)

The second-type modulus of exponential moment compactness Ξ± β ( Y · , N ) takes, according to formula (4.138), the following form, for β ≥ 0: 1

± βμ n+1 ( y ) + 2 β Ξ± β ( Y · , N ) = max sup e

2 2 σ n +1 ( y )

0≤ n ≤ N −1 y ∈R1

.

(4.139)

We are interested in the following one-dimensional analog of the condition E1 [β¯ ] assumed to hold for the log-price process Y n , for some β ≥ 0: E6 [ β ] : Ξ ± β ( Y · , N ) < K 18 , for some 1 < K 18 < ∞ . Lemma 4.4.1. The condition G1 implies that condition E6 [β ] holds for any β ≥ 0. Proof. The condition G1 and relations (4.138) and (4.139) imply that for any β ≥ 0, 1

± βμ n+1 ( y ) + 2 β Ξ± β ( Y · , N ) = max sup e

2 2 σ n +1 ( y )

0≤ n ≤ N −1 y ∈R1

1 2 1 + e K 17 (β+ 2 β ) ∈ (1, ∞) . (4.140) 2 Thus, the condition G1 implies that condition E6 [β ] holds for any β ≥ 0.

< K18 =

156 | 4 Upper bounds for option rewards Remark 4.4.1. The constant K18 is determined by the constant K17 via formula (4.140). Lemma 4.4.1 makes it possible to get upper bounds for the log-reward functions and the optimal expected rewards, for the above model of log-price process using Theo­ rems 4.1.1 and 4.1.2. The one-dimensional analog of the condition B1 [𝛾¯] takes the following form, for 𝛾 ≥ 0: y B4 [𝛾]: max0≤n≤N supy∈R1 |1g+(Ln,8 ee𝛾|)| y | < L 7 , for some 0 < L 7 < ∞ and 0 ≤ L 8 < ∞ . The following theorem is a corollary of Theorem 4.1.1. Theorem 4.4.1. Let the condition B4 [𝛾] holds for some 𝛾 ≥ 0 and also the condition G1 holds. Then, there exist the constants 0 ≤ M19 , M20 < ∞ such that the log-reward functions ϕ n (y) satisfy the following inequalities for y ∈ R1 , 0 ≤ n ≤ N, |ϕ n (y) | ≤ M19 + M20 e𝛾| y| .

(4.141)

Proof. The case 𝛾 = 0 is trivial. In this case, inequality (4.141) obviously hold with the constants M19 = L7 , M20 = L7 L8 . Let 𝛾 > 0. According to Lemma 4.4.1, the condition E6 [β ] (which is a one-dimensional version of the condition E1 [β¯ ]) holds for any β ≥ 𝛾. Thus, by Lemma 4.1.8, the corresponding one-dimensional versions of conditions of Theorem 4.1.1 hold if the condition B4 [𝛾] holds. In this case, the inequality given in Theorem 4.1.1 takes the forms of inequality (4.141). Remark 4.4.2. The constants M19 and M20 take, according to Remarks 4.1.3 and 4.1.7, the following form: M19 = L7 , 𝛾

M20 = L7 L8 I (𝛾 = 0) + L7 L8 (2K18 )N β I (𝛾 > 0) 1

= L7 L8 I (𝛾 = 0) + L7 L8 (1 + 2e K 17 (β+ 2 β ) ) 2

𝛾

Nβ

I (𝛾 > 0) .

(4.142)

One can take any β ≥ 𝛾, in formulas (4.142). The condition D1 [β¯ ] takes, in this case, the following forms, assumed to hold for some β > 0: D4 [β ]: Ee β| Y0| < K19 , for some 1 < K19 < ∞ . The following theorem, which is a corollary of Theorem 4.1.3, gives an upper bound for the optimal expected reward Φ = Φ(Mmax,N ). Theorem 4.4.2. Let the conditions B4 [𝛾], G1 , and D4 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, there exists a constant 0 ≤ M21 < ∞ such that the following inequality takes place: |Φ| ≤ M21 . (4.143)

4.4 Univariate Markov Gaussian LPP

| 157

Remark 4.4.3. The explicit formula for the constant M21 take according to formulas given in Remarks 4.1.5, 4.1.7, and 4.1.9, the following form: M21 = L7 + L7 L8 I (𝛾 = 0) 1

𝛾 β

+ L7 L8 (1 + 2e K 17 (β+ 2 β ) )K19 I (𝛾 > 0) . 2

(4.144)

4.4.2 Upper bounds for option rewards of univariate Markov Gaussian log-price processes with unbounded drift and volatility coefficients In this subsection, we give upper bounds for option rewards in the case, where the following condition holds: | μ ( y )|+ σ 2 ( y ) G2 : max0≤n≤N −1 supy∈R1 n+11+K 21| yn|+1 < K20 , for some 0 < K20 < ∞ and 0 ≤ K21 < ∞. Formula (4.138) gives a hint to use function A(β ) given by the following formula:   1 (4.145) A(β ) = K20 K21 β + β 2 , β ≥ 0 , 2 In this case, the second-type modulus of exponential moment compactness Ξ± β,A ( Y · , N ) takes the following form, for β ≥ 0: Ξ± β,A (Y · , N ) 1

=

max sup

0≤ n ≤ N −1 y ∈R1

e ± A N − n −1 ( β ) μ n +1 ( y ) + 2 A N − n −1 ( β ) σ n +1 ( y ) , e A(A N −n−1(β))| y| 2

2

(4.146)

where functions A n (β ), n = 0, 1, . . . , are generated by the above function A(β ) = K20 K21 (β + 21 β 2 ) according to the following recurrence formulas, for every β ≥ 0: ⎧ ⎨β

A n (β) = ⎩

A n−1 (β ) + A(A n−1 (β ))

for n = 0 , for n = 1, 2, . . . .

(4.147)

We are interested in the following variant of the condition E4 [β ] for the log price process Y n , assumed to hold for some β ≥ 0: E7 [ β ] : Ξ ± β,A ( Y · , N ) < K 22 , for some 1 < K 22 < ∞ . Lemma 4.4.2. The condition G2 implies that condition E7 [β ] holds for any β ≥ 0.

158 | 4 Upper bounds for option rewards Proof. The condition G2 imply that, for any y ∈ R1 , n = 0, . . . , N − 1 and β ≥ 0, 1 ± A N −n−1(β )μ n+1 (y) + A2N −n−1(β )σ 2n+1 (y) 2 1 ≤ A N −n−1(β )|μ n+1 (y)| + A2N −n−1 (β )σ 2n+1 (y) 2   1 < A N −n−1 (β ) + A2N −n−1(β ) (K20 + K20 K21 |y|) 2   1 = K20 A N −n−1(β ) + A2N −n−1 (β ) 2   1 + K20 K21 A N −n−1(β ) + A2N −n−1 (β ) |y| 2   1 2 = K20 A N −n−1(β ) + A N −n−1 (β ) + A(A N −n−1 (β ))|y| . 2

(4.148)

Relations (4.145)–(4.148) imply that, for β ≥ 0, Ξ± β,A ( Y · , N ) 1

2

2

e ± A N − n −1 ( β ) μ n +1 ( y ) + 2 A N − n −1 ( β ) σ n +1 ( y ) = max sup 0≤ n ≤ N −1 y ∈R1 e A(A N −n−1(β))| y| ≤

1

max e K 20(A N −n−1 (β)+ 2 A N −n−1 (β)) 2

0≤ n ≤ N − 1

1 2 1 + e K 20(A N −1 (β)+ 2 A N −1 (β)) = K22 ∈ (1, ∞) . 2 This inequality proves the lemma.


0) . (4.150) β Proof. The case 𝛾 = 0 is trivial. Let 𝛾 > 0. According to Lemma 4.4.2, the condition E7 [β ] holds for any β ≥ 𝛾. In this case inequality (4.76) given in Theorem 4.3.1 takes the form of inequality (4.150). Remark 4.4.5. The the constants M22 and M23 (β ) takes, according to Remarks 4.3.1, and 4.3.3, the following form: M22 = L7 , N

𝛾

M23 (β ) = L7 L8 I (𝛾 = 0) + L7 L8 (2K22 ) β N 𝛾 # 1 2 β = L7 L8 I (𝛾 = 0) + L7 L8 1 + 2e K 20 (A N −1 (β)+ 2 A N −1 (β)) I (𝛾 > 0) . One can take any β ≥ 𝛾, in formulas (4.151).

(4.151)

4.5 Multivariate modulated Markov Gaussian LPP

| 159

The condition D2 [β ], where function A(β ) = K20 K21 (β + 21 β 2 ) should be used, takes the following form, for some β ≥ 0: D5 [β ]: E exp{A N (β )|Y0 |} < K23 , for some 1 < K23 < ∞ . The following theorem gives upper bound for the optimal expected reward Φ = Φ(Mmax,N ). Theorem 4.4.4. Let the conditions B6 [𝛾], G2 , and D5 [β ] hold and 0 ≤ 𝛾 ≤ β < ∞. Then, there exists a constant 0 ≤ M24 < ∞ such that the following inequality takes place: |Φ| ≤ M24 .

(4.152)

Proof. The case 𝛾 = 0 is trivial. Let 𝛾 > 0. According to Lemma 4.4.2, the condition E7 [β ] holds for any β ≥ 𝛾. In this case, inequality (4.85) given in Theorem 4.3.2 takes the form of inequality (4.152). Remark 4.4.6. The explicit formula for the constant M18 takes, according to formulas given in Remarks 4.31, 4.3.3, and 4.3.5, the following form: M24 = L7 + L7 L8 I (𝛾 = 0) # N 𝛾 𝛾 1 2 β β + L7 L8 1 + 2e K 20 (A N −1 (β)+ 2 A N −1 (β)) K23 I (𝛾 > 0) .

(4.153)

4.5 Multivariate modulated Markov Gaussian LPP In this section, we consider multivariate modulated Markov Gaussian log-price pro­ cesses. This model is one of natural discrete time analogs for multivariate modulated diffusion log-price processes. We get upper bounds for option rewards for two classes of such log-price processes. The first one includes log-price with bounded drift and volatility coefficients. The second one includes models with drift and volatility coef­ ficients with not more than linear rate of growth as a function of space arguments. The latter class include various autoregressive and autoregressive stochastic volatility log-price processes with Gaussian noise terms.

160 | 4 Upper bounds for option rewards

4.5.1 Upper bounds for option rewards of multivariate modulated Markov Gaussian log-price processes with bounded characteristics In this model, a log-price process is given by the following stochastic transition dy­ namic relation: ⎧  =Y  n −1 +   n −1 , X n −1 ) + Σ n ( Y  n −1 , X n −1 ) W n , ⎪ Y μ n (Y ⎪ ⎪ ⎨ n X n = C n ( X n −1 , U n ) , (4.154) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0 , X0 ) = ((Y 0,1 , . . . , Y 0,k )), X0 ) is a random vector taking values in the where (a) (Y  n , U n ) = ((W n,1 , . . . , W n,k ), U n ), n = 1, 2, . . . , is a se­ space Z = Rk × X, (b) (W quence of independent random vectors taking values in the space Rk × U, where U is a measurable space with a σ-algebra of measurable subsets BU , (c) the random 0 , X0 ) and the random sequence (W  n , U n ), n = 1, 2, . . . , are independent, vector (Y  n = (W n,1 , . . . , W n,k ) has a standard multivariate normal (d) the random vector W distribution with EW n,i = 0, EW n,i W n,j = I (i = j), i, j = 1, . . . , k, for every n = 1, 2, . . . , (e) the random variables U1 , U2 , . . . , have a regular conditional distribution n = w G w (A) = P{U n ∈ A/W  }, n = 1, 2, . . . , (a probability measure in B ∈ BU , for every w  ∈ Rk , and a measurable function in w  ∈ Rk , for every B ∈ BU ), for every μ n ( y , x) = (μ n,1( y , x), . . . , μ n,k ( y , x)), n = 1, 2, . . . , are vector func­ n = 1, 2, . . . , (f)  tions, which components are measurable functions acting from the space Z = Rk × X to R1 , (g) Σ n ( y , x) = σ n,i,j( y , x) , n = 1, 2, . . . , are k × k matrix functions with el­ y , x), i, j = 1, . . . , k, which are measurable functions acting from the ements σ n,i,j( space Z = Rk × X to R1 , and (h) C n (x, u ) is a measurable function acting from the space X × U to the space X. Z n has an index component X n , and, therefore, a pay-off The log-price process  function g(n, ey , x) is also assumed to be a function of the argument  z = ( y , x) ∈ Z. In sequel, the log-reward functions ϕ n ( y , x) also are functions of ( y , x) ∈ Z. In this subsection, we give upper bounds for option rewards for the model, where the following condition holds: G3 : max0≤n≤N −1,i,j=1,...,k supz=(y ,x)∈Z(|μ n+1,i( y , x)| + σ 2n+1,i,j( y , x)) < K24 , for some 0 < K24 < ∞ .

z = ( y , x) ∈ Z, n = 0, 1, . . . , N − 1, i = 1, . . . , k, In this case, there exists, for every  the moment generation function Ez ,n e β(Y n+1,i −Y n,i ) , defined for β ∈ R1 by the following formula: Ez ,n e β(Y n+1,i −Y n,i ) = Ee β(μ n+1,i (y ,x)+ 1

k

= e βμ n+1,i (y ,x )+ 2 β

j =1

2

k

σ n+1,i,j ( y ,x ) W n+1,j )

j =1

σ 2n+1,i,j ( y ,x )

.

(4.155)

4.5 Multivariate modulated Markov Gaussian LPP |

161

The second-type modulus of exponential moment compactness Ξ± β ( Y · ,i , N ) takes, according to formula (4.155), the following form, for β ≥ 0: Ξ± β ( Y · ,i , N ) = max

sup

1

0≤ n ≤ N − 1  z =( y ,x )∈Z

e±βμ n+1,i (y ,x)+ 2 β

2

k

j =1

σ 2n+1,i,j ( y ,x )

.

(4.156)

We are interested in the following analog of the condition E1 [β¯ ], assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: E8 [β¯ ]: Ξ± β i ( Y · ,i , N ) < K 25,i , i = 1, . . . , k, for some 1 < K 25,i < ∞, i = 1, . . . , k . The following lemma is a variant of Lemma 4.4.1. Lemma 4.5.1. The condition G3 implies that condition E8 [β¯ ] holds for any vector pa­ rameter β¯ = (β 1 , . . . , β k ) with components β i ≥ 0, i = 1, . . . , k. Proof. It follows from the following relation, which holds for any β ≥ 0: Ξ± β ( Y i, · , N ) ≤ max

sup

0≤ n ≤ N − 1  z=( y ,x )∈Z


0) 1

= L1 L2,i I (𝛾i = 0) + L1 L2,i (1 + 2e K 24 (β i + 2 kβ i ) ) 2

𝛾

N βi

i

I (𝛾i > 0) .

(4.160)

One can take any β i ≥ 𝛾i , i = 1, . . . , k, in formulas (4.160). The following theorem, which is a corollary of Theorem 4.1.3, gives upper bounds for the optimal expected reward Φ. Theorem 4.5.2. Let the conditions B1 [𝛾¯], G3 , and D1 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exists a constant 0 ≤ M27 < ∞ such that the following inequality takes place: |Φ| ≤ M27 . (4.161) Remark 4.5.3. The explicit formula for the constant M27 takes, according to formulas given in Remarks 4.1.3, 4.1.7, and 4.1.9, the following form: M27 = L1 +





L1 L2,i +

i:𝛾i =0

1

L1 L2,i (1 + 2e K 24 (β i + 2 kβ i ) ) 2

𝛾

N βi

i

𝛾i β

K2,ii .

(4.162)

i:𝛾i >0

4.5.2 Upper bounds for option rewards for multivariate modulated Markov Gaussian log-price processes with unbounded characteristics In this subsection, we give upper bounds for option rewards for the model, where the following condition holds: G4 : max0≤n≤N −1, i,j=1,...,k supz=(y ,x)∈Z ∞ and 0 ≤ K27,1 , . . . , K27,k < ∞ .

| μ n+1,i ( y ,x )|+ σ 2n+1,i,j ( y ,x ) k 1+ l=1 K 27,l | y l |

< K26 , for some 0 < K26
0 j =1

(4.170)

i

Proof. According to Lemma 4.5.2, the condition E9 [β¯ ], which is the variant of the con­ dition E5 [β¯ ], holds, for any β i ≥ 𝛾i , i = 1, . . . , k. In this case, the condition C5 [β¯ ] holds by Lemma 4.3.16. Thus, conditions of Theorem 4.3.5 hold, if condition B1 [𝛾¯] holds. In this case, the inequalities given in Theorem 4.3.5, take the form of inequali­ ties (4.170). Remark 4.5.5. The explicit formulas for the constants M28 and M29,i , i = 1, . . . , k take, according to formulas given in Remarks 4.3.11, 4.3.13, 4.3.17, and 4.5.14, the following form: M28 = L1 , M29,i (β i ) = L1 L2,i I (𝛾i = 0) + L1 L2,i (2k K28,i )

𝛾

N βi

i

I (𝛾i > 0)

= L1 L2,i I (𝛾i = 0) # N 𝛾βi k 1 i k K 26 l=1 ( A N −1,l ( β¯ i )+ 2 k 2 A 2N −1,l ( β¯ i )) I (𝛾i > 0) . + L1 L2,i 1 + 2 e

(4.171)

In the above formula, one can take any β i ≥ 𝛾i , i = 1, . . . , k.   (β¯ ) = (K26 K27,1 · kl=1 (β l + 1 k 2 β 2l ), . . . , The condition D3 [β¯ ], where function A 2  K26 K27,k · kl=1 (β l + 21 k 2 β 2l ) should be used, takes the following form, for some β¯ = (β 1 , . . . , β k ) with nonnegative components and the corresponding vectors β¯ i = (β i,1 , . . . , β i,k ) with components β i,j = β j I (i = j), i, j = 1, . . . , k:  D6 [β¯ ]: E exp{ kj=1 A N,j (β¯ i )|Y0,j |} < K29,i , i = 1, . . . , k, for some 1 < K29,i < ∞, i = 1, . . . , k . The following theorem gives conditions and the upper bound for the optimal expected reward Φ = Φ(Mmax,N ).

166 | 4 Upper bounds for option rewards Theorem 4.5.4. Let the conditions B1 [𝛾¯], G4 , and D6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i < ∞, i = 1, . . . , k. Then, there exists a constant 0 ≤ M30 < ∞ such that the following inequality takes place: |Φ| ≤ M30 . (4.172) Proof. The case 𝛾i = 0 is trivial. Let 𝛾i > 0. According to Lemma 4.5.2, the condition E9 [β¯ ] holds for any β i ≥ 𝛾i . In this case inequality given in Theorem 4.3.7 takes the form of inequality (4.172). Remark 4.5.6. The explicit formula for the constant M24 takes, according to formulas given in Remarks 4.3.11, 4.3.13, 4.3.15, 4.3.17, and 4.5.2, the following form:

L1 L2,i M30 = L1 + i:𝛾i =0

+

i:𝛾i >0

# N 𝛾βi 𝛾i k 1 2 2 ¯ ¯ βi i L1 L2,i 1 + 2k e K 26 l=1 (A N −1,l (β i )+ 2 k A N −1,l (β i )) K29,i .

(4.173)

5 Convergence of option rewards – I In this chapter, we present our main convergence results for rewards of Americantype options with general perturbed pay-off functions for perturbed multivariate mod­ ulated Markov log-price processes with bounded characteristics. In Section 5.1, we give asymptotically uniform upper bounds for log-reward func­ tions and optimal expected rewards for multivariate modulated Markov log-price processes with bounded characteristics (conditional exponential moments for incre­ ments) and with general perturbed pay-off functions. In Section 5.2, we present our main convergence results for reward functions and optimal expected rewards for perturbed multivariate modulated Markov log-price pro­ cesses with bounded characteristics. In Section 5.3, we investigate convergence results for log-reward functions and optimal expected rewards for log-price processes represented by multivariate standard and modulated random walks, including Lévy random walks. Our main results are given in Theorems 5.2.1–5.2.4, for perturbed multivariate mod­ ulated Markov log-price processes, and Theorems 5.3.1–5.3.6 for perturbed log-price processes represented by random walks. What is important is that we impose minimal conditions of smoothness on the limiting pay-off functions and transition probabili­ ties. In the basic case, where the transition probabilities have densities with respect to some pivotal Lebesgue-type measure, it is usually required that the sets of weak discontinuity for the limiting transition probabilities and the sets of discontinuity for limiting pay-off functions are zero sets with respect to the above pivotal measure. In fact, such assumptions make it possible for the transition probabilities and pay-off functions to be very irregular. For example, the above discontinuity sets can be count­ able sets dense in the corresponding phase spaces. The results presented in this chapter are based on and generalize in several as­ pects the results obtained in the literature: for univariate modulated Markov log-price processes the results of Silvestrov, Jönsson, and Stenberg (2006, 2008, 2009), for multivariate Markov log-price processes the results of Lundgren and Silvestrov (2009, 2011) and of Silvestrov and Lundgren (2011). First, we consider multivariate modu­ lated log-price processes, i.e. combine together multivariate and modulation aspects together. Second, we consider pay-off functions, which also depend on an additional index component. Third, we improve moment compactness conditions (by taking the supremum of conditional exponential moments for increments in formulas defining the corresponding moduli of compactness over some special subsets of the phase space instead of the whole phase space). This generalization is important for some applications, for example, to skeleton approximation models. Fourth, we improve formulation of the corresponding conditions imposed on pay-off functions and the moment compactness condition on log-price processes by giving them in a natural asymptotic form.

168 | 5 Convergence of option rewards – I

5.1 Asymptotically uniform upper bounds for rewards – I In this section, we introduce the model of perturbed modulated Markov log-price pro­ cesses and give asymptotically uniform upper bounds for option rewards for such pro­ cesses. These upper bounds play an important role in proofs of convergence for option rewards for perturbed log-price processes.

5.1.1 Perturbed log-price processes We consider a discrete time multivariate modulated Markov log-price process  Z ε,n = ε,n , X ε,n) = ((Y ε,n,1, . . . , Y ε,n,k), X ε,n), n = 0, 1, . . . with a phase space Z = Rk × X, (Y z , A ). an initial distribution P ε,0(A), and transition probabilities P ε,n (  We assume that the process Z ε,n and, therefore, its transition probabilities P ε,n ( z , A) and initial distribution P ε,0(A) depend on the perturbation parameter ε ∈ [0, ε0 ], where ε0 > 0. We consider American-type options for the log-price process  Z ε,n and a pay-off function g ε (n, ey , x) also depending on the perturbation parameter ε ∈ [0, ε0 ]. We are going to formulate conditions that would provide convergence for the y , x) and optimal expected rewards Φ ε = corresponding log-reward functions ϕ ε,n ( (ε) Φ(Mmax,N ) for log-price processes  Z ε,n , respectively, to the log-reward functions (0) ϕ0,n ( y , x) and optimal expected rewards Φ0 = Φ(Mmax,N ) for log-price processes  Z0,n , as ε → 0. The prelimiting log-price processes  Z ε,n , for ε > 0, are usually simpler than the corresponding limiting log-price process  Z0,n . For example, the prelimiting processes can be atomic Markov chains with initial distributions and transition probabilities concentrated at a finite set of points. This let one compute effectively values of logreward functions and optimal expected rewards for prelimiting log-price processes using backward algorithms presented in Chapter 3. On the other hand, the limiting log-price process can be a nonatomic Markov chain with a continuous initial distribu­ tion and transition probabilities. In this case, the corresponding backward recurrence algorithms may not be effective. The convergence results may yield in such cases ef­ fective approximations for limiting option rewards. The same approach can be applied to pay-off functions. Prelimiting pay-off func­ tions can be constructed in such a way that they have a simpler structure than the cor­ responding limiting pay-off function. For example, prelimiting pay-off functions can be convex piecewise linear functions in price arguments. In such cases, one can de­ scribe effectively the structure of the corresponding optimal stopping domains. How­ ever, the corresponding limiting pay-off function can be a general convex function, with much more complex structure of the optimal stopping domain. The use of ap­

5.1 Asymptotically uniform upper bounds for rewards – I | 169

proximating pay-off functions may yield effective approximations for optimal stopping domains. We use this approach for designing of Monte Carlo-based approximation al­ gorithms for option rewards. These results will be presented in the second volume of this book. Conditions of convergence for option rewards, as usual, should be formulated in terms of “initial” characteristics of the model that are transition probabilities z , A), initial distributions P ε (A), and pay-off functions g ε (n, ey , x). Moreover, P ε,n ( these conditions should not involve these quantities in some complex combinations, in order to be really effective. The best variant is to formulate some separate and clearly formulated conditions for the above three quantities trying also to escape of making these conditions too restrictive. We systematically follow this approach. Naturally, these conditions should include some moment assumptions for each of the above quantities and also assumptions about convergence in some sense for the transition probabilities P ε,n ( z , A), the initial distribution P ε (A), and the pay-off  y function g ε (n, e , x), respectively, to the corresponding limit transition probabilities z , A), the initial distribution P0 (A), and the pay-off functions g0 (n, ey , x). P0,n ( These convergence assumptions let us interpret the log-price process  Z ε,n and the pay-off function g ε (n, ey , x) for ε > 0 as the perturbed versions of the log-price process  Z0,n and the pay-off function g0 (n, ey , x).

5.1.2 Asymptotically uniform upper bounds for rewards for log-price processes with bounded characteristics In this subsection, we present upper bounds for option rewards analogous to those given in Chapter 4, but modified in such a way that they become asymptotically uni­ form with respect to parameter ε → 0. Recall that we use the notations Pz ,n and Ez ,n for conditional probabilities and Z ε,n =  z. expectations under the condition  Let us recall the first-type modulus of exponential moment compactness for the ε,n = (Y ε,n,1, . . . , Y ε,n,k), for β ≥ 0, i = 1, . . . , k, components of the log-price process Y Δ β (Y ε,· ,i , N ) = max sup Ez ,n e β| Y ε,n+1,i −Y ε,n,i | . 0≤ n ≤ N − 1  z∈Z

(5.1)

The condition C1 [β¯ ] should be replaced by the following first-type condition of ex­ ponential moment compactness, which is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: C6 [β¯ ]: limε→0 Δ β i (Y ε,· ,i , N ) < K30,i , i = 1, . . . , k, for some 1 < K30,i < ∞, i = 1, . . . , k . The following lemmas give asymptotically uniform upper bounds for exponential mo­ ments of maximal absolute values for perturbed log-price processes.

170 | 5 Convergence of option rewards – I Lemma 5.1.1. Let the condition C6 [β¯ ] holds. Then there exist 0 < ε1 ≤ ε0 and the con­ stants 0 ≤ M31,i < ∞, i = 1, . . . , k such that the following inequality takes place for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε1 , and i = 1, . . . , k: $ % Ez ,n exp β i max |Y ε,r,i| ≤ M31,i e β i | y i | . (5.2) n≤r≤N

Proof. By the definition of the upper limit and the condition C6 [β¯ ], one can choose 0 < ε1 ≤ ε0 such that for 0 ≤ ε ≤ ε1 and i = 1, . . . , k, Δ β i (Y ε,· ,i(·), N ) < K30,i .

(5.3)

Relation (5.3) means that, for every ε ≤ ε1 , the condition C1 [β¯ ] (with the constants  ε,n. K30,i , i = 1, . . . , k, replacing the constants K1,i , i = 1, . . . , k) holds for the process Y  Therefore, Lemma 4.1.1 can be applied to the process Y ε,n , for every ε ≤ ε1 . This yields inequalities (5.2). For every i = 1, . . . , k, the constant M1,i penetrat­ ing the inequalities given in Lemma 4.1.1 is a function of the corresponding constant K1,i . The explicit expression for this function is given in Remark 4.1.1. The same for­ mula gives the expression for the constant M31,i as a function of the corresponding constant K30,i . Remark 5.1.1. The constants M31,i , i = 1, . . . , k are given by the following formu­ las, which follows from relation (5.3), and the corresponding formulas given in Re­ mark 4.1.1: N M31,i = K30,i . (5.4) Remark 5.1.2. The parameter ε1 is determined by relation (5.3). Let Zε = Zε,0 , . . . , Zε,N  be the sequence of measurable subsets of space Z and Δ β,Zε (Y ε,· ,i , N ) be the modified first-type modulus of exponential moment compact­ ness defined, for every ε ∈ [0, ε0 ] and β ≥ 0, by the following formula: Δ β,Zε (Y ε,· ,i , N ) = max

sup Ez ,n e β| Y ε,n+1,i −Y ε,n,i | .

0≤ n ≤ N − 1  z∈Z

(5.5)

ε,n

The following modified first-type condition of exponential moment compactness ¯ ]. It should be assumed to  n generalizes the condition C6 [β for the log-price process Y ¯ hold for some vector parameter β = (β 1 , . . . , β k ) with nonnegative components: C6 [β¯ ]: There exists a sequence of measurable sets Zε = Zε,0 , . . . , Zε,N , for every ε ∈ [0, ε0 ], such that   (a) limε→0 Δ β i ,Zε (Y ε,· ,i , N ) < K30,i , i = 1, . . . , k for some 1 < K30,i < ∞, i = 1, . . . , k; (b) Zε is a complete sequence of phase sets for the process Zε,n , i.e. Pε,n( z , Zε,n ) = 1,  z ∈ Zε,n−1, n = 1, . . . , N, for every ε ∈ [0, ε0 ] .

5.1 Asymptotically uniform upper bounds for rewards – I |

171

+ − Let us introduce, for every ε ∈ [0, ε0 ], sets Hε,n,k = (h− ε,n,1, h ε,n,1) × · · · × ( h ε,n,k , + − + h ε,n,k) × X, n = 0, . . . , N, where −∞ ≤ h ε,n,i ≤ h ε,n,i ≤ +∞, i = 1, . . . , k, n = 0, . . . , N. It is useful to note that in applications, for example to space skeleton models con­ sidered in Sections 1.7 and 1.8, the sets Zε,n, n = 0, . . . , N satisfy the following natural condition: ± H1 : Zε,n ⊇ Hε,n,k, n = 0, . . . , N, ε ∈ [0, ε0 ], where h± ε,n,i → h 0,n,i = ±∞ as ε → 0, for i = 1, . . . , k, n = 0, . . . , N .

z = ( y , x) = ((y1 , . . . , y k ), Remark 5.1.3. According to the condition H1 , for every  x), there exists ε1 ( z ) ∈ (0, ε0 ] such that the following relation holds for every ε ∈ [0, ε1 ( z )], + h− i = 1, . . . , k , n = 0, . . . , N . (5.6) ε,n,i < y i < h ε,n,i , Obviously,  z ∈ Zε,n , n = 0, . . . , N for all ε ∈ [0, ε1 ( z )]. Note that, in the case, where the condition H1 holds, the sets Z0,n = Z, n = 0, . . . , N. Lemma 5.1.2. Let the condition C6 [β¯ ] holds. Then there exists 0 < ε1 ≤ ε0 and the  constants 0 ≤ M31,i < ∞, i = 1, . . . , k such that the following inequality takes place for  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε1 , and i = 1, . . . , k, $ %  e βi |yi | . (5.7) Ez ,n exp β i max |Y ε,r,i| ≤ M31,i n≤r≤N

Proof. By the definition of the upper limit and the condition C6 [β¯ ], one can choose 0 < ε1 ≤ ε0 such that for 0 ≤ ε ≤ ε1 and i = 1, . . . , k,  Δ β i ,Zε (Y ε,· ,i(·), N ) < K30,i .

(5.8)

The following part of the proof repeats the proof of Lemma 5.1.1. The only differ­ ε,n , for every ε ≤ ε1 , instead ence is that Lemma 4.1.3 should be applied to the process Y ε,n , for every ε ≤ ε1 , in the proof of of Lemma 4.1.1, which was applied to the process Y Lemma 5.1.1.  This yields inequalities (5.7). For every i = 1, . . . , k, the constant M1,i penetrat­ ing the inequalities given in Lemma 4.1.3 is a function of the corresponding constant  . The explicit expression for this function is given in Remark 4.1.2. The same for­ K1,i  mula gives the expression for the constant M31,i as a function of the corresponding  constant K30,i .  Remark 5.1.4. The constants M31,i , i = 1, . . . , k are given by the following formu­ las, which follows from relation (5.8), and the corresponding formulas given in Re­ mark 4.1.2,   M31,i = (K30,i )N . (5.9)

172 | 5 Convergence of option rewards – I Remark 5.1.5. The parameter ε1 is determined by relation (5.8). Let us now give asymptotically uniform upper bounds for log-reward functions for Z ε,n . log-price processes  We assume that pay-off functions g ε (n, ey , x) also depends on perturbation pa­ rameter ε. The condition B1 [𝛾¯] should be replaced by the following condition assumed to hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components: B5 [𝛾¯]: limε→0 max0≤n≤N supz=(y ,x)∈Z 0 ≤ L10,1 , . . . , L10,k < ∞ .

1+

| g ( n,ey ,x )| kε 𝛾i |y i | i =1 L 10,i e

< L9 , for some 0 < L9 < ∞ and

The following two lemmas give asymptotically uniform upper bounds for condi­ tional expectations of maximal absolute values for perturbed pay-off processes g ε (n, e Yε,n , X ε,n ). Lemma 5.1.3. Let the conditions B5 [𝛾¯] and C6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, there exist 0 < ε3 ≤ ε0 and the constants 0 ≤ M32 , M33,i < ∞, i = 1, . . . , k such z = ((y1 , . . . , y k ), x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ that the following inequalities take place for  ε ≤ ε3 , k  # 

   Ez ,n max g ε n, e Y ε,n , X ε,n  ≤ M32 + M33,i e𝛾i | y i | . n≤r≤N

(5.10)

i =1

Proof. The condition B5 [𝛾¯] implies that there exists 0 < ε2 ≤ ε0 such that for any 0 ≤ ε ≤ ε2 , |g ε (n, ey , x)| < L9 . k 𝛾i | y i | 0≤ n ≤ N  z=( y ,x )∈Z 1 + i =1 L 10,i e

max

sup

(5.11)

Relation (5.11) means that, for every ε ≤ ε2 , the condition B1 [𝛾¯] (with the con­ stants L9 , L10,i , i = 1, . . . , k, replacing the constants L1 , L2,i , i = 1, . . . , k) holds for the ε,n . process Y Let us now take ε3 = min(ε1 , ε2 ), where ε1 was defined in relation (5.3). By the definition ε3 > 0. For every fixed 0 ≤ ε ≤ ε3 , both the conditions C1 [β¯ ] and B1 [𝛾¯] (with the con­ stants K30,i , i = 1, . . . , k, and L9 , L10,i , i = 1, . . . , k replacing, respectively, the con­ ε,n . stants K1,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process Y  Therefore, Lemma 4.1.4 can be applied to the process Y ε,n , for every 0 ≤ ε ≤ ε3 . This yields inequality (5.10). For every i = 1, . . . , k, the constants M2 and M3,i penetrating the inequalities given in Lemma 4.1.4 are functions of the corresponding constants K1,i and L1 , L2,i . The explicit expressions for these functions are given in Remark 4.1.3. The same formulas give the expressions for the constant M32 and M33,i as functions of the corresponding constants K30,i and L9 , L10,i .

5.1 Asymptotically uniform upper bounds for rewards – I | 173

Remark 5.1.6. The explicit formula for the constants M32 , M33,i , i = 1, . . . , k are given by the following formulas, which follow from the corresponding formulas given in Remark 4.1.3: 𝛾

M32 = L9 ,

N βi

M33,i = L9 L10,i I (𝛾i = 0) + L9 L10,i K30,ii I (𝛾i > 0) .

(5.12)

Remark 5.1.7. The parameter ε3 = min(ε1 , ε2 ), where the parameters ε1 and ε2 are determined, respectively, by relations (5.3) and (5.11). Lemma 5.1.4. Let the conditions B5 [𝛾¯] and C6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k.   Then, there exist 0 < ε3 ≤ ε0 and the constants 0 ≤ M32 , M33,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ((y1 , . . . , y k ), x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε3 : k  # 

     Ez ,n max g ε n, e Y ε,n , X ε,n  ≤ M32 + M33,i e𝛾i | y i | . n≤r≤N

(5.13)

i =1

Proof. It is analogous to the proof of Lemma 5.1.3. As was pointed out in the proof of this lemma, the condition B5 [𝛾¯] implies that there exists 0 < ε2 ≤ ε0 such that inequality (5.11) holds for 0 ≤ ε ≤ ε2 . This means that, for every ε ≤ ε2 , the condition B1 [𝛾¯] (with the constants L9 , L10,i , ε,n . i = 1, . . . , k, replacing the constants L1 , L2,i , i = 1, . . . , k) holds for the process Y Let us now take ε3 = min(ε1 , ε2 ), where ε1 was defined in relation (5.8). By the definition ε3 > 0. For every fixed 0 ≤ ε ≤ ε3 both the conditions C1 [β¯ ] and B1 [𝛾¯] (with the constants   K30,i , i = 1, . . . , k, and L9 , L10,i , i = 1, . . . , k replacing, respectively, the constants K1,i ,  ε,n. i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process Y ε,n , for every 0 ≤ ε ≤ ε3 . Therefore, Lemma 4.1.5 can be applied to the process Y  This yields inequality (5.10). For every i = 1, . . . , k, the constants M2 and M3,i penetrating the inequalities given in Lemma 4.1.5 are functions of the corresponding  and L1 , L2,i . The explicit expressions for these functions are given in constants K1,i   Remark 4.1.4. The same formulas give the expressions for the constant M32 and M33,i  as functions of the corresponding constants K30,i and L9 , L10,i .   Remark 5.1.8. The explicit formula for the constants M32 , M33,i , i = 1, . . . , k are given by the following formulas, which follows from the corresponding formulas given in Remark 4.1.4:  M32 = L9 ,

  M33,i = L9 L10,i I (𝛾i = 0) + L9 L10,i (K30,i )

𝛾

N βi

i

I (𝛾i > 0) .

(5.14)

Remark 5.1.9. The parameter ε3 = min(ε1 , ε2 ), where the parameters ε1 and ε2 are determined, respectively, by relations (5.8) and (5.11).

174 | 5 Convergence of option rewards – I Let us recall the second-type modulus of exponential moment compactness for the  ε,n = (Y ε,n,1 . . . , Y ε,n,k) defined for β ≥ 0, i = components of the log-price process Y 1, . . . , k, ± β( Y ε,n+1,i − Y ε,n,i ) . (5.15) Ξ± z ,n e β ( Y ε, · ,i , N ) = max sup E 0≤ n ≤ N − 1  z∈Z

The following second-type condition of the exponential moment compactness, based on moment-generating functions of increments for one-dimensional Markov ¯ ]. It is assumed to hold for some ε,n , replaces the condition E1 [β log-price processes Y vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: E10 [β¯ ]: limε→0 Ξ± β i ( Y ε, · ,i , N ) < K 31,i , i = 1, . . . , k, for some 1 < K 31,i < ∞, i = 1, . . . , k . Lemma 5.1.5. The condition E10 [β¯ ] implies the condition C6 [β¯ ] to hold. Proof. It follows from Lemma 4.1.8 that should be applied to the log-price processes Y ε,n,i. Remark 5.1.10. If the condition E10 [β¯ ] holds with the constants K31,i , i = 1, . . . , k, then the condition C6 [β¯ ] holds with the constants K30,i , i = 1, . . . , k given by the following formulas: (5.16) K30,i = 2K31,i , i = 1, . . . , k . Let Zε = Zε,0 , . . . , Zε,N  be a sequence of measurable sets. Recall the modified sec­ ond-type modulus of the exponential moment compactness for the components de­ fined, for every ε ∈ [0, ε0 ] and for β ≥ 0, i = 1, . . . , k, Ξ± β, Zε ( Y ε, · ,i , N ) = max

sup Ez ,n e±β(Y ε,n+1,i −Y ε,n,i ) .

0≤ n ≤ N − 1  z∈Z

(5.17)

ε,n

The following condition is a modified version of the condition E10 [β¯ ]. It is as­ sumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative com­ ponents: E10 [β¯ ]: There exists a sequence of measurable sets Zε = Zε,0 , . . . , Zε,N , for every ε ∈ [0, ε0 ], such that   (a) limε→0 Ξ± β i , Zε ( Y ε, · ,i , N ) < K 31,i , i = 1, . . . , k, for some 1 < K 31,i < ∞, i = 1, . . . , k; (b) Zε is a complete sequence of phase sets for the process Zε,n , i.e. Pε,n( z , Zε,n ) = 1,  z ∈ Zε,n−1, n = 1, . . . , N, for every ε ∈ [0, ε0 ]. Lemma 5.1.6. The condition E10 [β¯ ] implies the condition C6 [β¯ ] to hold. Proof. It follows from Lemma 4.1.9 that should be applied to the log-price processes Y ε,n,i.

5.1 Asymptotically uniform upper bounds for rewards – I | 175  , i = 1, . . . , k, then Remark 5.1.11. If the condition E10 [β¯ ] holds with the constants K31,i  ¯  the condition C6 [β ] holds with the constants K30,i , i = 1, . . . , k given by the following formulas:   = 2K31,i , i = 1, . . . , k . (5.18) K30,i

ε,n and pay-off functions g ε (n, ey , x) depend In the model, where log-price processes Y on a perturbation parameter ε ∈ [0, ε0 ], the corresponding log-reward functions also depend on this perturbation parameter. (ε) (ε) Let us denote by Fn,r = σ [ Z ε,l , n ≤ l ≤ r] and Mmax,n,N the class of all Markov Z ε,n such that (a) n ≤ τ ε,n ≤ N and (b) event moments τ ε,n for the log-price process  (ε) {τ ε,n = r} ∈ Fn,r , n ≤ r ≤ N. Let us assume that the conditions B5 [𝛾¯] and C6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. In this case, for Lemma 5.1.3 let us define log-reward functions for the Ameri­ can option for the log-price process  Z ε,n , for every  z = ( y , x) ∈ Z, 0 ≤ n ≤ N and 0 ≤ ε ≤ ε3 ,

ϕ ε,n ( z ) = ϕ ε,n ( y , x) =



Ez ,n g ε (τ ε,n , e Y τε,n , X τ εn ) .

sup (ε)

(5.19)

τ ε,n ∈Mmax,n,N

z ∈ Z for every 0 ≤ n ≤ N. These functions are measurable functions of  Moreover, for Lemma 5.1.3 let us obtain the following asymptotically uniform up­ per bounds for the log-reward functions ϕ ε,n ( y , x ). Theorem 5.1.1. Let the conditions B10 [𝛾¯] and C6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the log-reward functions ϕ ε,n ( y , x) satisfy the following inequalities for  z = y , x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε3 : ( |ϕ ε,n ( y , x)| ≤ M32 +

k

M33,i e𝛾i | y i | .

(5.20)

i =1

Proof. It is analogous to the proof of Theorem 4.1.1. Using Lemma 5.1.3 and the defi­ z = ( y , x) ∈ nition of the log-reward functions, we get the following inequalities for  Z, 0 ≤ n ≤ N and 0 ≤ ε ≤ ε3 : 

|ϕ ε,n ( y , x)| ≤ Ez ,n max |g ε (r, e Y ε,r , X ε,r )| ≤ M32 + n≤r≤N

k

M33,i e𝛾i | y i | ,

(5.21)

i =1

which prove the above theorem. A more complicated is the case, where the conditions B10 [𝛾¯] and C6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. For relation (5.19) let us, according to Lemma 5.1.4, define log-reward functions Z ε,n , for every 0 ≤ ε ≤ ε3 and  z= for the American option for the log-price process  y , x) ∈ Zε,n , 0 ≤ n ≤ N, and get the following upper bounds. (

176 | 5 Convergence of option rewards – I Theorem 5.1.2. Let the conditions B5 [𝛾¯] and C6 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the log-reward functions ϕ ε,n ( y , x) satisfy the following inequalities for  z =  y , x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε3 : (  |ϕ ε,n ( y , x)| ≤ M32 +

k

 M33,i e𝛾i | y i | .

(5.22)

i =1

Proof. It is analogous to the proof of Theorem 5.1.1. In this case Lemma 5.1.4 should be used instead of Lemma 5.1.3. An inequality analogous to (5.21) can be written down. The differences are that it holds for  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε3 , and   with the constants M32 , M33,i , i = 1, . . . , k. Remark 5.1.12. It is useful to note that the log-reward function ϕ ε,N ( y , x) = g ε (N, ey , x),  z = ( y , x) ∈ Zε,N always exists. The corresponding upper bounds for this function are provided by the condition B5 [𝛾¯]. However, there is no guarantee that, under conditions of Theorem 5.1.2, the log-reward functions ϕ ε,n ( y , x) exist for points  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N − 1. Even if these functions do exist for these points, Theorem 5.1.2 does not provide upper bounds for the log-reward functions in these points. However, there is an important case where this problem is not essential. Let us assume that the condition H1 holds. Then, as was pointed out in Remark 5.1.3, for any point  z = ( y , x) ∈ Z, there exists ε1 ( z) ∈ (0, ε0 ] such that  z = ( y , x) ∈ Hε,n,k ⊆ Zε,n, n = 0, . . . , N for ε ∈ [0, ε1 ( z )]. The quantity ε1 ( z) is determined by relation (5.6) y , x) given in Theorem 5.1.2 given in Remark 5.1.3. Thus, the upper bounds for ϕ ε,n ( take place for ε ∈ [0, ε1 ( z )]. Let us now get asymptotically uniform upper bounds for optimal expected re­ wards. The condition D1 [β¯ ] should be replaced, respectively, by the following condition assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative com­ ponents: D7 [β¯ ]: limε→0 Ee β i | Y ε,0,i | < K32,i , i = 1, . . . , k, for some 1 < K32,i < ∞, i = 1, . . . , k . The following lemma gives explicit upper bounds for the expectation of maximum of absolute values for the pay-off processes g ε (n, e Yε,n , X ε,n). Lemma 5.1.7. Let the conditions B5 [𝛾¯], C6 [β¯ ] and D7 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, there exist 0 < ε5 ≤ ε0 and a constant 0 ≤ M34 < ∞ such that the following inequality takes place, for 0 ≤ ε ≤ ε5 : 

E max |g ε (n, e Y ε,n , X ε,n )| ≤ M34 . 0≤ n ≤ N

(5.23)

Proof. The condition D7 [β¯ ] implies that there exists 0 < ε4 ≤ ε0 such that for 0 ≤ ε ≤ ε4 and i = 1, . . . , k, Ee β i | Y ε,0,i | < K32,i . (5.24)

5.1 Asymptotically uniform upper bounds for rewards – I |

177

Relation (5.24) means that, for every ε ≤ ε4 , the condition D1 [β¯ ] (with the con­ stants K32,i , i = 1, . . . , k, replacing the constants K2,i , i = 1, . . . , k) holds for the pro­  ε,n. cess Y Let us now take ε5 = min(ε1 , ε2 , ε4 ), where ε1 and ε2 were defined in relations (5.3) and (5.11). By the definition ε5 > 0. For every fixed 0 ≤ ε ≤ ε5 , the conditions C1 [β¯ ], B1 [𝛾¯], and D1 [β¯ ] (with the con­ stants K30,i , K32,i , i = 1, . . . , k, and L17 , L18,i , i = 1, . . . , k replacing, respectively, the ε,n . constants K1,i , K2,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process Y ε,n , for every 0 ≤ ε ≤ ε5 . Therefore, Lemma 4.1.6 can be applied to the process Y This yields inequality (5.23). For every i = 1, . . . , k, the constant M4 penetrat­ ing the inequality given in Lemma 4.1.6 is a function of the constants K1,i , K2,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k. The explicit expression for this function is given in Remark 4.1.5. The same formula gives the expression for the constant M58 as a function of the constants K30,i , K32,i , i = 1, . . . , k and L17 , L18,i , i = 1, . . . , k. Remark 5.1.13. The explicit formula for the constant M34 is given by the following for­ mula, which follows from the corresponding formula given in Remark 4.1.5: M34 = L9 +



L9 L10,i +

i:𝛾i =0



𝛾

N βi

𝛾i β

i L9 L10,i K30,ii K32,i .

(5.25)

i:𝛾i >0

Remark 5.1.14. The parameter ε5 = min(ε1 , ε2 , ε4 ), where the parameters ε1 , ε2 , and ε4 are determined, respectively, by relations (5.3), (5.11), and (5.24). Note also that one can replace the first-type condition of the exponential moment boundedness D7 [β¯ ] by the following condition of the exponential moment bounded­ ness based on moment-generating functions for the initial value of log-price processes  ε,0, assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative Y components: F4 [β¯ ]: limε→0 Ee±β i Y ε,0,i < K33,i , i = 1, . . . , k, for some 1 ≤ K33,i < ∞, i = 1, . . . , k . Lemma 5.1.8. The condition F4 [β¯ ] implies the condition D7 [β¯ ] to hold. Proof. It follows from the following inequality that holds for any β ≥ 0: Ee β| Y ε,0,i | ≤ Ee+βY ε,0,i + Ee−βY ε,0,i .

(5.26)

Remark 5.1.15. As follows from inequality (5.26), if the condition F4 [β¯ ] holds with the constants K33,i , i = 1, . . . , k, then the condition D7 [β¯ ] holds with the constants K32,i , i = 1, . . . , k given by the following formulas: K32,i = 2K33,i ,

i = 1, . . . , k .

(5.27)

The condition D1 [β¯ ] should be replaced by the following condition assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components:

178 | 5 Convergence of option rewards – I   D7 [β¯ ]: (a) limε→0 Ee β i | Y ε,0,i | < K32,i , i = 1, . . . , k, for some 1 < K32,i < ∞, i = 1, . . . , k; ε,0 ∈ Zε,0} = 1, for every ε ∈ [0, ε0 ], where Zε,0 is the set penetrating (b) P{Z the condition C6 [β¯ ] .

The following lemma gives alternative explicit upper bounds for the expectation of maximal absolute values for the pay-off processes g ε (n, e Yε,n , X ε,n ). Lemma 5.1.9. Let the conditions B5 [𝛾¯], C6 [β¯ ], and D7 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i , i =  1, . . . , k. Then, there exist 0 < ε5 ≤ ε0 and a constant 0 ≤ M58 < ∞ such that the  following inequality takes place for 0 ≤ ε ≤ ε5 : 

 . E max |g ε (n, e Y ε,n , X ε,n )| ≤ M34

0≤ n ≤ N

(5.28)

Proof. The condition D7 [β¯ ] implies that there exists 0 < ε4 ≤ ε0 such that 0 ≤ ε ≤ ε4 and i = 1, . . . , k,  Ee β i | Y ε,0,i | < K32,i . (5.29) Relation (5.29) means that, for every ε ≤ ε4 , the condition D1 [β¯ ] (with the con­   , i = 1, . . . , k, replacing the constants K2,i , i = 1, . . . , k) holds for the pro­ stants K32,i  cess Y ε,n . Let us now take ε5 = min(ε1 , ε2 , ε4 ), where ε1 and ε2 were defined in relations (5.8) and (5.11). By the definition ε5 > 0. For every fixed 0 ≤ ε ≤ ε5 , the conditions C1 [β¯ ], B1 [𝛾¯], and D1 [β¯ ] (with the con­   stants K30,i , K32,i i = 1, . . . , k, and L17 , L18,i , i = 1, . . . , k replacing, respectively, the    ε,n. , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process Y constants K1,i , K2,i ε,n , for every 0 ≤ ε ≤ ε5 . Therefore, Lemma 4.1.7 can be applied to the process Y This yields inequality (5.28). For every i = 1, . . . , k, the constant M4 penetrat­   , K2,i ,i = ing the inequality given in Lemma 4.1.7 is a function of the constants K1,i 1, . . . , k and L1 , L2,i , i = 1, . . . , k. The explicit expression for this function is given in  Remark 4.1.6. The same formulas give the expression for the constant M34 as a function   of the constants K30,i , K32,i , i = 1, . . . , k and L17 , L18,i , i = 1, . . . , k.  Remark 5.1.16. The explicit formula for the constant M34 is given by the following formula, which follows from the corresponding formulas given in Remark 4.1.6: 𝛾 𝛾i



N i    M34 = L9 + L9 L10,i + L9 L10,i (K30,i ) βi (K32,i ) βi . (5.30)

i:𝛾i =0

i:𝛾i >0

ε5

Remark 5.1.17. The parameter = min(ε1 , ε2 , ε4 ), where the parameters ε1 , ε2 , and ε4 are determined, respectively, by relations (5.8), (5.11), and (5.29). Note also that one can replace the second-type condition of the exponential moment boundedness D7 [β¯ ] by the following condition of the exponential moment bounded­ ness based on moment-generating functions for the initial value of log-price processes ε,0, and assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnega­ Y tive components:

5.1 Asymptotically uniform upper bounds for rewards – I | 179   , i = 1, . . . , k, for some 1 ≤ K33,i < ∞, i = 1, . . . , k; F4 [β¯ ]: (a) limε→0 Ee±β i Y ε,0,i < K33,i ε,0 ∈ Zε,0} = 1, for every ε ∈ [0, ε0 ], where Zε,0 is the set penetrating (b) P{Z the condition C6 [β¯ ] .

Lemma 5.1.10. The condition F4 [β¯ ] implies the condition D7 [β¯ ] to hold. Proof. It follows from inequality (5.26).  Remark 5.1.18. If the condition F4 [β¯ ] holds with the constants K33,i , i = 1, . . . , k, then  ¯  the condition D7 [β ] holds with the constants K32,i , i = 1, . . . , k given by the following formulas:   K32,i = 2K33,i , i = 1, . . . , k . (5.31)

Let us assume that either conditions of Lemma 5.1.7 or Lemma 5.1.9 hold. In this case, the optimal expected rewards for the American option for the log-price process  Z ε,n can be defined, respectively, for every 0 ≤ ε ≤ ε5 , or, for every 0 ≤ ε ≤ ε5 , (ε)

Φ ε = Φ(Mmax,N ) =

sup (ε) τ ε,0 ∈Mmax,N



Eg ε (τ ε,0 , e Y ε,τε,0 , X ε,τ ε,0 ) .

(5.32)

Also, as follows from Theorem 2.3.4, the following formula takes place, respec­ tively, for every 0 ≤ ε ≤ ε5 , or for every 0 ≤ ε ≤ ε5 : ε,0 , X ε,0) . Φ ε = Eϕ ε,0 (Y

(5.33)

The following two theorems give asymptotically uniform upper bounds for the optimal expected rewards Φ ε . Theorem 5.1.3. Let the conditions B10 [𝛾¯], C6 [β¯ ], and D7 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the following inequality takes place, for 0 ≤ ε ≤ ε5 : |Φ ε | ≤ M34 .

(5.34)

Proof. It follows from the definition of Φ ε and Lemma 5.1.7 that, for 0 ≤ ε ≤ ε5 : |Φ ε | ≤

max

(ε) τ ε,0 ∈Mmax,N



E|g ε (τ ε,0 , e Y τε,0 , X τ ε,0 )| 

≤ E max |g ε (n, e Y ε,n , X ε,n )| ≤ M34 . 0≤ n ≤ N

(5.35)

This inequality proves the theorem. Theorem 5.1.4. Let the conditions B5 [𝛾¯], C6 [β¯ ], and D17 [β¯ ] hold, and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the following inequality takes place, for 0 ≤ ε ≤ ε5 :  |Φ ε | ≤ M34 .

(5.36)

Proof. It is analogous to the proof of Theorem 5.1.3. The only difference is that Lemma 5.1.9 is used instead of Lemma 5.1.7. In this case, the inequality analogous to (5.35) can be written for 0 ≤ ε ≤ ε5 .

180 | 5 Convergence of option rewards – I

5.2 Modulated Markov LPP with bounded characteristics In this section, we present results about the convergence of option rewards for per­ turbed multivariate modulated Markov log-price processes with bounded characteris­ tics.

5.2.1 Locally uniform convergence for functions and probability measures In order to formulate the convergence conditions, we should introduce notions of lo­ cally uniform convergence for functions and probability measures. We assume that the phase space of the index component X is a Polish metric space, i.e. complete, separable, metric space, with a metric dX (x , x ). In this case, the phase space Z = Rk × X is also a Polish space with the metric 1 z ,  z  ) = (| y − y  |2 + d2X (x , x )) 2 ,  z  = ( y  , x ),  z  = ( y  , x ) ∈ Z, where dZ ( 1   2   2 2        | y − y | = (|y1 − y1 | +· · ·+|y k − y k | ) ,  y = ( y 1 , . . . , y k ),  y = (y 1 , . . . , y k ) ∈ Rk is the Euclidean metric in the space Rk . Let us denote by BX the corresponding Borel σ-algebra of subsets of X (the mini­ mal σ-algebra containing all balls R d (x) = {x ∈ X : dX (x, x ) ≤ d} in the space X) and let BZ be the corresponding Borel σ-algebra of subsets of Z. z ) be real-valued functions defined on the space Z and depending on the Let f ε ( parameter ε ∈ [0, ε0 ]. U We use the symbol f ε ( z0 ) −→ f0 ( z0 ) as ε → 0 to indicate that the functions f ε ( z) converge to a function f0 ( z ) locally uniformly in point  z0 as ε → 0. This means that, zε →  z0 as ε → 0 (in the sense that dZ ( zε ,  z0 ) → 0 as ε → 0), for any  z ε ) → f0 ( z0 ) f ε (

as

ε → 0.

(5.37)

The term used above for the locally uniform convergence is clarified by the follow­ ing useful proposition. Lemma 5.2.1. Relation (5.37) holds if and only if the following relation holds: lim lim

sup

0< d →0 ε →0 d ( z 0 )≤ d Z z ,

|f ε ( z ) − f0 ( z0 )| = 0 .

(5.38)

Proof. Let relation (5.37) holds but relation (5.38) does not. This assumption im­ plies that (a) there exist c > 0 and a sequence 0 < d n → 0 as n → ∞ such that limε→0 supdZ (z ,z0)≤d n |f ε ( z ) − f0 ( z0 )| ≥ c. Relation (a) implies that (b) there exists z ) − f0 ( z0 )| ≥ c/2. Relations 0 < ε n → 0, n = 1, 2, . . . such that supdZ(z ,z0)≤d n |f ε n ( (b) imply that (c) there exists  z n , n = 1, 2, . . . such that dZ ( zn ,  z0 ) ≤ d n , n = 1, 2, . . . and |f ε n ( z n ) − f0 ( z0 )| ≥ c/4. Finally, relations (c) imply that (d) f ε n ( z n )  →f0 ( z0 ) as

5.2 Modulated Markov LPP with bounded characteristics

| 181

n → ∞. Obviously, (d) contradicts to the assumption that relation (5.37) holds. Thus, relation (5.37) implies relation (5.38). Now, let us assume that relation (5.38) holds. Let us take an arbitrary c > 0. Rela­ tion (5.38) implies that (e) there exists d = d(c) > 0 such that limε→0 supdZ(z ,z0)≤d(c) |f ε ( z ) − f0 ( z0 )| ≤ c. Let us take an arbitrary  zε →  z0 as ε → 0. Obviously, there ex­ ists ε(c) > 0 such that (f) dZ ( z,  z0 ) ≤ d(c) for ε ≤ ε(c). Relations (e) and (f) imply z ε ) − f0 ( z0 )| ≤ supdZ (z ,z0)≤d(c) |f ε ( z) − f0 ( z0 )| ≤ c for ε ≤ ε(c). Since an that (g) |f ε ( arbitrary choice of c > 0, relations (g) imply that relation (5.37) holds. Thus, relation (5.38) implies relation (5.37). It is useful to mention two particular cases. The locally uniform convergence claimed by relation (5.37) implies convergence of functions f ε ( z ) to the function f0 ( z) as ε → 0, in point  z0 , i.e. f ε ( z0 ) → f0 ( z0 ) as ε → 0. Indeed, it is enough to choose  zε ≡  z0 in relation (5.37) to see this. The locally uniform convergence claimed by relation (5.37) implies continuity of z ), in point  z0 . function f0 ( Indeed, the relation ε → 0 in (5.38) includes also the case, where ε ≡ 0. For the set A ∈ BZ , let us denote by A− , the subset of internal points  z ∈ A, for z) ⊆ A. Also, let us denote A+ = which there exists δz > 0 such that the ball R δz ( − Z \ A , the closure of set A (which contains all points from the set A plus all limits of convergent sequences of points from A). Finally, let us denote ∂A = A+ \ A− , the boundary of set A. Let P ε (A) be the probability measures defined on the σ-algebra BZ and depending on the parameter ε ∈ [0, ε0 ]. We use the symbol P ε (·) ⇒ P0 (·) as ε → 0 to indicate that the measures P ε (A) weakly converge to a measure P0 (A) as ε → 0. This means that P ε (A) → P0 (A) as ε → 0 for all the Borel sets A such that P0 (∂A) = 0. z , A),  z ∈ Z be a family of probability measures defined on the σ-algebra Let P ε ( BZ and depending on the parameter ε ∈ [0, ε0 ]. U

We use the symbol P ε ( z0 , ·) =⇒ P0 ( z0 , ·) as ε → 0 to indicate that measures P ε ( z , A) weakly converge to a measure P0 ( z , A) locally uniformly for point  z0 as ε → 0. zε →  z0 as ε → 0 (in the sense that dZ ( zε ,  z0 ) → 0 as ε → 0), This means that, for any  z ε , ·) ⇒ P0 ( z0 , ·) P ε (

as

ε → 0.

(5.39)

Again, it is useful to mention two particular cases. The locally uniform weak convergence claimed by relation (5.39) implies the weak z0 , ·) ⇒ P0 ( z0 , ·) as ε → 0. convergence of measures P ε ( Indeed, it is enough to choose  zε ≡  z0 in (5.39) to see this. The locally uniform convergence claimed by relation (5.39) implies weak continu­ z , A), in point  z0 , i.e. that P0 ( z ε , ·) ⇒ P0 ( z0 , ·) as ity for the family of measures P0 ( ε → 0 for any  zε →  z0 as ε → 0.

182 | 5 Convergence of option rewards – I Indeed, relation (5.39) implies that, for any Borel set A such that P0 (∂A) = 0 U functions P0 ( z ε , A) −→ P0 ( z0 , A) as ε → 0. By the remarks above this relation implies that the function P0 ( z , A) is continuous in point  z0 .

5.2.2 Convergence of log-reward functions for multivariate modulated Markov log-price processes with bounded characteristics Let us now consider the log-reward functions ϕ ε,n ( y , x) introduced in Subsection 5.1.2 and give conditions for their convergence. We impose on the pay-off functions g ε (n, ey , x) the following condition of locally uniform convergence: I1 : There exist measurable sets Zn ⊆ Z, n = 0, . . . , N, such that the pay-off functions U

g ε (n, ey0 , x0 ) −→ g0 (n, ey0 , x0 ) as ε → 0, i.e. g ε (n, ey ε , x ε ) → g0 (n, ey0 , x0 ) as ε → 0 for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn as ε → 0, and n = 0, . . . , N . z , A ) = P{  Z ε,n ∈ We also impose on transition probabilities of log-price processes P ε,n ( A/ Z ε,n−1 =  z } the following condition of locally uniform weak convergence: J1 : There exist measurable sets Z n ⊆ Z, n = 0, . . . , N such that U

z0 , ·) =⇒ P0,n ( z0 , ·) as ε → 0, i.e. P ε,n ( z ε , ·) ⇒ P0,n ( z0 , ·) as ε → 0, for (a) P ε,n (  any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn−1 as ε → 0, and n = 1, . . . , N;  (b) P0,n ( z0 , Zn ∩ Z z0 ∈ Zn−1 ∩ Z n −1 and n = 1, . . . , N, where Z n , n ) = 1, for every  n = 0, . . . , N are sets introduced in the condition I1 . 



A typical example is where the sets Zn , Zn , n = 1, . . . , N are empty sets. Then the condition J1 (b) obviously holds.   Another typical example is where the sets Z n , Zn , n = 1, . . . , N are at most finite or countable sets. Then the assumption that the measures P0,n ( z 0 , A ),  z0 ∈ Zn−1 ∩ Z n −1   have no atoms at points from the sets Zn , Zn , for every n = 1, . . . , N, implies that the condition J1 (b) holds. One more example is where measures P0,n ( z0 , A ) ,  z0 ∈ Zn−1 ∩ Z n −1 , n = 1, . . . , N are absolutely continuous with respect to some σ-finite measure P(A) on BZ and   P(Zn ), P(Zn ) = 0, n = 1, . . . , N. This assumption also implies that the condition J1 (b) holds. Theorem 5.2.1. Let the conditions B5 [𝛾¯] and C6 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions I1 and J1 hold. Then, for every n = 0, 1, . . . , N, the z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn ∩ Z following relation takes place for any  n: ϕ ε,n ( y ε , x ε ) → ϕ0,n ( y0 , x0 )

as

ε → 0.

(5.40)

5.2 Modulated Markov LPP with bounded characteristics

|

183

Proof. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. By Theorem 5.1.1, the conditions B5 [𝛾¯] and C6 [β¯ ] imply that there exist ε3 > 0 and the constants M32 , M33,i , i = 1, . . . , k such that the following upper bounds take place for the log-reward functions ϕ ε,n ( y , x) for  z = ( y , x) ∈ Z, 0 ≤ n ≤ N and ε ≤ ε3 : |ϕ ε,n ( y , x)| ≤ M32 +

k

M33,i e𝛾i | y i | .

(5.41)

i =1

z = ( y , x) ∈ Using inequalities (5.41) and the condition C6 [β¯ ] we get for any  Z, 0 ≤ n ≤ N − 1 and ε ≤ ε3 ,  # α  ε,n+1 , X ε,n+1   Ez ,n ϕ ε,n+1 Y ⎛ ⎞α k

≤ Ez ,n ⎝M32 + M33,i e𝛾i | Y ε,n+1,i | ⎠ i =1

⎛

≤ Ez ,n (k + 1)

α −1 ⎝

α M32

α ≤ (k + 1)α−1 ⎝M32 +

k

⎞

α M33,i Ez ,n e𝛾i α| y i | e𝛾i α| Y ε,n+1,i −y i | ⎠

i =1

⎛ ≤ (k + 1)

⎞ α M33,i e𝛾i α| Y ε,n+1,i | ⎠

i =1

⎛

α −1 ⎝

+

k

α M32

+

⎛ α ≤ (k + 1)α−1 ⎝M32 +

k

⎞ α M33,i e𝛾i α| y i| Ez ,n e β i | Y ε,n+1,i −Y ε,n,i | ⎠

i =1 k

⎞ α M33,i e𝛾i α| y i| K30,i ⎠ = Q( y) < ∞ .

(5.42)

i =1

y , x) are defined by the fol­ According to Theorem 2.3.1, the reward functions ϕ ε,n ( lowing boundary relation: y , x) = g ε (N, ey , x) , ϕ ε,N (

 z = ( y , x) ∈ Z ,

(5.43)

and backward recurrence relations ε,n+1 , X ε,n+1)), ϕ ε,n ( y , x) = max(g(n, ey , x), Ez ,n ϕ ε,n+1(Y  z = ( y , x) ∈ Z ,

n = N − 1, . . . , 0 .

(5.44)

We shall prove asymptotic relations (5.40) using recursive reasoning based on re­ lations (5.42)–(5.44). The condition I1 implies, due to the boundary relation (5.43) for the log-reward function ϕ ε,N ( y , x), that the following relation holds for an arbitrary  z ε = ( yε , xε ) →    z0 = ( y0 , x0 ) ∈ ZN ∩ ZN as ε → 0: ϕ ε,N ( y ε , x ε ) → ϕ0,N ( y0 , x0 )

as

ε → 0.

(5.45)

184 | 5 Convergence of option rewards – I Let us prove that the following relation, similar to (5.45), holds for an arbitrary  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ ZN −1 ∩ Z N −1 as ε → 0: y ε , x ε ) → ϕ0,N −1( y0 , x0 ) ϕ ε,N −1(

as

ε → 0.

(5.46)

ε,n( Z ε,n ( z) = (Y z ), X ε,n( z )) be, for every n = 1, . . . , N and  z ∈ Z, a random Let   vector such that P{Z ε,n ( z ) ∈ A} = P ε,n ( z , A), A ∈ BZ . Let us prove that the following relation takes place: ε,N ( 0,N ( ϕ ε,N (Y z ε ), X ε,N ( z ε )) −→ ϕ0,N (Y z0 ), X0,N ( z0 )) d

as

ε → 0.

(5.47)

Take an arbitrary sequence ε r → ε0 = 0 as r → ∞. The condition J1 implies that d  Z ε r ,N ( z ε r ) −→  Z ε0 ,N ( z ε0 )

as

r → ∞,

(5.48)

and ε0 ,N ( z ε0 ) ∈ ZN ∩ Z P{ Z N} = 1 .

(5.49)

According to the Skorokhod representation theorem, relation (5.48) let one con­ ε ,N ( Z ε r ,N ( z ε r ) = (Y z ε r ), X ε r ,N ( z ε r )), r = 0, 1, . . . on some struct the random variables  r probability space (Ω, F , P) such that, for every r = 0, 1, . . . , ε ,N ( z ε r ) ∈ A } = P{  Z ε r ,N ( z ε r ) ∈ A} , P{ Z r

A ∈ BZ ,

(5.50)

Z ε r ,N ( z ε r ) converge almost sure to the random variable and such that random variables    Z ε0 ,N ( z ε0 ) as r → ∞, i.e. a.s.  Z ε r ,N ( z ε r ) −→  Z ε0 ,N ( z ε0 )

as

r → ∞.

(5.51)

Let us denote Z ε r ,N ( z ε r , ω) →  Z ε0 ,N ( z ε0 , ω) as r → ∞} A N = {ω ∈ Ω :  and B N = {ω ∈ Ω :  Z ε0 ,N ( z ε0 , ω) ∈ ZN ∩ Z N} . Relation (5.51) implies that P(A N ) = 1. Also relations (5.49) and (5.50) imply that P(B N ) = 1. Thus, P(A N ∩ B N ) = 1. By relation (5.45) and the definition of the sets A N and B N , the following relation takes place, for every ω ∈ A N ∩ B N : ε ,N ( z ε r , ω), X ε r ,N ( z ε r , ω)) ϕ ε r ,N (Y r ε ,N ( → ϕ ε0 ,N (Y z ε0 , ω), X ε0 ,N ( z ε0 , ω)) 0

as

r → ∞.

(5.52)

Since P(A N ∩ B N ) = 1, relation (5.52) implies that the following relation holds: ε ,N ( z ε r ), X ε r ,N ( z ε r )) ϕ ε r ,N (Y r a.s.

ε ,N ( −→ ϕ ε0 ,N (Y z ε0 ), X ε0 ,N ( z ε0 )) 0

as

r → ∞.

(5.53)

5.2 Modulated Markov LPP with bounded characteristics

|

185

Relation (5.50) implies that, for every r = 0, 1, . . . , ε ,N ( P{ϕ ε r ,N (Y z ε r ), X ε r ,N ( z ε r )) ∈ A} r ε r ,N ( z ε r ), X ε r ,N ( z ε r )) ∈ A} , = P{ϕ ε r ,N (Y

A ∈ BZ .

(5.54)

Relations (5.53) and (5.54) imply that the following relation holds: ε r ,N ( z ε r ), X ε r ,N ( z ε r )) ϕ ε r ,N (Y ε0 ,N ( −→ ϕ ε0 ,N (Y z ε0 ), X ε0 ,N ( z ε0 )) d

as

r → ∞.

(5.55)

Since an arbitrary choice of a sequence ε r → ε0 , relation (5.55) implies the relation of weak convergence (5.47). ε,n( Z ε,n ( z) = ( Y z ), X ε,n( z )), the following By the definition of the random vectors  relation take place for every  z ∈ Z: ε,N ( ε,N , X ε,N )|α z), X ε,N ( z ))|α = Ez ,N −1|ϕ ε,N (Y E|ϕ ε,N (Y

(5.56)

The function Q( y ) defined in (5.42) is a continuous function of argument  y ∈ Rk . Therefore, relation (5.42) implies that the following relation takes place: ε,N ( lim E|ϕ ε,N (Y z ε ), X ε,N ( z ε ))|α ≤ Q( y0 ) < ∞ . ε →0

(5.57)

Relations (5.47) and (5.57) imply that the following relation holds: ε,N ( ε,N , X ε,N ) Eϕ ε,N (Y z ε ), X ε,N ( z ε )) = Ez ε ,N −1 ϕ ε,N (Y 0,N ( → Eϕ ε,N (Y z0 ), X0,N ( z0 )) 0,N , X0,N ) = Ez0 ,N −1 ϕ0,N (Y

as

ε → 0.

(5.58)

Finally, relation (5.58) and the condition I1 imply that the following relation, equivalent to (5.46), holds: ε,N , X ε,N )) y ε , x ε ) = max(g ε (N − 1, ey ε , x ε ), Ez ε ,N −1 ϕ ε,N (Y ϕ ε,N −1( → ϕ0,N −1 ( y0 , x0 ) = max(g0 (N − 1, ey0 , x0 ), 0,N , X0,N )) Ez0 ,N −1 ϕ0,N (Y

ε → 0.

as

(5.59)

By repeating the recursive procedure described above we prove relations (5.40) for n = N − 2, . . . , 0. Remark 5.2.1. According to Lemma 5.1.5, the conditions C6 [β¯ ] can be replaced in The­ orems 5.2.1 by the condition E10 [β¯ ]. Remark 5.2.2. As follows from the remarks made in Subsection 5.2.1, the following point-wise convergence relation takes place for the log-reward functions ϕ ε,n ( y , x ), z = ( y , x) ∈ Zn ∩ Z , n = 0, 1, . . . , N: for every  n ϕ ε,n ( y , x) → ϕ0,n ( y , x)

as

ε → 0.

(5.60)

186 | 5 Convergence of option rewards – I Also, the reward function ϕ0,n ( y , x) is continuous in points  z = ( y , x) ∈ Zn ∩ Z n , for every n = 0, 1, . . . , N. The following theorem gives alternative conditions for the convergence of log-re­ ward functions for the American-type options. Theorem 5.2.2. Let the conditions B5 [𝛾¯] and C6 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions H1 , I1 , and J1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn ∩ Z n: ϕ ε,n ( y ε , x ε ) → ϕ0,n ( y0 , x0 )

as

ε → 0.

(5.61)

Proof. It repeats the proof of Theorem 5.2.1, with some minor changes. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. By Theorem 5.1.2, the conditions B5 [𝛾¯] and C6 [β¯ ] imply that there exist ε3 > 0 and   the constants M32 , M33,i , i = 1, . . . , k such that the following upper bounds take place for the log-reward functions ϕ ε,n ( y , x), for  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N, ε ≤ ε3 :  |ϕ ε,n ( y , x)| ≤ M32 +

k

 M33,i e𝛾i | y i | .

(5.62)

i =1

ε,n+1 ∈ Zε,n+1} = 1 for  z∈ Recall that the condition C6 [β¯ ] (b) implies that Pz ,n {Z Zε,n , n = 0, . . . , N − 1. Using these relations, inequalities (5.62) and the condition z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N − 1, ε ≤ ε3 , C6 [β¯ ], we get for any   # α  ε,n+1 , X ε,n+1   Ez ,n ϕ ε,n+1 Y ⎛ ⎞α k

  ≤ Ez ,n ⎝M32 + M33,i e𝛾i | Y ε,n+1,i | ⎠ i =1

⎛

≤ Ez ,n (k + 1)

α −1 ⎝

 α (M32 )

+

k

⎞  (M33,i )α e𝛾i α| Y ε,n+1,i | ⎠

i =1

⎛

⎞ k

 α  ≤ (k + 1)α−1 ⎝(M32 ) + (M33,i )α Ez ,n e𝛾i α| y i | e𝛾i α| Y ε,n+1,i −y i | ⎠ i =1

⎛ ≤ (k + 1)

α −1 ⎝

⎛

 α (M32 )

⎞ k

 α 𝛾i α | y i | β i | Y ε,n+1,i − Y ε,n,i | ⎠ + (M33,i ) e Ez ,n e i =1

⎞ k

 α   ⎠ ≤ (k + 1)α−1 ⎝(M32 ) + (M33,i )α e𝛾i α| y i | K30,i y) < ∞ . = Q (

(5.63)

i =1

The following part of the proof repeats the corresponding part of the proof for The­ orem 5.2.1. The only difference is that additional reasoning is required for the relation, which should replace relation (5.57).

5.2 Modulated Markov LPP with bounded characteristics

| 187

z (·) =  zε , ε ∈ As in the proof of Theorem 5.2.1, we choose an arbitrary function  zε →  z0 ∈ ZN −1 ∩ Z as ε → 0, and should prove that [0, ε0 ] such that  N −1 ε,N ( lim E|ϕ ε,N (Y z ε ), X ε,N ( z ε ))|α ≤ Q ( y0 ) < ∞ . ε →0

(5.64)

In this case, we should use the condition H1 , due to which the point  z0 = y0 , x0 ) ∈ Zε,N −1, for every ε ∈ [0, ε1 ( z0 )]. The parameter ε1 ( z0 ) ∈ (0, ε0 ] has ( been defined in relation (5.6) given in Remark 5.1.3. zε →  z0 as ε → 0, this implies that there exists ε1 ( z (·)) ∈ (0, ε1 ( z0 )] such Since  that  z ε = ( y ε , x ε ) ∈ Zε,n , n = 0, . . . , N, for every ε ∈ [0, ε1 ( z (·))]. ε,N , X ε,N )|α given in relation (5.63) Therefore, the upper bounds for Ez ,N −1|ϕ ε,N (Y   z (·)))]. hold for every ε ∈ [0, min(ε3 , ε1 ( Due to the above remarks and continuity of the function Q ( y), relation (5.63) im­ plies that relation (5.64), replacing relation (5.57), holds. The rest of the proof repeats the corresponding part of the proof of Theorem 5.2.1.

5.2.3 Convergence of optimal expected rewards for modulated Markov log-price processes with bounded characteristics (ε)

Let us now consider the optimal expected rewards Φ ε = Φ(Mmax,N ) introduced in Subsection 5.1.2 and give conditions for their convergence. We should additionally impose a condition of weak convergence on the initial dis­ tributions P ε,0(A) = P{ Z ε,0 ∈ A}: K1 : (a) P ε,0(·) ⇒ P0,0 (·) as ε → 0;   (b) P0,0 (Z0 ∩ Z 0 ) = 1, where Z0 and Z0 are the sets introduced in the conditions I1 and J1 . 



A typical example is where the sets Z0 , Z0 , n = 1, . . . , N are empty sets. Then the condition K1 (b) obviously holds.   Another typical example is where sets Z0 , Z0 are at most finite or countable sets.   Then the assumption that measure P0,0 (·) has no atoms at points of sets Z0 , Z0 im­ plies that the condition K1 (b) holds. One more example is where measure P0,0 (·) is absolutely continuous with respect   to some σ-finite measure P(A) on BZ and P(Z0 ), P(Z0 ) = 0. This assumption also implies that the condition K1 (b) holds. Theorem 5.2.3. Let the conditions B5 [𝛾¯], C6 [β¯ ], and D7 [β¯ ] hold with the vector parame­ ters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions I1 , J1 , and K1 hold. Then the following relation takes place: Φ ε → Φ0 as ε → 0 . (5.65)

188 | 5 Convergence of option rewards – I Proof. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. According to Theorem 5.1.1, the conditions B5 [𝛾¯] and C6 [β¯ ] imply that there exist ε3 > 0 and the constants M32 , M33,i , i = 1, . . . , k such that the following upper bounds take place for the reward functions ϕ ε,0 ( y , x) for  z = ( y , x) ∈ Z and ε ≤ ε3 : |ϕ ε,0 ( y , x)| ≤ M32 +

k

M33,i e𝛾i | y i | .

(5.66)

i =1

As was pointed out in the proof of Lemma 5.1.7, the condition D7 [β¯ ] implies that there exists 0 < ε4 ≤ ε0 such that 0 ≤ ε ≤ ε4 and i = 1, . . . , k, Ee β i | Y ε,0,i | < K32,i .

(5.67)

Using relations (5.66), (5.67), and (5.69), we get that for any ε ≤ ε5 = min(ε3 , ε4 ), ⎛ ⎞α k

α 𝛾 | Y | ε,0 , X ε,0)| ≤ E ⎝M32 + M33,i e i ε,0,i ⎠ E|ϕ ε,0 (Y i =1

⎛

α ≤ E(k + 1)α−1 ⎝M32 +

≤ (k + 1)

⎞ α M33,i e𝛾i α| Y ε,0,i | ⎠

i =1

⎛ α −1 ⎝

k

α M32

+

k

⎞

α M33,i Ee β i | Y ε,0,i | ⎠

i =1

⎛ α ≤ (k + 1)α−1 ⎝M32 +

k

⎞ α M33,i K32,i ⎠ = Q < ∞ .

(5.68)

i =1

Also, as follows from Theorem 2.3.4, the following formula takes place, for every 0 ≤ ε ≤ ε5 : ε,0 , X ε,0) . Φ ε = Eϕ ε,0 (Y (5.69) Let us prove that the following relation takes place: ε,0 , X ε,0) −d→ ϕ0,0 (Y 0,0 , X0,0 ) ϕ ε,0 (Y

as

ε → 0.

(5.70)

Take an arbitrary sequence ε r → ε0 = 0 as r → ∞. The condition K1 implies that d  Z ε r ,0 −→  Z ε0 ,0

as

r → ∞,

(5.71)

and ε0 ,0 ∈ Z0 ∩ Z P{ Z 0} = 1.

(5.72)

According to the Skorokhod representation theorem, for relation (5.71) let us con­ ε ,0 , X ε ,0), r = 0, 1, . . . on some probability struct the random variables  Z ε r ,0 = (Y r r space (Ω, F , P) such that, for every r = 0, 1, . . . , ε ,0 ∈ A} = P{ P{ Z Z ε r ,0 ∈ A} , r

A ∈ BZ ,

(5.73)

5.2 Modulated Markov LPP with bounded characteristics

|

189

and a.s.   Z ε r ,0 −→  Z ε0 ,0

as

r → ∞.

(5.74)

Let us denote Z ε r ,0 (ω) →  Z ε0 ,0(ω) as r → ∞} A = {ω ∈ Ω :  and Z ε0 ,0 (ω) ∈ Z0 ∩ Z B = {ω ∈ Ω :  0}. Relation (5.74) implies that P(A) = 1. Also relations (5.72) and (5.73) imply that P(B) = 1. Thus, P(A ∩ B) = 1. By relation (5.40) given in Theorem 5.2.1, which require the conditions B5 [𝛾¯], C6 [β¯ ], I1 , and J1 to hold, and the definition of the sets A and B, for every ω ∈ A ∩ B, ε ,0 (ω), X ε ,0 (ω)) → ϕ ε0 ,0 (Y ε ,0 (ω), X ε ,0 (ω)) ϕ ε r ,0 (Y r r 0 0

as

r → ∞.

(5.75)

Since P(A ∩ B) = 1, relation (5.75) implies the following relation: a.s. ε ,0 , X ε ,0 ) − ε ,0 , X ε ,0 ) ϕ ε r ,N (Y → ϕ ε0 ,N (Y r r 0 0

as

r → ∞.

(5.76)

Relation (5.73) implies that, for every r = 0, 1, . . . , ε ,0 , X ε ,0 ) ∈ A} = P{ϕ ε r ,0(Y ε r ,0 , X ε r ,0 ) ∈ A} , P{ϕ ε r ,0 (Y r r

A ∈ BZ .

(5.77)

Relations (5.76) and (5.77) imply that the random variables ε r ,0 , X ε r ,0 ) −d→ ϕ ε0 ,0 (Y ε0 ,0 , X ε0 ,0) ϕ ε r ,0 (Y

as

r → ∞.

(5.78)

Since an arbitrary choice of a sequence ε r → ε0 , relation (5.78) implies the relation of weak convergence (5.70). Relations (5.68), (5.69), and (5.70) imply relation (5.65). Remark 5.2.3. According to Lemmas 5.1.5 and 5.1.8, the conditions C6 [β¯ ] and D7 [β¯ ] can be replaced in Theorem 5.2.3, respectively, by the conditions E10 [β¯ ] and F5 [β¯ ]. The following theorem gives alternative conditions of convergence for optimal ex­ pected rewards. Theorem 5.2.4. Let the conditions B5 [𝛾¯], C6 [β¯ ], and D7 [β¯ ] hold with the vector pa­ rameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions H1 , I1 , J1 , and K1 hold. Then the following relation takes place: Φ ε → Φ0

as

ε → 0.

(5.79)

190 | 5 Convergence of option rewards – I Proof. It repeats the proof of Theorem 5.2.3 with some minor changes. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. According to Theorem 5.1.2, the conditions B5 [𝛾¯] and C6 [β¯ ] imply that there exist    ε3 > 0 and the constants M32 , M33,i , i = 1, . . . , k such that the following upper bounds take place for the reward functions ϕ ε,0 ( y , x) for  z = ( y , x) ∈ Zε,0, ε ≤ ε3 :  |ϕ ε,0 ( y , x)| ≤ M32 +

k

 M33,i e𝛾i | y i | .

(5.80)

i =1

As was pointed out in the proof of Lemma 5.1.9, the condition D7 [β¯ ] implies that there exists 0 < ε4 ≤ ε0 such that 0 ≤ ε ≤ ε4 and i = 1, . . . , k,  . Ee β i | Y ε,0,i | < K32,i

(5.81)

ε,0 ∈ Zε,0} = 1 for  Recall that the condition D7 [β¯ ] (b) implies that Pz ,n {Z z ∈ Zε,0. Using this relation and relations (5.69), (5.80) and (5.81), we get that for any ε ≤ ε5 = min(ε3 , ε4 ), ⎛ ⎞α k

α   𝛾 | Y | ε,0 , X ε,0)| ≤ E ⎝M32 + M33,i e i ε,0,i ⎠ E|ϕ ε,0 (Y i =1

⎛

≤ E(k + 1)

α −1 ⎝

 α (M32 )

+

k

⎞  (M33,i )α e𝛾i α| Y ε,0,i | ⎠

i =1

⎛

⎞ k

 α  ≤ (k + 1)α−1 ⎝(M32 ) + (M33,i )α Ee β i | Y ε,0,i | ⎠ i =1

⎛ ≤ (k + 1)

α −1 ⎝

 α (M32 )

⎞ k

 α  ⎠ + (M33,i ) K32,i = Q < ∞ .

(5.82)

i =1

The following part of the proof repeats the corresponding part of the proof of The­ orem 5.2.3. The only difference is that relation (5.82) should be used instead of relation (5.68). Also, relation analogous to (5.75) does require reference to relation (5.61) given in Theorem 5.2.2 instead of relation (5.40) given in Theorem 5.2.1.

5.2.4 Convergence of log-rewards for modulated Markov log-price processes with a discrete phase space of the modulator Some useful remarks can be made for the case, where the modulator X ε,n has the dis­ crete phase space X = {x1 , x2 , . . . }. In order to simplify notations one can assume that x i = i, i = 1, 2, . . . .

5.2 Modulated Markov LPP with bounded characteristics

| 191

The space X equipped with the metric dX (i, j) = I (i ≠ j) is a complete and sepa­ rable metric space. The moment conditions B5 [𝛾¯], C6 [β¯ ], and D7 [β¯ ] as well as the conditions E10 [β¯ ] and F5 [β¯ ] do not change their formulations. Convergence conditions can be simplified if we take into consideration that the convergence relation, dX (i ε , i0 ) → 0 as ε → 0, means that i ε ≡ i0 for all ε small enough. The condition I1 can be replaced by the following simpler condition: I2 : There exist measurable sets Yi,n ⊆ Y, i ∈ X, n = 0, . . . , N, such that the payoff yε →  y0 ∈ Yi,n , i ∈ X and function g ε (n, ey ε , i) → g0 (n, ey0 , i) as ε → 0 for any  n = 0, . . . , N . The condition I2 implies that the condition I1 holds with the sets Zn = ∪i∈X (Yi,n ×{i}), n = 0, . . . , N. The condition J1 can be replaced by the following simpler condition: J2 : There exist measurable sets Y i,n ⊆ Y, i ∈ X, n = 0, . . . , N such that (a) P ε,n (( y ε , i), ·) ⇒ P0,n (( y0 , i), ·) as ε → 0, for any  yε →  y0 ∈ Y i,n −1 as ε → 0, i ∈ X and n = 1, . . . , N;  y0 , i), (Yj,n ∩ Y y0 ∈ Yi,n−1 ∩ Y (b) j ∈X P 0,n (( j,n ) × {j }) = 1, for every  i,n −1,  i ∈ X, n = 1, . . . , N, where Yi,n , i ∈ X, n = 0, . . . , N are sets introduced in the condition I2 . The condition J2 implies that the condition J1 holds with the sets Zn , n = 0, . . . , N  defined above and sets Z n = ∪i ∈X (Y i,n × {i }), n = 0, . . . , N. The condition K1 should be replaced by the following simpler condition: K2 : (a) P ε,0(·) ⇒ P0,0 (·) as ε → 0;      (b) i ∈X P 0,0 (Y i,0 ∩ Y i,0 ) = 1, where Y i,0 , i ∈ X and Y i,0 , i ∈ X are sets intro­ duced, respectively, in the conditions I2 and J2 . The condition K2 implies that the condition K1 holds with the sets Z0 , Z 0 introduced above.

5.2.5 Convergence of log-reward functions for modulated Markov log-price processes given in a dynamic form Let us consider the case where a discrete time multivariate modulated Markov log-price ε,n , X ε,n ), n = 0, 1, . . . with a phase space Z = Rk × X, an initial dis­ Z ε,n = (Y process  tribution P ε (A), and transition probabilities P ε,n ( z , A), is given for every ε ∈ [0, ε0 ], by the following stochastic transition dynamic relation:  Z ε,n = A ε,n ( Z ε,n−1, U ε,n ) ,

n = 1, 2, . . . ,

(5.83)

192 | 5 Convergence of option rewards – I where (a)  Z ε,0 is a random variable taking value in the space Z, (b) U ε,n , n = 1, 2, . . . , is a sequence of independent random variables taking values in the measurable space U (with a σ-algebra of measurable subsets BU ), with distributions G n (A) = P{U ε,n ∈ A}, n = 1, 2, . . . , (c) the random variable  Z ε,0 and the sequence of random variables U ε,n , n = 1, 2, . . . are independent, and (d) A ε,n ( z , u ), n = 1, 2, . . . are measurable functions acting from the space Z × U to the space Z. Z ε,0 can also be given in the dynamic form,  Z ε,0 = A ε,0(U ε,0), The random variable  where: (e) (b) U ε,0 is a random variable taking values in the space U, with the distribu­  ε,0 and the sequence of random tion G0 (A) = P{U ε,0 ∈ A}; (f) the random variable U variables U ε,n , n = 0, 1, . . . are independent; (g) A ε,0(u ) is a measurable function acting from the space U to the space Z. Note that we can always assume that the distributions of random variables U ε,n , n = 0, 1, . . . do not depend on ε. This can be achieved by an appropriate modification of the functions A ε,n ( z , u ) and A ε,0(u ). Moreover, according to our initial model assumptions, the phase space Z is a Pol­ z , A) and initial dis­ ish metric space. In this case, for any transition probabilities P ε,n ( tribution P ε (A), it is possible to construct a Markov chain  Z ε,n given in the dynamic form (5.83) with the space U = [0, 1] and i.i.d. random variables U ε,n , n = 0, 1, . . . uniformly distributed in the interval U. The moment and convergence conditions B5 [𝛾¯] and I1 for the pay-off functions g ε (n, ey , x) do not change. The moment conditions for C6 [β¯ ], C6 [β¯ ], and D7 [β¯ ], D7 [β¯ ] as well as the condi­ tions E10 [β¯ ], E10 [β¯ ] and F4 [β¯ ], F4 [β¯ ] for log-price processes  Z ε,n do not change their formulations. The convergence condition J1 for the transition probabilities of the log-price pro­ cesses  Z ε,n can be replaced by the following condition imposed on the transition dy­ namic functions A ε,n ( z , u ): z) ⊆ U ,  z ∈ Z, n = J3 : There exist measurable sets Z n ⊆ Z, n = 0, . . . , N and U n ( 1, . . . , N such that (a) A ε,n ( z ε , u ) → A0,n ( z0 , u ) as ε → 0, for any  zε →  z0 ∈ Z n −1 as ε → 0 and u ∈ Un ( z0 ), n = 1, . . . , N, where (b) P{U0,n ∈ Un ( z0 )} = 1,  z0 ∈ Z z0 , U0,n ) ∈ Zn ∩Z z0 ∈ Zn−1 ∩Z n −1 , n = 1, . . . , N; (c) P{A 0,n ( n −1 , n } = 1, for every   n = 1, . . . , N, where Zn , n = 0, . . . , N are sets introduced in the condition I1 . Lemma 5.2.2. The condition J3 implies that the condition J1 holds for the log-price pro­ cesses  Z ε,n defined by the dynamic relation (5.83). Proof. The condition J3 (a) implies that for any real-valued, bounded, and continuous function f ( z ) defined on the space Z, (a) functions f (A ε,n ( z ε , u )) → f (A0,n ( z0 , u )) as ε → 0, for any  zε →  z0 ∈ Z as ε → 0 and u ∈ U ( z ) , n = 1, . . . , N. n 0 n −1

5.2 Modulated Markov LPP with bounded characteristics

| 193

Since the distributions G n (A) = P{U ε,n ∈ A} do not depend on ε, and by the z0 )) = P{U ε,n ∈ Un ( z0 )} = 1,  z0 ∈ Z condition J3 (b) G n (Un ( n −1 , n = 1, . . . , N. Thus, relation (a) implies, by the Lebesgue theorem, that (b) Ef (A ε,n ( z ε , U ε,n )) →  Ef (A0,n ( z0 , U0,n )) as ε → 0, for any  zε →  z0 ∈ Zn−1 as ε → 0, and n = 1, . . . , N. z ), Since an arbitrary choice of real-valued bounded, and continuous function f ( relation (b) is equivalent to the following relation of weak convergence (c) A ε,n ( zε , U ε,n ) ⇒ A0,n ( z0 , U0,n ) as ε → 0, for any  zε →  z0 ∈ Z as ε → 0, and n = 1, . . . , N. n −1 This relation is, in fact, a reformulation of the condition J1 (a). It remain to note that the condition J3 (c) is a reformulation of the condition J1 (b). Remark 5.2.4. The condition J3 (c) is equivalent to the assumption that there exist measurable sets Un ⊆ U, n = 1, . . . , N such that P{A0,n ( z0 , u ) ∈ Zn ∩ Z n } = 1 for    z0 ∈ Zn−1, u ∈ Un , n = 1, . . . , N, and P{U0,n ∈ Un } = 1, n = 1, . . . , N. every  The convergence condition K1 for the initial distribution of the log-price processes can be replaced by the following condition imposed on the transformation function Z ε,0 is also given in the dynamic A ε,0(u ), in the case where the random variable  form: K3 : There exist a measurable set U0 ⊆ U such that (a) A ε,0(u ) → A0,0 (u ) as ε → 0, for u ∈ U0 , (b) P{A0,0 (U0,0 ) ∈ Z0 ∩ Z 0 } = 1. Indeed, the condition K3 (a) implies that for any real-valued bounded, and continuous function f ( z ) defined on the space Z, (d) functions f (A ε,0(u )) → f (A0,0 (u )) as ε → 0, for any u ∈ U0 . Since the distributions G0 (A) = P{U ε,0 ∈ A} do not depend on ε, and, by the condition K3 (b), G0 (U0 ) = P{U ε,0 ∈ U0 } = 1, relation (d) implies, by the Lebesgue theorem, that (e) Ef (A ε,0 (U ε,0)) → Ef (A0,0 (U0,0 )) as ε → 0. Since an arbitrary choice of real-valued bounded, and continuous function z ), relation (e) is equivalent to the following relation of weak convergence (f) f ( A ε,0(U ε,0) ⇒ A0,0 (U0,0 ) as ε → 0. This relation is, in fact, a reformulation of the condition K1 (a). Remark 5.2.5. The condition K3 (b) is equivalent to the assumption that there exist  measurable sets U0 ⊆ U such that P{A0,n ( z0 , u ) ∈ Z0 ∩ Z 0 } = 1 for every u ∈ U0 and P{U0,0 ∈ U0 } = 1.

194 | 5 Convergence of option rewards – I

5.3 LPP represented by modulated random walks In this section, we present results about the convergence of option rewards for log-price processes represented by standard and modulated random walks.

5.3.1 Convergence of rewards for price processes represented by standard random walks ε,n = (Y ε,n,1 . . . , Let us consider the model where the multivariate Markov log-price Y Y ε,n,k) is, for every ε ∈ [0, ε0 ], a standard random walk  ε,n−1 + W  ε,n , ε,n = Y Y

n = 1, 2, . . . .

(5.84)

ε,0 = (Y ε,0,1 , . . . , Y ε,0,k) is a k-dimensional random vector, (b) W  ε,n = where (a) Y (W ε,n,1 , . . . , W ε,n,k), n = 1, 2, . . . is a sequence of k-dimensional independent random ε,0 and the random sequence W  ε,n , n = 1, 2, . . . vectors, and (c) the random vector Y are independent. In this case, the model without index component is considered. In fact, one can always define a “virtual” constant index component X ε,n which takes the constant value x0 for every n = 0, 1, . . . , and, the space X = {x0 } containing the only point x0 . In this case, pay-off functions g ε (n, ey , x) = g ε (n, ey ) do not depend on the index argument x, for every ε ∈ [0, ε0 ]. y , x) = ϕ ε,n ( y ) also do not de­ Correspondingly, the log-reward functions ϕ ε,n ( pend on the index argument x, for every ε ∈ [0, ε0 ]. The moment conditions B5 [𝛾¯], C6 [β¯ ] and E10 [β¯ ], formulated in Subsection 1.6.1, are simplified for this model. The first-type modulus of the exponential moment compactness Δ β (Y ε,· ,i , N ) for ε,n , expressed in terms of the components of the components of the log-price process Y  jumps W n , takes the following form, for β ≥ 0:

Δβ (Y ε,· ,i , N ) = max Ee β| W ε,n+1,i | . 0≤ n ≤ N − 1

(5.85)

The condition C6 [β¯ ], assumed to hold for the vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, takes the following form: C7 [β¯ ]: limε→0 Δβ i (Y ε,· ,i , N ) < K34,i , i = 1, . . . , k, for some 1 < K34,i < ∞, i = 1, . . . , k . The second-type modulus of the exponential moment compactness Ξ± β ( Y ε, · ,i , N ) for  ε,n , expressed in terms of components of jumps components of the log-price process Y  n , takes the following form for β ≥ 0: W ± βW ε,n+1,i Ξ± . β ( Y ε, · ,i , N ) = max Ee

0≤ n ≤ N − 1

(5.86)

5.3 LPP represented by modulated random walks

| 195

The condition E10 [β¯ ], assumed to hold for the vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, takes the following form: E11 [β¯ ]: limε→0 Ξ± β i ( Y ε, · ,i , N ) < K 35,i , i = 1, . . . , k, for some 1 < K 35,i < ∞, i = 1, . . . , k . The following lemma is a direct corollary of Lemma 5.1.5. Lemma 5.3.1. The condition E11 [β¯ ] implies the condition C7 [β¯ ] to hold. The condition B5 [𝛾¯], assumed to hold for the vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components, takes the following form: ey )| B6 [𝛾¯]: limε→0 max0≤n≤N supy∈Rk 1+| gkε (n, 𝛾i |y i | < L 11 , for some 0 < L 11 < ∞ and 0 ≤ i =1 L 12,i e L12,1 , . . . , L12,k < ∞ . Conditions D7 [β¯ ] and F4 [β¯ ] do not change. Convergence conditions I1 and J1 are also simplified in this model. The condition I1 should be replaced by the following simpler condition: I3 : There exist measurable sets Yn ⊆ Rk , n = 0, . . . , N, such that the pay-off function U

g ε (n, ey0 ) −→ g0 (n, ey0 ) as ε → 0, i.e. g ε (n, ey ε ) → g0 (n, ey0 ) as ε → 0 for any  yε →  y0 ∈ Yn as ε → 0, and n = 0, . . . , N . Obviously, the condition I3 implies that the condition I1 holds with the sets Zn = Yn × {x0 }, n = 0, . . . , N. y ∈ Rk , the set Let us define, for a Borel set A ⊆ Rk and a point  



y = { y : y + y ∈ A} . A −

(5.87)

ε,n ∈ A/Y ε,n−1 =  The transition probabilities P ε,n ( y , A ) = P{ Y y } of the Markov ˜  ε,n ∈ A} of the  process Y ε,n can be expressed in terms of distributions P ε,n (A) = P{W  jumps W ε,n , for y ∈ Rk , A ∈ Bk , n = 1, . . . ,

˜ ε,n (A −   ε,n ∈ A} = P P ε,n ( y , A) = P{ y+W y) .

(5.88)

Let us assume that the following condition holds: ˜ 0,n (·) as ε → 0, for n = 1, . . . , N; ˜ ε,n (·) ⇒ P J4 : (a) P ˜ 0,n (Yn −  (b) P0,n ( y , Yn ) = P y ) = 1,  y ∈ Yn−1 , for n = 1, . . . , N, where Yn , n = 0, . . . , N are the sets introduced in the conditions I3 . A typical example is where the sets Yn , n = 1, . . . , N are empty sets. Then the condi­ tion J4 (b) obviously holds. Another typical example is where the sets Yn are at most finite or countable sets. ˜ 0,n (·), n = 1, . . . , N have no atoms implies Then the assumption that the measures P that the condition J4 (b) holds. One more example is where the sets Y n have zero Lebesgue measure. Then, the as­ ˜ 0,n (·), n = 1, . . . , N are absolutely continuous with respect sumption that measures P to the Lebesgue measure in Rk , implies that the condition J4 (b) holds.

196 | 5 Convergence of option rewards – I U

Lemma 5.3.2. The condition J4 implies that P ε,n ( y0 , ·) ⇒ P0,n ( y0 , ·) as ε → 0, i.e. y ε , ·) ⇒ P0,n ( y0 , ·) as ε → 0, for any  yε →  y0 ∈ Rk as ε → 0, and n = 1, . . . , N. P ε,n ( Proof. The condition J4 implies that, for any  yε →  y0 ∈ Rk and n = 1, . . . , N, random d   vectors  y ε + W ε,n −→  y0 + W0,n as ε → 0. This is equivalent to the proposition of the lemma, due to relation (5.88). By Lemma 5.3.2, the condition J4 implies the condition J1 to hold. In this case, the role of sets Zn is played by sets Yn × {x0 } while the role of the set Z n is played by the set Rk × {x0 }, for every n = 0, 1, . . . , N. The condition K1 imposed on the initial distributions P ε,0(A) should be replaced by the following condition: K4 : (a) P ε,0 (·) ⇒ P0,0 (·) as ε → 0; (b) P0,0 (Y0 ) = 1, where Y0 is the set introduced in the condition I3 . Obviously, the condition K4 implies that the condition K1 holds with the sets Z0 = Y0 × {x0 } and Z 0 = Rk × {x 0 }. Theorem 5.2.1 takes in this case the following form.  ε,n is represented, for every ε ∈ [0, ε0 ], Theorem 5.3.1. Let the log-price process Y by the standard random walk given by the stochastic transition dynamic relation (5.84). Let also the conditions B6 [𝛾¯] and C7 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions I3 and J4 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any  yε →  y0 ∈ Yn :

y ε ) → ϕ0,n ( y0 ) ϕ ε,n (

as

ε → 0.

(5.89)

Theorem 5.2.3 takes in this case the following form.  ε,n is represented, for every ε ∈ [0, ε0 ], by Theorem 5.3.2. Let the log-price process Y the standard random walk given by the stochastic transition dynamic relation (5.84). Let also the conditions B6 [𝛾¯], C7 [β¯ ], and D7 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0 for i = 1, . . . , k, and also the conditions I3 , J4 , and K4 hold. Then the following relation takes place:

Φ ε → Φ0

as

ε → 0.

(5.90)

5.3 LPP represented by modulated random walks

| 197

5.3.2 Convergence of option rewards for log-price processes represented by Lévy random walks Let us consider the model where, a log-price Y ε,n be, for every ε ∈ [0, ε0 ], a standard univariate random walk, Y ε,n = Y ε,n−1 + W ε,n ,

n = 1, 2, . . . .

(5.91)

where (a) Y ε,0 is a real-valued random variable, (b) W ε,n , n = 1, 2, . . . is a sequence of independent random variables, (c) the random variable Y ε,0 and the random se­ quence W ε,n , n = 1, 2, . . . are independent, and (d) random variable W ε,n has, for every ε ∈ [0, ε0 ] and n = 1, 2, . . . , an infinitely divisible distribution, i.e. its charac­ teristic function has the following form: ψ ε,n (s) = E exp{isW ε,n } ⎧ ⎪

# ⎨  1 isy = exp iμ ε,n s − σ 2ε,n s2 + e − 1 − isy Π ε,n (dy) ⎪ 2 ⎩ | y | 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions I1 , J7 , and K6 , hold. Then the following relation takes place:

Φ ε → Φ0

as

ε → 0.

(5.105)

6 Convergence of option rewards – II In this chapter, we present our main convergence results for rewards of American-­ type options with general perturbed pay-off functions for perturbed multivariate mod­ ulated Markov log-price processes with unbounded characteristics. In Section 6.1, we give asymptotically uniform upper bounds for log-reward func­ tions and optimal expected rewards for univariate modulated Markov log-price pro­ cesses with unbounded characteristics (conditional exponential moments for incre­ ments). In Section 6.2, we present convergence results for log-reward functions and opti­ mal expected rewards for perturbed univariate modulated Markov log-price processes with unbounded characteristics. In Section 6.3, we give asymptotically uniform upper bounds for log-reward func­ tions and optimal expected rewards for multivariate modulated Markov log-price pro­ cesses with unbounded characteristics. In Section 6.4, we present convergence results for log-reward functions and op­ timal expected rewards for perturbed multivariate modulated Markov log-price pro­ cesses with unbounded characteristics. In Section 6.5, we show in which way conditions of convergence for rewards of American-type options can be reformulated in terms of modulated price Markov pro­ cesses instead of modulated Markov log-price processes. Our main results are given in Theorems 6.2.1–6.2.4 for perturbed univariate mod­ ulated Markov log-price processes with unbounded characteristics, and in Theorems 6.4.1–6.4.4 for perturbed multivariate modulated Markov log-price processes with un­ bounded characteristics. What is important that we impose minimal conditions of smoothness on the lim­ iting pay-off functions and transition probabilities of limiting log-price processes. For the basic case, where the transition probabilities have densities with respect to some pivotal Lebesgue-type measure, it is usually required that the sets of weak disconti­ nuity for the limiting transition probabilities and the sets of discontinuity for pay-off functions are zero sets with respect to the above pivotal measure. In fact, such as­ sumptions make it possible for the transition probabilities and pay-off functions to be very irregular. The results presented in this chapter are new. They essentially improve results given in Chapter 5. The main improvement is that we consider much more general moduli of exponential moment compactness with normalizing coefficients for condi­ tional exponential moments of increments. For example, this makes it possible to get the corresponding convergence results for Markov Gaussian log-price processes with drift and volatility coefficients with not more than linear rate of growth as a function of price arguments. The corresponding applications are presented in Chapters 7 and 8.

204 | 6 Convergence of option rewards – II

6.1 Asymptotically uniform upper bounds for rewards – II In this section, we present asymptotically uniform upper bounds for option rewards for univariate modulated Markov log-price processes with unbounded characteristics.

6.1.1 Upper bounds for reward functions for univariate modulated Markov log-price processes with unbounded characteristics We consider a one-dimensional modulated Markov log-price process Z ε,n = (Y ε,n , X ε,n ), n = 0, 1, . . . , with a phase space Z = R1 × X, an initial distribution P ε (A), and transition probabilities P ε,n (z, A) = P{Z ε,n ∈ A/Z ε,n−1 = z}, depending on the perturbation parameter ε ∈ [0, ε0 ]. We assume that the phase space of the index component X is a Polish metric space, i.e. complete, separable, metric space, with a metric dX (x , x ). Let us recall the class A of measurable functions A(β ) defined for β ≥ 0 such that the function A(β ) is nondecreasing in β and A(0) = 0. As pointed out in Sub­ section 4.3.1, any function A(β ) ∈ A generates the sequence of functions A n (β ), n = 0, 1, . . . , from the class A, which are defined by the following recurrence re­ lations, for β ≥ 0, ⎧ ⎨β for n = 0 , (6.1) A n (β) = ⎩ A n−1(β ) + A(A n−1 (β )) for n = 1, 2, . . . . Note that by the definition A0 (β ) ≤ A1 (β ) ≤ A2 (β ) ≤ · · · , for every β ≥ 0. Let us also recall the first-type A-modulus of exponential moment compactness for the log-price process Y ε,n , defined for β ≥ 0 Ez,n e A N −n−1 (β)| Y ε,n+1 −Y ε,n | . 0≤ n ≤ N −1 z =( y,x )∈Z e A(A N −n−1(β))| y|

Δ β,A (Y ε,· , N ) = max

sup

(6.2)

Recall that we use the notations Pz,n and Ez,n for conditional probabilities and expectations under the condition Z ε,n = z. The condition C4 [β ] should be replaced by the following first-type condition of exponential moment compactness, assumed to hold for some β ≥ 0: C9 [β ]: limε→0 Δ β,A (Y ε,· , N ) < K41 , for some 1 < K41 < ∞ . The following lemma gives asymptotically uniform upper bounds for exponential mo­ ments of maxima for perturbed log-price processes.

6.1 Asymptotically uniform upper bounds for rewards – II | 205

Lemma 6.1.1. Let the condition C9 [β ] holds. Then there exist 0 < ε6 ≤ ε0 and the constants 0 ≤ M35 < ∞ such that the following inequality takes place for z = (y, x) ∈ Z, 0 ≤ n ≤ N and 0 ≤ ε ≤ ε6 : % $ Ez,n exp β max |Y ε,r | ≤ M35 exp{A N −n (β )|y|} . (6.3) n≤r≤N

Proof. By the condition C9 [β ], one can choose 0 < ε6 ≤ ε0 such that for 0 ≤ ε ≤ ε6 Δ β,A (Y ε,· , N ) < K41 .

(6.4)

Relation (6.4) means that, for every ε ≤ ε6 , the condition C4 [β ] (with the constant K41 replacing the constants K9 ) holds for the process Y ε,n . Therefore, Lemma 4.3.1 can be applied to the process Y ε,n , for every ε ≤ ε6 . This yields inequalities (6.3). The constant M11 penetrating the inequality given in Lemma 4.3.1 is a function of the corresponding constant K9 . The explicit expression for this function is given in Remark 4.3.1. The same formula gives the expression for the constant M35 as a function of the corresponding constant K41 . Remark 6.1.1. The constant M35 is given by the following formula, which follows from the corresponding formula given in Remark 4.3.1: N M35 = K41 .

(6.5)

Remark 6.1.2. The parameter ε6 is determined by relation (6.4). Let Zε = Zε,0 , . . . , Zε,N  be a sequence of measurable sets. Let us introduce a modi­ fied first-type A-modulus of exponential moment compactness defined, for every ε ∈ [0, ε0 ] and β ≥ 0 Ez,n e A N −n−1 (β)| Y ε,n+1 −Y ε,n | . 0≤ n ≤ N −1 z =( y,x )∈Zε,n e A(A N −n−1(β))| y|

Δ β,A,Zε (Y ε,· , N ) = max

sup

(6.6)

The following first-type condition of exponential moment compactness for the log-price process Y ε,n generalizes the condition C9 [β ]. It should be assumed to hold for some β ≥ 0: C9 [β ]: There exists a sequence of measurable sets Zε = Zε,0 , . . . , Zε,N , for every ε ∈ [0, ε0 ], such that   , for some 1 < K41 < ∞; (a) limε→0 Δ β,A,Zε (Y ε,· , N ) < K41 (b) Zε is a complete sequence of phase sets for the process  Z ε,n , i.e. Pε,n ( z, Zε,n ) = 1,  z ∈ Zε,n−1, n = 1, . . . , N, for every ε ∈ [0, ε0 ] . + Let us introduce, for every ε ∈ [0, ε0 ], the sets Hε,n = (h− ε,n , h ε,n ) × X, n = 0, . . . , N, + where −∞ ≤ h− ε,n ≤ h ε,n ≤ +∞, n = 0, . . . , N. It is useful to note that in applications, for example to space-skeleton models con­ sidered in Chapters 7 and 8, the sets Zε,n , n = 0, . . . , N satisfy the following condition analogous to H1 :

206 | 6 Convergence of option rewards – II ± H2 : Zε,n ⊇ Hε,n , n = 0, . . . , N , ε ∈ [0, ε0 ], where h± ε,n → h 0,n = ±∞ as ε → 0, for n = 0, . . . , N .

Remark 6.1.3. According to the condition H2 , for every z = (y, x), there exists ε2 (z) ∈ (0, ε0 ] such that the following relation holds for every ε ∈ [0, ε2 (z)]: + h− ε,n < y < h ε,n ,

n = 0, . . . , N .

(6.7)

Obviously, z ∈ Zε,n , n = 0, . . . , N for all ε ∈ [0, ε2 (z)]. Note that in the case where the condition H2 holds, the sets Z0,n = Z, n = 0, . . . , N. Lemma 6.1.2. Let the condition C9 [β ] holds. Then there exist 0 < ε6 ≤ ε0 and a con­  stant 0 ≤ M35 < ∞ such that the following inequalities take place for z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε6 : $ %  exp{A N −n (β )|y|} . (6.8) Ez ,n exp β max |Y ε,r | ≤ M35 n≤r≤N

Proof. By the condition C9 [β ], one can choose 0 < ε6 ≤ ε0 such that for 0 ≤ ε ≤ ε6 ,  . Δ β,A,Zε (Y ε,·(·), N ) < K41

(6.9)

The following part of the proof repeats the proof of Lemma 6.1.1. The only differ­ ε,n , for every ε ≤ ε6 , instead ence is that Lemma 4.3.2 should be applied to the process Y ε,n , for every ε ≤ ε6 , in the proof of of Lemma 4.3.1 that was applied to the process Y Lemma 6.1.1.  penetrating the inequalities given This yields inequalities (6.8). The constant M11 in Lemma 4.3.2 is a function of the corresponding constant K9 . The explicit expression for this function is given in Remark 4.3.2. The same formula gives the expression for   as a function of the corresponding constant K41 . the constant M35  Remark 6.1.4. The constant M35 is given by the following formula, which follows from the corresponding formula given in Remark 4.3.2:   N = (K41 ) . M35

(6.10)

Remark 6.1.5. The parameter ε6 is determined by relation (6.9).

6.1.2 Asymptotically uniform upper bounds for option rewards for univariate modulated Markov log-price processes Let us now give asymptotically uniform upper bounds for log-reward functions for log-price processes Z ε,n .

6.1 Asymptotically uniform upper bounds for rewards – II | 207

In this case, the pay-off functions g ε (n, e y , x) are functions of the argument z = (y, x) ∈ Z = R1 × X depending on the perturbation parameter ε. The condition B3 [𝛾] should be replaced by the following condition, assumed to hold for some 𝛾 ≥ 0: y ε ( n,e ,x )| B8 [𝛾]: limε→0 max0≤n≤N sup(y,x)∈Z | g1+ < L15 , for some 0 < L15 < ∞ and 0 ≤ L 16 e 𝛾|y| L16 < ∞ . The following lemma gives asymptotically uniform upper bounds for conditional ex­ pectations for maximal absolute values for the perturbed pay-off processes g ε (n, e Y ε,n , X ε,n ). Lemma 6.1.3. Let the conditions B8 [𝛾] and C9 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, there exist 0 < ε8 ≤ ε0 and the constants 0 ≤ M36 , M37 < ∞ such that the following inequalities take place for z = (y, x) ∈ Z, 0 ≤ n ≤ N and 0 ≤ ε ≤ ε8 :  #    Ez,n max g ε n, e Y ε,n , X ε,n  n≤r≤N ( ) A N − n ( β )𝛾 ≤ M36 + M37 I (𝛾 = 0) + M61 exp |y| I (𝛾 > 0) . (6.11) β Proof. The condition B8 [𝛾] implies that there exists 0 < ε7 ≤ ε0 such that for any 0 ≤ ε ≤ ε7 , |g ε (n, e y , x)| max sup < L15 . (6.12) 0≤ n ≤ N z =( y,x )∈Z 1 + L 16 e 𝛾| y | Relation (6.12) means that for every ε ≤ ε7 , the condition B3 [𝛾] (with the constants L15 , L16 , replacing the constants L5 , L6 ) holds for the process Y ε,n. Let us now take ε8 = min(ε6 , ε7 ), where ε6 was defined in relation (6.4). By the definition ε8 > 0. For every fixed 0 ≤ ε ≤ ε8 both the conditions C4 [β ] and B3 [𝛾] (with the con­ stants K41 and L15 , L16 replacing, respectively, the constants K9 and L5 , L6 ) hold for the process Y ε,n. Therefore, Lemma 4.3.3 can be applied to the process Y ε,n, for every 0 ≤ ε ≤ ε8 . This yields inequality (6.11). The constants M12 and M13 penetrating the inequali­ ties given in Lemma 4.3.3 are functions of the corresponding constants K9 and L5 , L6 . The explicit expressions for these functions are given in Remark 4.3.3. The same formu­ las give the expressions for the constant M36 and M37 as functions of the corresponding constants K41 and L21 , L22 . Remark 6.1.6. The constants M36 and M37 are given by the following formulas: 𝛾

M36 = L15 ,

Nβ

M37 = L15 L16 I (𝛾 = 0) + L15 L16 K41 I (𝛾 > 0) .

(6.13)

Remark 6.1.7. The parameter ε8 = min(ε6 , ε7 ), where the parameters ε6 and ε7 are determined, respectively, by relations (6.4) and (6.12).

208 | 6 Convergence of option rewards – II Lemma 6.1.4. Let the conditions B8 [𝛾] and C9 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, there exist   0 < ε8 ≤ ε0 and the constants 0 ≤ M36 , M37 < ∞ such that the following inequalities take place for z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε8 :  #    Ez,n max g ε n, e Y ε,n , X ε,n  n≤r≤N ( ) A N − n ( β )𝛾    ≤ M36 + M37 I (𝛾 = 0) + M37 exp |y| I (𝛾 > 0) . (6.14) β Proof. It is analogous to the proof of Lemma 6.1.3. As was pointed out in the proof of this lemma, the condition B8 [𝛾] implies that there exists 0 < ε7 ≤ ε0 such that inequality (6.12) holds for 0 ≤ ε ≤ ε7 . Relation (6.12) means that for every ε ≤ ε7 , the condition B3 [𝛾] (with the constants L15 , L16 , replacing the constants L5 , L6 ) holds for the process Y ε,n . Let us now take ε8 = min(ε6 , ε7 ), where ε6 was defined in relation (6.9). By the definition ε8 > 0. For every fixed 0 ≤ ε ≤ ε8 both the conditions C4 [β ] and B3 [𝛾] (with the con­  stants K41 and L15 , L16 replacing, respectively, the constants K9 and L5 , L6 ) hold for the process Y ε,n . Therefore, Lemma 4.3.4 can be applied to the process Y ε,n , for every 0 ≤ ε ≤ ε8 .   This yields inequality (6.14). The constants M12 and M13 penetrating the inequali­ ties given in Lemma 4.3.4 are functions of the corresponding constants K9 and L5 , L6 . The explicit expressions for these functions are given in Remark 4.3.4. The same for­   mulas give the expressions for the constant M36 and M37 as functions of the corre­  sponding constants K41 and L15 , L16 .   Remark 6.1.8. The constants M36 and M37 are given by the following formulas:  = L15 , M36

𝛾

  Nβ M37 = L15 L16 I (𝛾 = 0) + L15 L16 (K41 ) I (𝛾 > 0) .

(6.15)

Remark 6.1.9. The parameter ε8 = min(ε6 , ε7 ), where the parameters ε6 and ε7 are determined, respectively, by relations (6.9) and (6.12). Let us recall the modified second-type A-modulus of exponential moment compact­ ness for the log-price process Y ε,n, defined for β ≥ 0, Ez,n e±A N −n−1 (β)(Y ε,n+1 −Y ε,n ) . 0≤ n ≤ N −1 z =( y,x )∈Z e A(A N −n−1 (β))| y|

Ξ± β,A ( Y ε, · , N ) = max

sup

(6.16)

The following second-type condition of exponential moment compactness, based on moment-generating functions for increments of log-price processes Y ε,n , replaces the condition E4 [β ]. It is assumed to hold for some β ≥ 0: E13 [β ]: limε→0 Ξ± β,A ( Y ε, · , N ) < K 42 , for some 1 < K 42 < ∞ . Lemma 6.1.5. The condition E13 [β ] implies that the condition C9 [β ] holds.

6.1 Asymptotically uniform upper bounds for rewards – II | 209

Proof. It follows from Lemma 4.3.7 that should be applied to the log-price processes Y ε,n. Remark 6.1.10. If the condition E15 [β ] holds with the constant K42 , then the condition C9 [β ] holds with the constant K41 given by the following formula: K41 = 2K42 .

(6.17)

Let Zε = Zε,0 , . . . , Zε,N  be a sequence of measurable sets. Let us introduce and a modified second-type A-modulus of exponential moment compactness defined, for every ε ∈ [0, ε0 ] and β ≥ 0, Ξ± β,A, Zε ( Y ε, · , N ) = max

sup

0≤ n ≤ N −1 z =( y,x )∈Zε,n

Ez,n e±A N −n−1 (β)(Y ε,n+1 −Y ε,n ) . e A(A N −n−1 (β))| y|

(6.18)

The following second-type condition of exponential moment compactness is based on moment-generating functions for increments of log-price processes Y ε,n . It generalizes the condition E13 [β ] and is assumed to hold for some β ≥ 0: E13 [β ]: For every ε ∈ [0, ε0 ], there exists a sequence of measurable sets Zε = Zε,0, . . . , Zε,N  such that   (a) limε→0 Ξ± β,A, Zε ( Y ε, · , N ) < K 42 , for some 1 < K 42 < ∞; (b) Zε is a complete sequence of phase sets for the process Z ε,n , i.e. Pz,n {Z ε,n ∈ Zn } = 1, z ∈ Zε,n−1, n = 1, . . . , N . Lemma 6.1.6. The condition E13 [β ] implies the condition C9 [β ] holds. Proof. It follows from inequality Lemma 4.3.8 that should be applied to the log-price processes Y ε,n .  Remark 6.1.11. If the condition E13 [β ] holds with constant K42 , then the condition   C9 [β ] holds with the constant K41 given by the following formula:   K41 = 2K42 .

(6.19)

Let us assume that conditions B12 [𝛾] and C9 [β ] hold and 0 ≤ 𝛾 ≤ β. In this case, for Lemma 6.1.3 let us define log-reward functions for American op­ tion for the modulated log-price process Z ε,n , for every 0 ≤ n ≤ N, and 0 ≤ ε ≤ ε8 , ϕ ε,n (z) = ϕ ε,n (y, x) =

sup (ε) τ ε,n ∈Mmax,n,N

Ez,n g ε (τ ε,n , e Y ε,τε,n , X ε,τ ε,n ) ,

z = (y, x) ∈ Z .

(6.20)

These functions are the measurable functions of z ∈ Z, for every 0 ≤ n ≤ N. Moreover, for Lemma 6.1.3 let us get the following asymptotically uniform upper bounds for the log-reward functions ϕ ε,n (y, x).

210 | 6 Convergence of option rewards – II Theorem 6.1.1. Let the conditions B8 [𝛾] and C9 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, the logreward functions ϕ ε,n (y, x) satisfy the following inequalities for z = (y, x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε8 : |ϕ ε,n (y, x)| ≤ M36 + M37 I (𝛾 = 0) ( ) A N − n ( β )𝛾 |y| I (𝛾 > 0) . + M37 exp β

(6.21)

Proof. It is analogous to the proof of Theorem 5.1.1. Using Lemma 6.1.3 and the defi­ nition of the log-reward functions, we get the following inequalities for z = (y, x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε8 : |ϕ ε,n (y, x)| ≤ Ez,n max |g ε (r, e Y ε,r , X ε,r )| n≤r≤N

≤ M36 + M37 I (𝛾 = 0) + M37 exp

(

) A N − n ( β )𝛾 |y| I (𝛾 > 0) , β

(6.22)

which proves the above theorem. Let us assume that the conditions B8 [𝛾] and C9 [β ] hold and 0 ≤ 𝛾 ≤ β. In this case, for Lemma 6.1.4 let us define the log-reward functions ϕ ε,n (y, x) using relation (6.20), only for z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε3 . Also, for Lemma 6.1.4 let us get the following upper bounds for these log-reward functions. Theorem 6.1.2. Let the conditions B8 [𝛾] and C9 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, the logreward functions ϕ ε,n (y, x) satisfy the following inequalities for z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε8 :   |ϕ ε,n (y, x)| ≤ M36 + M37 I (𝛾 = 0) ( ) (β + A N −n (β ))𝛾  |y| I (𝛾 > 0) . + M37 exp β

(6.23)

Proof. It is analogous to the proof of Theorem 6.1.1. In this case, Lemma 6.1.4 should be used instead of Lemma 6.1.3. An inequality analogous to (6.22) can be written. The difference is that it holds for z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε8 , and with the   constants M60 , M61 . Remark 6.1.12. It is useful to note that the log-reward function ϕ ε,N (y, x) = g ε (N, e y , x), z = (y, x) ∈ Zε,N always exists. The corresponding upper bounds for this function are given by the condition B8 [𝛾]. However, there is no guarantee that, under the conditions of Theorem 6.1.2, the log-reward functions ϕ ε,n (y, x) exist for points z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N − 1. Even, if these functions do exist for these points, Theorem 6.1.2 does not give upper bounds for the log-reward function in these points.

6.1 Asymptotically uniform upper bounds for rewards – II | 211

However, there is an important case, where this problem is not essential. Let us assume that the condition H2 holds. Then, as was pointed out in Remark 6.1.3, for any point z = (y, x) ∈ Z, there exists ε2 (z) ∈ (0, ε0 ] such that z = (y, x) ∈ Hε,n ⊆ Zε,n , n = 0, . . . , N, for ε ∈ [0, ε2 (z)]. The quantity ε2 (z) is determined by relation (6.7) given in Remark 6.1.3. Thus, the upper bounds for ϕ ε,n (y, x) given in Theorem 6.1.2 take place for ε ∈ [0, ε2 (z)].

6.1.3 Upper bounds for optimal expected rewards for univariate modulated Markov log-price processes with unbounded characteristics We can also get asymptotically uniform upper bounds for optimal expected rewards, for modulated Markov log-price processes Z ε,n . The condition D2 [β ] should be replaced by the following condition, assumed to hold for some β ≥ 0: D8 [β ]: limε→0 E exp{A N (β )|Y ε,0|} < K43 , for some 1 < K43 < ∞ . The following lemma gives asymptotically uniform upper bounds for the expectation of maximal absolute values for the pay-off processes g ε (n, e Y ε,n , X ε,n ). Lemma 6.1.7. Let the conditions B8 [𝛾], C9 [β ], and D8 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, there exist 0 < ε10 ≤ ε0 and the constant 0 ≤ M38 < ∞ such that for the following inequality takes place for 0 ≤ ε ≤ ε10 : E max |g ε (n, e Y ε,n , X ε,n )| ≤ M38 . 0≤ n ≤ N

(6.24)

Proof. The condition D8 [β ] implies that there exists 0 < ε9 ≤ ε0 such that for 0 ≤ ε ≤ ε9 , Ee A N (β)| Y ε,0 | < K43 . (6.25) Relation (6.25) means that for every ε ≤ ε9 , the condition D2 [β ] (with the constant K86 replacing constant K10 ) holds for the process Y ε,n . Let us now take ε10 = min(ε6 , ε7 , ε9 ), where ε6 and ε7 were defined in relations (6.4) and (6.12). By the definition ε10 > 0. For every fixed 0 ≤ ε ≤ ε10 , the conditions C4 [β ], B3 [𝛾], and D2 [β ] (with the con­ stants K41 , K43 , and L15 , L16 replacing, respectively, the constants K9 , K10 and L5 , L6 ) hold for the process Y ε,n. Therefore, Lemma 4.3.5 can be applied to the process Y ε,n , for every 0 ≤ ε ≤ ε10 . This yields inequality (6.24). The constant M14 penetrating the inequality given in Lemma 4.3.5 is a function of the constants K9 , K10 , and L5 , L6 . The explicit expressions for these functions are given in Remark 4.3.5. The same formulas give the expressions for constant M38 as a function of the corresponding constants K41 , K43 and L15 , L16 .

212 | 6 Convergence of option rewards – II Remark 6.1.13. The constant M38 is given by the following formula, which follows from the corresponding formula given in Remark 4.1.5: 𝛾

Nβ

𝛾 β

M38 = L15 + L15 L16 I (𝛾 = 0) + L15 L16 K41 K43 I (𝛾 > 0) .

(6.26)

Remark 6.1.14. The parameter ε10 = min(ε6 , ε7 , ε9 ), where the parameters ε6 , ε7 , and ε9 are determined, respectively, by relations (6.4), (6.12), and (6.25). Note also that one can replace the first-type condition of exponential moment bound­ edness D8 [β ] by the following second-type condition of exponential moment bound­ edness based on moment-generating functions for the initial values of log-price pro­ cesses Y ε,0, assumed to hold for some β ≥ 0: F5 [β ]: limε→0 E exp{±A N (β )Y ε,0} < K44 , for some 1 < K44 < ∞ . Lemma 6.1.8. The condition F5 [β ] implies that the condition D8 [β ] holds. Proof. It follows from the following inequality that holds for any β ≥ 0, Ee A N (β)| Y ε,0 | ≤ Ee+A N (β)Y ε,0 + Ee−A N (β)Y ε,0 .

(6.27)

Remark 6.1.15. As follows from inequality (6.27), if the condition F5 [β ] holds with the constant K41 , then the condition D8 [β ] holds with constant K43 given by the following formulas: (6.28) K43 = 2K44 . We can also get alternative asymptotically uniform upper bounds for optimal expected rewards in the case, where the condition C9 [β ] is used instead of the condition C9 [β ]. The condition D2 [β¯ ] should be replaced by the following condition, assumed to hold for some β ≥ 0:   , for some 1 < K43 < ∞; D8 [β ]: (a) limε→0 E exp{A N (β )|Y ε,0|} < K43 (b) P{Z ε,0 ∈ Zε,0} = 1, for every ε ∈ [0, ε0 ], where Zε,0 is the set penetrating condition C9 [β ] . The following lemma gives alternative asymptotically uniform upper bounds for the expectation of maximum of absolute values for the pay-off processes. Lemma 6.1.9. Let the conditions B8 [𝛾], C9 [β ], and D8 [β ] hold and 0 ≤ 𝛾 ≤ β. Then,  there exist 0 < ε10 ≤ ε0 and a constant 0 ≤ M38 < ∞ such that for the following  inequality takes place for 0 ≤ ε ≤ ε10 ,  E max |g ε (n, e Y ε,n , X ε,n )| ≤ M38 .

0≤ n ≤ N

(6.29)

Proof. The condition D8 [β ] implies that there exists 0 < ε9 ≤ ε0 such that for 0 ≤ ε ≤ ε9 ,  Ee A N (β)| Y ε,0 | < K43 . (6.30)

6.1 Asymptotically uniform upper bounds for rewards – II |

213

Relation (6.30) means that, for every ε ≤ ε9 , the condition D2 [β ] (with the con­   stant K43 replacing constant K10 ) holds for the process Y ε,n .  Let us now take ε10 = min(ε6 , ε7 , ε9 ), where ε6 and ε7 were defined in relations (6.9) and (6.12). By the definition ε10 > 0. For every fixed 0 ≤ ε ≤ ε10 , the conditions C4 [β ], B3 [𝛾], and D2 [β ] (with the con­   stants K41 , K43 , and L15 , L16 replacing, respectively, the constants K9 , K10 and L5 , L6 ) hold for the process Y ε,n. Therefore, Lemma 4.3.6 can be applied to the process Y ε,n, for every 0 ≤ ε ≤ ε10 .  This yields inequality (6.29). The constant M14 penetrating the inequalities given  and L5 , L6 . The in Lemma 4.3.6 is a function of the corresponding constants K9 , K10 explicit expressions for these functions are given in Remark 4.3.6. The same formulas  give the expressions for constant M38 as a function of the corresponding constants   K41 , K43 and L15 , L16 .  Remark 6.1.16. The constant M38 is given by the following formula, which follows from the corresponding formula given in Remark 4.3.6: 𝛾

𝛾

  Nβ  β = L15 + L15 L16 I (𝛾 = 0) + L15 L16 (K41 ) (K43 ) I (𝛾 > 0) . M38

(6.31)

Remark 6.1.17. The parameter ε10 = min(ε6 , ε7 , ε9 ), where parameters ε6 , ε7 , and ε9 are determined, respectively, by relations (6.9), (6.12), and (6.30). Note also that one can replace the first-type condition of exponential moment bound­ edness D16 [β ] by the following second-type condition of exponential moment bound­ edness based on moment-generating functions for the initial value of log-price pro­ cesses Y ε,0, assumed to hold for some β ≥ 0:   F5 [β ]: (a) limε→0 E exp{±A N (β )Y ε,0} < K44 , for some 1 ≤ K44 < ∞; (b) P{Z ε,0 ∈ Zε,0} = 1, for every ε ∈ [0, ε0 ], where Zε,0 is the set penetrating condition C6 [β¯ ] . Lemma 6.1.10. The condition F5 [β ] implies that the condition D8 [β ] holds. Proof. It follows from inequality (6.27).  Remark 6.1.18. If the condition F5 [β ] holds with the constant K44 , then the condition   D8 [β ] holds with the constant K43 given by the following formulas:   K43 = 2K44 .

(6.32)

Let us assume that either conditions of Lemma 6.1.7 or Lemma 6.1.9 hold. In this case, the optimal expected log-reward for American option for the log-price process Z ε,n can be defined, respectively, for every 0 ≤ ε ≤ ε10 or 0 ≤ ε ≤ ε10 , (ε)

Φ ε = Φ(Mmax,N ) =

sup (ε) τ ε,0 ∈Mmax,N

Eg ε (τ ε,0 , e Y ε,τε,0 , X ε,τ ε,0 ) .

(6.33)

214 | 6 Convergence of option rewards – II Also, as follows from Theorem 2.3.4, the following formula takes place, respec­ tively, for every 0 ≤ ε ≤ ε10 or 0 ≤ ε ≤ ε10 , Φ ε = Eϕ ε,0 (Y ε,0 , X ε,0) .

(6.34)

The following two theorems give asymptotically uniform upper bounds for the optimal expected rewards Φ ε . Theorem 6.1.3. Let the conditions B8 [𝛾], C9 [β ], and D8 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, the following inequality takes place, for 0 ≤ ε ≤ ε10 : |Φ ε | ≤ M38 .

(6.35)

Proof. It follows from Lemma 6.1.7 that for 0 ≤ ε ≤ ε10 , |Φ ε | ≤

max

(ε)

τ ε,0 ∈Mmax,N

E|g ε (τ ε,0 , e Y τε,0 , X τ ε,0 )|

≤ E max |g ε (n, e Y ε,n , X ε,n )| ≤ M38 . 0≤ n ≤ N

(6.36)

This inequality proves the theorem. Theorem 6.1.4. Let the conditions B8 [𝛾], C9 [β ] and D8 [β ] hold and 0 ≤ 𝛾 ≤ β. Then, the following inequality takes place for 0 ≤ ε ≤ ε10 :  |Φ ε | ≤ M38 .

(6.37)

Proof. It is analogous to the proof of Theorem 6.1.3. The only difference that Lemma 6.1.9 is used instead of Lemma 6.1.7. In this case, the inequality analogous to (6.36) can be written for 0 ≤ ε ≤ ε10 .

6.2 Univariate modulated LPP with unbounded characteristics In this section, we present conditions of convergence of option rewards for univariate modulated Markov log-price processes with unbounded characteristics.

6.2.1 Convergence of option rewards for univariate modulated Markov log-price processes Let us consider the log-reward functions ϕ ε,n (y, x) introduced in Subsection 6.1.1 and give conditions for their convergence. We impose the following condition of locally uniform convergence for the pay-off functions g ε (n, e y , x):

6.2 Univariate modulated LPP with unbounded characteristics

|

215

I5 : There exist measurable sets Zn ⊆ Z = R1 × X, n = 0, . . . , N, such that the U

pay-off function g ε (n, e y0 , x0 ) −→ g0 (n, e y0 , x0 ) as ε → 0, i.e. g ε (n, e y ε , x ε ) → g0 (n, e y0 , x0 ) as ε → 0 for any z ε = (y ε , x ε ) → z0 = (y0 , x0 ) ∈ Zn as ε → 0, and n = 0, . . . , N . We also impose the following condition of locally uniform weak convergence on the transition probabilities of log-price processes P ε,n (z, A) = P{Z ε,n ∈ A/Z ε,n−1 = z}: J8 : There exist measurable sets Z n ⊆ Z, n = 0, . . . , N such that: U

(a) P ε,n (z0 , ·) =⇒ P0,n (z0 , ·) as ε → 0, i.e. P ε,n (z ε , ·) ⇒ P0,n (z0 , ·) as ε → 0, for any z ε = (y ε , x ε ) → z0 = (y0 , x0 ) ∈ Z n −1 as ε → 0, and n = 1, . . . , N;    (b) P0,n (z0 , Zn ∩ Zn ) = 1, for every z0 ∈ Zn−1 ∩ Z n −1 and n = 1, . . . , N, where Z n , n = 0, . . . , N are sets introduced in the condition I5 . 



A typical example is where the sets Zn , Zn , n = 1, . . . , N are empty sets. Then the condition J8 (b) obviously holds.   Another typical example is where the sets Zn , Zn , n = 1, . . . , N are at most finite or countable sets. Then the assumption that measures P0,n (z0 , A), z0 ∈ Zn−1 ∩ Z n −1 ,   n = 1, . . . , N have no atoms at points of the sets Zn , Zn , for every n = 1, . . . , N implies that the condition J8 (b) holds. One more example is, where the measures P0,n (z0 , A), z0 ∈ Zn−1 ∩ Z n −1 , n = 1, . . . , N are absolutely continuous with respect to some σ-finite measure P(A) on BZ   and P(Zn ), P(Zn ) = 0, n = 1, . . . , N. This assumption implies that the condition J8 (b) holds. Theorem 6.2.1. Let the conditions B8 [𝛾] and C9 [β ] hold with the parameters 𝛾, β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions I5 and J8 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any z ε = (y ε , x ε ) → z0 = (y0 , x0 ) ∈ Zn ∩ Z n: ϕ ε,n (y ε , x ε ) → ϕ0,n (y0 , x0 )

as

ε → 0.

(6.38)

Proof. It is analogous to the proof of Theorem 5.2.1. Due to the above conditions imposed on the parameters 𝛾 and β, one can always find α > 1 such that 𝛾α ≤ β. According to Theorem 6.1.1, conditions B8 [𝛾] and C9 [β ] imply that there exist ε8 > 0 and the constant M36 , M37 such that the following upper bounds take place for the log-reward functions ϕ ε,n (y, x), for z = (y, x) ∈ Z, n = 0, 1, . . . , N and ε ≤ ε8 , |ϕ ε,n (y, x)| ≤ M36 + M37 I (𝛾 = 0) ) ( A N − n ( β )𝛾 |y| I (𝛾 > 0) . + M37 exp β

(6.39)

216 | 6 Convergence of option rewards – II Using inequalities (6.39) and the condition C9 [β ], we get for any z = (y, x) ∈ Z, n = 0, 1, . . . , N − 1, ε ≤ ε8 , Ez,n |ϕ ε,n+1(Y ε,n+1, X ε,n+1)|α &

) 'α A N − n −1 ( β )𝛾 ≤ Ez ,n M36 + M37 I (𝛾 = 0) + M37 exp |Y ε,n+1| I (𝛾 > 0) β ) ' & ( A N −n−1(β )𝛾α α −1 α α α ≤ Ez,n 2 M36 + M37 I (𝛾 = 0) + M37 exp |Y ε,n+1| I (𝛾 > 0) β & ( ) A N −n−1(β )𝛾α α α α ≤ 2α−1 M36 + M37 I (𝛾 = 0) + M37 exp |y| β ) ' ( A N −n−1(β )𝛾α |Y ε,n+1 − y| I (𝛾 > 0) × Ez,n exp β & ( ) A N −n−1(β )𝛾α α −1 α α α ≤2 M36 + M37 I (𝛾 = 0) + M37 exp |y| β '   × Ez,n exp A N −n−1 (β )|Y ε,n+1 − Y ε,n| I (𝛾 > 0) & ( ) A N −n−1(β )𝛾α α α α + M37 I (𝛾 = 0) + M37 exp |y| ≤ 2α−1 M36 β ' × e A(A N −n−1 (β))| y| K41 I (𝛾 > 0)

(

= Q A (y) < ∞ .

(6.40)

The following part of the proof repeats the corresponding part in the proof of The­ orem 5.2.1. The only difference is that relation (5.57) is replaced by the analogous rela­ tion lim E|ϕ ε,N (Y ε,N (z ε ), X ε,N (z ε ))|α ≤ Q A (y0 ) < ∞ , (6.41) ε →0

where the function Q(y) defined in relation (5.42) is replaced by the function Q A (y) defined in relation (6.40). Remark 6.2.1. According to Lemma 6.1.5, the condition C9 [β ] can be replaced in The­ orem 6.2.1 by the condition E13 [β ]. Remark 6.2.2. As follows from the remarks made in Subsection 5.2.1, the following point-wise convergence relation takes place for the reward functions, for any z = (y, x) ∈ Zn ∩ Z n , n = 0, 1, . . . , N, ϕ ε,n (y, x) → ϕ0,n (y, x)

as

ε → 0.

(6.42)

Also, the reward function ϕ0,n (y, x) is continuous in points z = (y, x) ∈ Zn ∩ Z n , for every n = 0, 1, . . . , N. The following theorem gives alternative conditions for the convergence of log-re­ ward functions for American-type options, in the above model.

6.2 Univariate modulated LPP with unbounded characteristics

| 217

Theorem 6.2.2. Let the conditions B8 [𝛾] and C9 [β ] hold with the parameters 𝛾, β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions H2 , I5 , and J8 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any z ε = (y ε , x ε ) → z0 = (y0 , x0 ) ∈ Zn ∩ Z n: ϕ ε,n (y ε , x ε ) → ϕ0,n (y0 , x0 )

as

ε → 0.

(6.43)

Proof. It is analogous to the proof of Theorem 5.2.2. Due to the above conditions imposed on the parameters 𝛾 and β, one can always find α > 1 such that 𝛾α ≤ β. By Theorem 6.1.2, conditions B8 [𝛾] and C9 [β ] imply that there exist ε8 > 0 and   the constants M36 , M37 such that the following upper bounds take place for the reward functions ϕ ε,n (y, x) for z = (y, x) ∈ Zε,n, n = 0, 1, . . . , N, ε ≤ ε8 ,   |ϕ ε,n (y, x)| ≤ M36 + M37 I (𝛾 = 0)  + M37 exp{

A N − n ( β )𝛾 |y|}I (𝛾 > 0) . β

(6.44)

Recall that the condition C9 [β ] (b) implies that Pz,n {Z ε,n+1 ∈ Zε,n+1} = 1, for z ∈ Zε,n , n = 0, . . . , N − 1. Using these relations, inequalities (6.44) and the condition C9 [β ], we get for any z = (y, x) ∈ Zε,n , 0 ≤ n ≤ N − 1 and ε ≤ ε8 , Ez,n |ϕ ε,n+1 (Y ε,n+1 , X ε,n+1 )|α &

) 'α A N − n − 1 ( β )𝛾 ≤ Ez ,n + 0) + exp |Y ε,n+1 | I (𝛾 > 0) β ) ' & ( A N −n−1 (β)𝛾α α −1  α  α  α ≤ Ez,n 2 |Y ε,n+1 | I (𝛾 > 0) (M36 ) + (M37 ) I (𝛾 = 0) + (M37 ) exp β &  M36

 M37 I (𝛾 =

(

 M37

 α  α ) + (M37 ) I (𝛾 = 0) ≤ 2α−1 (M36

(  α ) + (M37

exp

&

) ) ' ( A N −n−1 (β)𝛾α A N −n−1 (β)𝛾α |y| × Ez,n exp |Y ε,n+1 − y| I (𝛾 > 0) β β

 α  α ) + (M37 ) I (𝛾 = 0) ≤ 2α−1 (M36

(  α ) + (M37

exp

&

) '   A N −n−1 (β)𝛾α |y| × Ez,n exp A N −n−1 (β)|Y ε,n+1 − Y ε,n | I (𝛾 > 0) β

 α  α ) + (M37 ) I (𝛾 = 0) ≤ 2α−1 (M36

(  α ) + (M37

exp

= QA (y) < ∞ .

) ' A N −n−1 (β)𝛾α A(A N −n−1 (β))|y|  |y| × e K 41 I (𝛾 > 0) β

(6.45)

The following part of the proof repeats the corresponding part in the proof of The­ orem 5.2.2. The only difference is that relation (5.64) should be slightly modified.

218 | 6 Convergence of option rewards – II As in the proof of Theorem 5.2.2, we choose an arbitrary function z(·) =  z ε , e ∈ [0, ε0 ] such that z ε → z0 ∈ ZN −1 ∩Z N −1 as ε → 0, and should prove that the following relation, which replaces relation (5.64), holds: lim E|ϕ ε,N (Y ε,N (z ε ), X ε,N (z ε ))|α ≤ QA (y0 ) < ∞ . ε →0

(6.46)

In this case, we should use the condition H2 , due to which the point z0 = (y0 , x0 ) ∈ Zε,N −1, for every ε ∈ [0, ε2 (z0 )]. The parameter ε2 (z0 ) ∈ (0, ε0 ] has been defined in relation (6.7) given in Remark 6.1.3. Since z ε → z0 as ε → 0, this implies that there exists ε2 (z(·)) ∈ (0, ε2 (z0 )] such that z ε = (y ε , x ε ) ∈ Zε,N −1 for every ε ∈ [0, ε2 (z(·))]. Therefore, upper bounds for Ez,N −1|ϕ ε,N (Y ε,N , X ε,N )|α given in relation (6.45) hold for every ε ∈ [0, min(ε8 , ε2 (z(·)))]. Due to the above remarks and continuity of function QA (y), relation (6.45) implies relation (6.46), which replaces relation (5.64). The remaining part of the proof repeats the corresponding part in the proof of Theorem 5.2.2. 

6.2.2 Convergence of optimal expected rewards for univariate modulated Markov log-price processes (ε)

Let us now consider optimal expected rewards Φ ε = Φ(Mmax,N ) introduced in Sub­ section 6.1.1 and give conditions for their convergence. We should impose the following condition of weak convergence on the initial dis­ tributions P ε,0(A) = P{Z ε,0 ∈ A}: K7 : (a) P ε,0(·) ⇒ P0,0 (·) as ε → 0;   (b) P0,0 (Z0 ∩ Z 0 ) = 1, where Z0 and Z0 are the sets introduced in conditions I5 and J8 . 



A typical example is where the sets Z0 , Z0 , n = 1, . . . , N are empty sets. Then the condition K7 (b) obviously holds.   Another typical example is where the sets Z0 , Z0 are at most finite or countable sets. Then the assumption that the measure P0,0 (·) has no atoms at points of the sets   Z0 , Z0 implies that the condition K7 (b) holds. One more example is, where measure P0,0 (·) is absolutely continuous with re­   spect to some σ-finite measure P(A) on BZ and P(Z0 ), P(Z0 ) = 0. This assumption implies that the condition K7 (b) holds. The following theorem gives conditions for convergence of optimal expected re­ wards Φ ε . Theorem 6.2.3. Let the conditions B8 [𝛾], C9 [β ], and D8 [β ] hold with the vector param­ eters 𝛾, β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also conditions I5 , J8 , and K7

6.2 Univariate modulated LPP with unbounded characteristics

|

219

hold. Then the following relation takes place: Φ ε → Φ0

ε → 0.

as

(6.47)

Proof. It is analogous to the proof of Theorem 5.2.3. Due to the above conditions imposed on the parameters 𝛾 and β, one can always find α > 1 such that 𝛾α ≤ β. According to Theorem 6.1.1, conditions B8 [𝛾] and C9 [β ] imply that there exist ε8 > 0 and the constants M36 , M37 such that the following upper bounds take place for the log-reward functions ϕ ε,0(y, x) for z = (y, x) ∈ Z and ε ≤ ε8 , |ϕ ε,0(y, x)| ≤ M36 + M37 I (𝛾 = 0) ( ) A N ( β )𝛾 |y| I (𝛾 > 0) . + M37 exp β

(6.48)

Using inequalities (6.48) and the condition D8 [β ], we get for any ε ≤ ε10 , E|ϕ ε,0 (Y ε,0 , X ε,0)|α &

(

) 'α A N ( β )𝛾 ≤ E M36 + M37 I (𝛾 = 0) + M37 exp |Y ε,0| I (𝛾 > 0) β & ( ) ' A N (β )𝛾α α α α ≤ E2α−1 M36 + M37 I (𝛾 = 0) + M37 exp |Y ε,0| I (𝛾 > 0) β & ' α α + M37 I (𝛾 = 0) ≤ 2α−1 M36

α + M37 E exp{A N (β )|Y ε,0|}I (𝛾 > 0)

 α α α + M37 I (𝛾 = 0) + M37 K43 I (𝛾 > 0) = Q A < ∞ . ≤ 2α−1 M36

(6.49)

The following proof repeats the proof of Theorem 5.2.3. The only difference is that relation (6.49) should be used instead of relation (5.68) used in Theorem 5.2.3. Remark 6.2.3. According to Lemmas 6.1.5 and 6.1.8, the conditions C9 [β ] and D8 [β ] can be replaced in Theorem 6.2.3, respectively, by conditions E13 [β ] and F6 [β ]. The following theorem gives alternative conditions for convergence of optimal ex­ pected rewards Φ ε . Theorem 6.2.4. Let the conditions B8 [𝛾], C9 [β ], and D8 [β ] hold with the vector param­ eters 𝛾, β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions H2 , I5 , J8 , and K7 hold. Then the following relation takes place: Φ ε → Φ0

as

ε → 0.

(6.50)

220 | 6 Convergence of option rewards – II Proof. It is analogous to the proof of Theorem 5.2.4. Due to the above conditions imposed on the parameters 𝛾, β, one can always find α > 1 such that 𝛾α ≤ β. According to Theorem 6.1.4, conditions B8 [𝛾] and C9 [β ] imply that there exist ε8 >   0 and the constants M36 , M37 such that the following upper bounds take place for the reward functions ϕ ε,0 (y, x) for z = (y, x) ∈ Zε,0, ε ≤ ε8 ,   |ϕ ε,0 (y, x)| ≤ M36 + M37 I (𝛾 = 0)  + M37 exp{

A N ( β )𝛾 |y|}I (𝛾 > 0) . β

(6.51)

Recall that the condition D8 [β ] (b) implies that Pz,n {Z ε,0 ∈ Zε,0} = 1 for z ∈ Zε,0. Using this relation, inequalities (6.51) and the condition D16 [β ], we get for any ε ≤ ε10 , E|ϕ ε,0 (Y ε,0 , X ε,0)|α &

(

) 'α A N ( β )𝛾 |Y ε,0| I (𝛾 > 0) β & ( ) ' A N (β )𝛾α α −1  α  α  α (M36 ) + (M37 ) I (𝛾 = 0) + (M37 ) exp ≤ E2 |Y ε,0| I (𝛾 > 0) β   α  α  α ) + (M37 ) I (𝛾 = 0) + (M37 ) E exp{A N (β )|Y ε,0|}I (𝛾 > 0) ≤ 2α−1 (M36   α  α  α  ≤ 2α−1 (M36 ) + (M37 ) I (𝛾 = 0) + (M37 ) K43 I (𝛾 > 0) = QA < ∞ . (6.52)    ≤ E M36 + M37 I (𝛾 = 0) + M37 exp

The following part of the proof repeats the corresponding part in the proof of The­ orem 5.2.4. The only difference is that relation (6.52) should be used instead of relation (5.82). Also, relation analogous to (5.75) does require reference to relation (6.43) given in Theorem 6.2.2 instead of relation (6.38) given in Theorem 6.2.1.

6.3 Asymptotically uniform upper bounds for rewards – III In this section, we present asymptotically uniform upper bound for option rewards for multivariate modulated Markov log-price processes with unbounded characteristics.

6.3.1 Moduli of exponential moment compactness for multivariate modulated Markov log-price processes with unbounded characteristics ε,n , X ε,n ) = We consider a multivariate modulated Markov log-price process  Z ε,n = (Y ((Y ε,n,1, . . . , Y ε,n,k), X ε,n ), n = 0, 1, . . . , with a phase space Z = Rk × X, an initial

6.3 Asymptotically uniform upper bounds for rewards – III | 221

distribution P ε (A), and the transition probabilities P ε,n (z, A) = P{Z ε,n ∈ A/Z ε,n−1 = z}, which depend on a perturbation parameter ε ∈ [0, ε0 ].  (β¯ ) = Let us recall the notion of a class Ak of measurable, vector functions A ¯ ¯ ¯ (A1 (β ), . . . , A k (β)) defined for β = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k such that func­ tions A i (β¯ ), i = 1, . . . , k are nondecreasing in every argument β i , i = 1, . . . , k and such ¯ ) = 0, i = 1, . . . , k. As pointed out in Section 4.3 any function A  (β¯ ) ∈ Ak that A i (0 ¯ ¯ ¯  generates the sequence of functions A n (β ) = (A n,1(β ), . . . , A n,k (β)), n = 0, 1, . . . from the class Ak that are defined by the following recurrence relation, for any β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k: ⎧ ⎨ β¯ for n = 0 ,  n (β¯ ) = A (6.53) ⎩A  (A  n−1 (β¯ )) for n = 1, 2, . . . .  n−1 (β¯ ) + A Note that by the definition A0,j (β¯ ) ≤ A1,j (β¯ ) ≤ · · · for every β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k and j = 1, . . . , k.  -modulus of exponential moment compactness Let us also recall the first-type A  ε,n . It is defined, for a vector parameter β¯ = (β 1 , . . . , β k ) for the log-price process Y with nonnegative components  Δ β, ¯ A  ( Y ε, · , N ) =

max

sup

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Z

Ez ,n e

k

j =1

e

k

A N −n−1,j ( β¯ )| Y ε,n+1,j − Y ε,n,j |

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(6.54)

Recall that we use the notations Pz ,n and Ez ,n for the conditional probabilities and expectations under the condition  Z ε,n =  z. The condition C5 [β¯ ] should be replaced by the following condition, assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components and vectors β¯ i = (β i,1 , . . . , β i,k ) = (0, . . . , 0, β i , 0, . . . , 0), i = 1, . . . , k (the vector β¯ i has the ith component equals β i and other components equal to 0): ε, · , N ) < K45,i , i = 1, . . . , k, for some 1 < K45,i < ∞, i = C10 [β¯ ]: limε→0 Δ β¯ i , A (Y 1, . . . , k . The following lemma gives asymptotically uniform upper bounds for moments of max­ ima for components of perturbed multivariate modulated Markov log-price processes. Lemma 6.3.1. Let the condition C10 [β¯ ] holds. Then there exists 0 < ε11 ≤ ε0 and the constants 0 ≤ M39,i < ∞, i = 1, . . . , k such that the following inequalities take place, for  z = ( y , x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε11 , and i = 1, . . . , k, ⎫ ⎧ $ % k ⎬ ⎨

(6.55) Ez ,n exp β i max |Y ε,r,i| ≤ M39,i exp ⎩ A N −n,j(β¯ i )|y j |⎭ . n≤r≤N

j =1

Proof. The condition C10 [β¯ ] implies that there exists 0 < ε11 ≤ ε0 such that, for 0 ≤ ε ≤ ε11 and i = 1, . . . , k, ε, · , N ) < K45,i . Δ β¯ i , A¯ (Y (6.56)

222 | 6 Convergence of option rewards – II Relation (6.56) means that, for every ε ≤ ε11 , the condition C5 [β¯ ] (with the con­ stants K45,i , i = 1, . . . , k replacing the constants K13,i , i = 1, . . . , k) holds for the pro­ ε,n . cess Y ε,n , for every ε ≤ ε11 . Therefore, Lemma 4.3.11 can be applied to the process Y This yields inequalities (6.55). For every i = 1, . . . , k, the constant M9,i penetrat­ ing the inequality given in Lemma 4.3.11 is a function of the corresponding constant K13,i . The explicit expression for this function is given in Remark 4.3.11. The same for­ mula gives the expression for the constant M39,i as a function of the corresponding constant K45,i . Remark 6.3.1. The constants M39,i , i = 1, . . . , k are given by the following formulas, which follows from the corresponding formulas given in Remark 4.3.11, N . M39,i = K45,i

(6.57)

Remark 6.3.2. The parameter ε11 is determined by relation (6.56). Let Zε = Zε,0 , . . . , Zε,N  be a sequence of measurable sets. Let us introduce a modi­  -modulus of exponential moment compactness, defined for every ε ∈ fied first-type A [0, ε0 ] and vectors β¯ = (β 1 , . . . , β k ) with nonnegative components by the following formula:  Δ β, ¯ A  , Zε ( Y ε, · , N ) = max

sup

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Zε,n

Ez ,n e

k

j =1

e

k

A N −n−1,j ( β¯ )| Y ε,n+1,j − Y ε,n,j |

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(6.58)

The following first-type condition of exponential moment compactness for the log-price process Y ε,n generalizes the condition C10 [β¯ ]. It is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: C10 [β¯ ]: There exists a sequence of measurable sets Zε = Zε,0 , . . . , Zε,N , for every ε ∈ [0, ε0 ], such that   ε, · , N ) < K45,i , i = 1, . . . , k, for some 1 < K45,i < ∞, i = (a) limε→0 Δ β¯ i , A ,Zε (Y 1, . . . , k; (b) Zε is a complete sequence of phase sets for the process  Z ε,n , i.e. Pε,n( z , Zε,n ) = 1,  z ∈ Zε,n−1, n = 1, . . . , N, for every ε ∈ [0, ε0 ] . Lemma 6.3.2. Let the condition C10 [β¯ ] holds. Then there exists 0 < ε11 ≤ ε0 and the  constants 0 ≤ M39,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε11 , and i = 1, . . . , k, ⎫ ⎧ $ % k ⎬ ⎨

 exp ⎩ A N −n,j(β¯ i )|y j |⎭ . (6.59) Ez ,n exp β i max |Y ε,r,i| ≤ M39,i n≤r≤N

j =1

Proof. By the condition C10 [β ], one can choose 0 < ε11 ≤ ε0 such that, for 0 ≤ ε ≤ ε11 and i = 1, . . . , k, ε, · , N ) < K45,i . Δ β¯ i , A (Y (6.60)

6.3 Asymptotically uniform upper bounds for rewards – III | 223

The following part of the proof repeats the corresponding part in the proof of Lemma 6.3.1. The only difference is that Lemma 4.3.12 should be applied to the pro­ ε,n , for every ε ≤ ε11 , instead of Lemma 4.3.11 that was applied to the process cess Y  Y ε,n, for every ε ≤ ε11 , in the proof of Lemma 6.3.1.  This yields inequalities (6.59). The constants M15,i , i = 1, . . . , k penetrating the  inequalities given in Lemma 4.3.12 are functions of the corresponding constant K13,i , i = 1, . . . , k. The explicit expression for these functions is given in Remark 4.3.8. The  same formulas give expressions for the constant M39,i , i = 1, . . . , k as functions of the  corresponding constant K45,i , i = 1, . . . , k.  Remark 6.3.3. The constants M39,i , i = 1, . . . , k are given by the following formulas, which follows from the corresponding formula given in Remark 4.3.8:   = (K45,i )N . M39,i

(6.61)

Remark 6.3.4. The parameter ε11 is determined by relation (6.60). Let us now give asymptotically uniform upper bounds for log-reward functions for the Z ε,n. multivariate modulated Markov log-price processes  In this case, the pay-off functions g ε (n, ey , x) are functions of the argument  z= ( y , x) ∈ Z = Rk × X and depend on perturbation parameter ε. We impose on pay-off functions the condition B5 [𝛾¯], which should be assumed to hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components. The following lemmas give asymptotically uniform upper bounds for the condi­ tional expectations for the maximal absolute values for perturbed pay-off processes g(n, e Yε,n , X ε,n ). Lemma 6.3.3. Let the conditions B5 [𝛾¯] and C10 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, there exist 0 < ε12 ≤ ε0 and the constants 0 ≤ M40 , M41,i < ∞, i = 1, . . . , k such y , x) ∈ Z, 0 ≤ n ≤ N and 0 ≤ ε ≤ ε12 , that the following inequalities take place for z = ( 

Ez ,n max |g ε (r, e Y ε,r , X ε,r )| n≤r≤N

⎧⎛ ⎞ ⎫ k ⎨

𝛾⎬ M41,i + M41,i exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ . ≤ M40 + βi i:𝛾 =0 i:𝛾 >0 j =1

i



(6.62)

i

Proof. As pointed out in the proof of Lemma 5.1.3, the condition B5 [𝛾¯] implies that there exists 0 < ε2 ≤ ε0 such that, for every ε ≤ ε2 , the condition B1 [𝛾¯] (with the constants L19 , L20,i , i = 1, . . . , k, replacing the constants L1 , L2,i , I = 1, . . . , k) holds ε,n . for the process Y Let us now take ε12 = min(ε11 , ε2 ), where ε11 was defined in relation (6.56). By the definition ε12 > 0. For every fixed 0 ≤ ε ≤ ε12 both the conditions C5 [β¯ ] and B1 [𝛾¯] (with the con­ stants K45,i , i = 1, . . . , k, and L19 , L20,i , i = 1, . . . , k replacing, respectively, the con­  ε,n. stants K13,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process Y

224 | 6 Convergence of option rewards – II  ε,n, for every 0 ≤ ε ≤ ε12 . Therefore, Lemma 4.3.13 can be applied to the process Y This yields inequality (6.62). The constants M16 and M17,i , i = 1, . . . , k penetrating the inequalities given in Lemma 4.3.13 are functions of the corresponding constants K13,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k. The explicit expressions for these functions are given in Remark 4.3.13. The same formulas give the expressions for the constant M40 and M41,i , i = 1, . . . , k as functions of the corresponding constants K88,i , i = 1, . . . , k and L19 , L20,i , i = 1, . . . , k.

Remark 6.3.5. The constants M40 and M41,i , i = 1, . . . , k are given by the following formulas: M40 = L19 , 𝛾

N βi

M41,i = L19 L20,i I (𝛾i = 0) + L19 L20,i K45,ii I (𝛾i > 0) .

(6.63)

Remark 6.3.6. The parameter ε12 = min(ε11 , ε2 ), where the parameters ε11 and ε2 are determined, respectively, by relations (6.56) and (5.11). Lemma 6.3.4. Let the conditions B5 [𝛾¯] and C10 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k.   Then, there exist 0 < ε12 ≤ ε0 and the constants 0 ≤ M40 , M41,i < ∞, i = 1, . . . , k such that the following inequalities take place for  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε12 , and i = 1, . . . , k:  #     Ez ,n max g ε r, e Y ε,r , X ε,r  n≤r≤N

≤

 M40

+

i:𝛾i =0

 M41,i

+

i:𝛾i >0

⎧⎛ k ⎨

 M41,i exp ⎩⎝

⎫ ⎬ 𝛾 i A N −n,j(β¯ i )|y j |⎠ ⎭ . βi j =1 ⎞

(6.64)

Proof. It is analogous to the proof of Lemma 6.3.3. As was pointed out in the proof of this lemma, the condition B5 [𝛾¯] implies that there exists 0 < ε2 ≤ ε0 such that, for every ε ≤ ε2 , the condition B1 [𝛾¯] (with the constants L9 , L10,i , I = 1, . . . , k, replacing  ε,n. constants L1 , L2 ) holds for the process Y   Let us now take ε12 = min(ε11 , ε2 ), where ε11 was defined in relation (6.60). By the definition ε12 > 0. For every fixed 0 ≤ ε ≤ ε12 both the conditions C5 [β¯ ] and B1 [𝛾¯] (with the con­  stants K45,i , i = 1, . . . , k, and L9 , L10,i , i = 1, . . . , k replacing, respectively, the con­  ε,n . stants K13,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process Y ε,n , for every 0 ≤ ε ≤ ε12 . Therefore, Lemma 4.3.14 can be applied to the process Y   This yields inequality (6.64). The constants M16 and M17,i , i = 1, . . . , k penetrating the inequalities given in Lemma 4.3.14 are functions of the corresponding constants  , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k. The explicit expressions for these functions K13,i  are given in Remark 4.3.14. The same formulas give expressions for the constant M40   and M41,i , i = 1, . . . , k as functions of the corresponding constants K45,i , i = 1, . . . , k and L9 , L10,i , i = 1, . . . , k.

6.3 Asymptotically uniform upper bounds for rewards – III | 225   and M41,i , i = 1, . . . , k are given by the following Remark 6.3.7. The constants M40 formulas:  M40 = L9 ,   M41,i = L9 L10,i I (𝛾i = 0) + L9 L10,i (K45,i )

𝛾

N βi

i

I (𝛾i > 0) .

(6.65)

Remark 6.3.8. The parameter ε12 = min(ε11 , ε2 ), where the parameters ε11 and ε2 are determined, respectively, by relations (6.60) and (5.11).  -modulus of exponential moment compactness for the Let us recall the second-type A  ε,n, defined for vector β¯ = (β 1 , . . . , β k ) with components of the log-price process Y nonnegative components and vectors ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , where Ik = {¯ı k : ı j = +, −, j = 1, . . . , k }, ¯ı k  Δ β, ¯ A  ( Y ε, · ,

N ) = max

sup

Ez ,n e

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Z

k

¯

j =1 ı j A N −n−1,j ( β )( Y n+1,j − Y n,j )

e

k

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(6.66)

The following second-type condition of exponential moment compactness based  ε,n replaces on moment-generating functions for increments of log-price processes Y the condition E5 [β¯ ]. It is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: ¯ı k ε, · , N ) < K46,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k, for some E14 [β¯ ]: limε→0 Ξ β¯ , A (Y i 1 < K46,i < ∞, i = 1, . . . , k . Lemma 6.3.5. The condition E14 [β¯ ] implies that the condition C10 [β¯ ] holds. Proof. It follows from Lemma 4.3.17 that should be applied to the log-price processes  ε,n. Y Remark 6.3.9. If the condition E14 [β¯ ] holds with constant K46,i , i = 1, . . . , k, then the condition C10 [β¯ ] holds with the constant K45,i , i = 1, . . . , k given by the following formulas: (6.67) K45,i = 2k K46,i , i = 1, . . . , k . Let Zε = Zε,0 , . . . , Zε,N  be a sequence of measurable sets. Let us introduce a modi­  -modulus of exponential moment compactness, defined for vector fied second-type A β¯ = (β 1 , . . . , β k ) with nonnegative components and vectors ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , where Ik = {¯ı k : ı j = +, −, j = 1, . . . , k }, ¯ı k  Ξ β, ¯ A  , Z ( Y ε, · ,

N ) = max

sup

0≤ n ≤ N − 1  z =(( y 1 ,...,y k ) , x )∈Zε,n

Ez ,n e

k

¯

j =1 ı j A N −n−1,j ( β )( Y ε,n+1,j − Y ε,n,j )

e

k

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(6.68)

The following modified second-type condition of exponential moment compact­ ness is based on moment-generating functions for increments of log-price processes

226 | 6 Convergence of option rewards – II Y ε,n . It is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnega­ tive components: E14 [β¯ ]: There exists, for every ε ∈ [0, ε0 ], a sequence of measurable sets Zε = < Zε,0, . . . , Zε,N > such that ¯ı  (a) limε→0 Ξ β¯k , A ,Z (Y ε,· , N ) < K46,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k, for some i

ε

 1 < K46,i < ∞, i = 1, . . . , k; (b) Zε is a complete sequence of phase sets, i.e. Pn (z, Zε,n ) = 1, z ∈ Zε,n−1, n = 1, . . . , N .

Lemma 6.3.6. The condition E14 [β¯ ] implies that the condition C10 [β¯ ] holds. Proof. It follows from Lemma 4.3.18 that should be applied to the log-price processes ε,n . Y  Remark 6.3.10. If the condition E14 [β¯ ] holds with constant K46,i , i = 1, . . . , k, then  ¯  condition C10 [β ] holds with the constant K45,i , i = 1, . . . , k given by the following formulas:   = 2k K46,i , i = 1, . . . , k . (6.69) K45,i

6.3.2 Asymptotically uniform upper bounds for rewards for multivariate modulated Markov log-price processes with unbounded characteristics Let us assume that the conditions B5 [𝛾¯] and C10 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Lemma 6.3.3 lets us define log-reward functions ϕ ε,n ( z) = ϕ ε,n ( y , x ),  z = ( y , x) ∈  Z for the log-price process Z ε,n , for every 0 ≤ n ≤ N and 0 ≤ ε ≤ ε12 , z ) = ϕ ε,n ( y , x) ϕ ε,n ( =



sup (ε) τ ε,n ∈Mmax,n,N

Ez ,n g ε (τ ε,n , e Y ε,τε,n , X ε,τ ε,n ) ,

 z = ( y , x) ∈ Z .

(6.70)

z ∈ Z, for every 0 ≤ n ≤ N. These functions are the measurable functions of  Also, Lemma 6.3.3 lets us get the following asymptotically uniform upper bounds y , x ). for the log-reward functions ϕ ε,n ( Theorem 6.3.1. Let the conditions B5 [𝛾¯] and C10 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the log-reward functions ϕ ε,n ( y , x) satisfy the following inequalities for  z = y , x) ∈ Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε12 : ( ⎧⎛ ⎞ ⎫ k ⎨



𝛾⎬ ⎝ ¯ |ϕ ε,n ( y , x)| ≤ M40 + M41,i + M41,i exp ⎩ A N −n,j(β i )|y j |⎠ i ⎭ . (6.71) βi i:𝛾 =0 i:𝛾 >0 j =1 i

i

6.3 Asymptotically uniform upper bounds for rewards – III | 227

Proof. It is analogous to the proof of Theorem 5.1.1. Using Lemma 6.3.3 and the defi­ z = ( y , x) ∈ nition of the log-reward functions, we get the following inequalities for  Z, 0 ≤ n ≤ N, 0 ≤ ε ≤ ε12 : 

|ϕ ε,n ( y , x)| ≤ Ez ,n max |g ε (r, e Y ε,r , X ε,r )| n≤r≤N

⎧⎛ ⎞ ⎫ k ⎨

𝛾⎬ ≤ M40 + M41,i + M41,i exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ , (6.72) βi i:𝛾 =0 i:𝛾 >0 j =1

i

i

which prove the above theorem. Let us now assume that the conditions B5 [𝛾¯] and C10 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Also for Lemma 6.3.4, let us define log-reward functions ϕ ε,n ( y , x) using relation  (6.70), only for  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε12 . Also for Lemma 6.3.4, let us obtain the following upper bounds for the log-reward functions. Theorem 6.3.2. Let the conditions B5 [𝛾¯] and C10 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. y , x) satisfy the following inequalities for 0 ≤ n ≤ Then, the log-reward functions ϕ ε,n ( N, z = ( y , x) ∈ Zε,n, 0 ≤ ε ≤ ε12 : ⎧⎛ ⎞ ⎫ k ⎨



𝛾⎬    ⎝ |ϕ ε,n ( y , x)| ≤ M40 + M41,i + M41,i exp ⎩ A N −n,j(β¯ i )|y j |⎠ i ⎭ . (6.73) βi i:𝛾 =0 i:𝛾 >0 j =1 i

i

Proof. It is analogous to the proof of Theorem 6.3.1. In this case Lemma 6.3.4 should be used instead of Lemma 6.3.3. An inequality analogous to (6.72) can be written down. The differences are that it holds for  z = ( y , x) ∈ Zε,n , 0 ≤ n ≤ N, 0 ≤ ε ≤ ε12 , and   with the constants M40 , M41,i , i = 1, . . . , k. Remark 6.3.11. It is useful to note that the log-reward function ϕ ε,N ( y , x) = g ε (N,  y e , x ),  z = ( y , x) ∈ Zε,N always exists. The corresponding upper bounds for this func­ tion are given by condition B5 [𝛾¯]. However, there is no guarantee that, under condi­ y , x) exist for points  z = ( y , x) ∈ tions of Theorem 6.3.2, the log-reward functions ϕ ε,n ( Zε,n , 0 ≤ n ≤ N − 1. Even, if these functions do exist for these points, Theorem 6.3.2 does not give upper bounds for the log-reward functions in these points. However, it is the important case where this problem is not essential. Let us assume that the condition H1 holds. Then, as was pointed out in Remark 5.1.3, for any point  z = ( y , x) ∈ Z, there exists ε1 (z) ∈ (0, ε0 ] such that  z = ( y , x) ∈ Hε,n ⊆ Zε,n , n = 0, . . . , N for ε ∈ [0, ε1 ( z)]. The quantity ε1 ( z ) is determined by relation (5.6) given in Remark 5.1.3. Thus, the upper bounds for ϕ ε,n ( y , x) given in Theorem 6.3.2 z )]. take place for ε ∈ [0, ε1 ( Let us now give asymptotically uniform upper bounds for the optimal expected reward for the modulated Markov log-price processes  Z ε,n .

228 | 6 Convergence of option rewards – II The condition D3 [β¯ ] should be replaced, respectively, by the following first-type condition of exponential moment boundedness, assumed to hold for some vector pa­ rameter β¯ = (β 1 , . . . , β k ) with nonnegative components: k D9 [β¯ ]: limε→0 E exp{ j=1 A N,j (β¯ i )|Y ε,0,j|} < K47,i , i = 1, . . . , k, for some 1 < K47,i < ∞, i = 1, . . . , k . The following lemma gives explicit upper bounds for the expectation of maximal ab­ solute values for the pay-off processes g ε (n, e Yε,n , X ε,n ). Lemma 6.3.7. Let the conditions B5 [𝛾¯], C10 [β¯ ], and D9 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, there exist 0 < ε14 ≤ ε0 and a constant 0 ≤ M42 < ∞ such that for the following inequality takes place for 0 ≤ ε ≤ ε14 : 

E max |g ε (n, e Y ε,n , X ε,n )| ≤ M42 .

(6.74)

0≤ n ≤ N

Proof. The condition D9 [β¯ ] implies that there exists 0 < ε13 ≤ ε0 such that for 0 ≤ ε ≤ ε13 and i = 1, . . . , k, ⎫ ⎧ k ⎬ ⎨

E exp ⎩ A N,j (β¯ i )|Y ε,0,j|⎭ < K47,i . (6.75) j =1

Relation (6.75) means that, for every ε ≤ ε13 , the condition D3 [β¯ ] (with constants K47,i , ε,n . i = 1, . . . , k replacing the constants K14,i , i = 1, . . . , k) holds for the process Y Let us now take ε14 = min(ε11 , ε2 , ε13 ), where ε11 and ε2 were defined in relations (6.56) and (5.11). By the definition ε14 > 0. For every fixed 0 ≤ ε ≤ ε14 , the conditions C5 [β¯ ] and B1 [𝛾¯], and D3 [β¯ ] (with the constants K45,i , K47,i , i = 1, . . . , k, and L9 , L10,i , i = 1, . . . , k replacing, respectively, the constants K13,i , K14,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process ε,n . Y ε,n , for every 0 ≤ ε ≤ ε14 . Therefore, Lemma 4.3.15 can be applied to the process Y This yields inequality (6.74). The constant M18 penetrating the inequalities given in Lemma 4.3.15 is a function of the corresponding constants K13,i , K14,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k. The explicit expression for these function is given in Re­ mark 4.3.15. The same formula gives the expressions for constant M42 as a function of the corresponding constants K45,i , K47,i , i = 1, . . . , k, and L9 , L10,i , i = 1, . . . , k. Remark 6.3.12. The constant M42 is given by the following formula, which follows from the corresponding formula given in Remark 4.3.15: M42 = L9 +

i:𝛾i =0

L9 L10,i +



𝛾

N βi

𝛾i β

i L9 L10,i K45,ii K47,i .

(6.76)

i:𝛾i >0

Remark 6.3.13. The parameter ε14 = min(ε11 , ε2 , ε13 ), where ε11 and ε2 were defined in relations (6.56), (5.11), and (6.75).

6.3 Asymptotically uniform upper bounds for rewards – III |

229

Note also that one can replace the following second-type condition of exponential mo­ ment boundedness D9 [β¯ ] by the following condition of exponential moment bound­ edness based on moment-generating functions for the initial value of log-price pro­ ε,0. It is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with cesses Y nonnegative components:  F6 [β¯ ]: limε→0 E exp{ kj=1 ı j A N,j (β¯ i )Y ε,0,j} < K48,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k for some 1 < K48,i < ∞, i = 1, . . . , k . Lemma 6.3.8. The condition F6 [β¯ ] implies that the condition D9 [β¯ ] holds. Proof. It follows from the following inequality that holds for any β¯ = (β 1 , . . . , β k ), β 1 , . . . , β k ≥ 0, k k

¯ ¯ Ee j=1 ı j A N,j (β i )Y0,j . (6.77) Ee j=1 A N,j (β i )| Y0,j | ≤ ¯ı k ∈Ik

Remark 6.3.14. If the condition F6 [β¯ ] holds with the constant K48,i , i = 1, . . . , k, then the condition D9 [β¯ ] holds with the constant K47,i , i = 1, . . . , k given by the following formulas: (6.78) K47,i = 2k K48,i , i = 1, . . . , k . The condition D3 [β¯ ] should be replaced, respectively, by the following condition, as­ sumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative compo­ nents: k  , i = 1, . . . , k, for some 1 < D9 [β¯ ]: (a) limε→0 E exp{ j=1 A N,j (β¯ i )|Y ε,0,j|} < K47,i  K47,i < ∞, i = 1, . . . , k; ε,0 ∈ Zε,0} = 1, for every ε ∈ [0, ε0 ], where Zε,0 is the set penetrating (b) P{Z condition C9 [β ] . The following lemma gives alternative explicit upper bounds for the expectation of maximal absolute values for the pay-off processes g ε (n, e Yε,n , X ε,n ). Lemma 6.3.9. Let the conditions B5 [𝛾¯], C10 [β¯ ] and D9 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i =  1, . . . , k. Then, there exist 0 < ε14 ≤ ε0 and a constant 0 ≤ M42 < ∞ such that for the  following inequality takes place for 0 ≤ ε ≤ ε14 : 

 . E max |g ε (n, e Y ε,n , X ε,n )| ≤ M42

0≤ n ≤ N

(6.79)

Proof. The condition D9 [β¯ ] implies that there exists 0 < ε13 ≤ ε0 such that for 0 ≤ ε ≤ ε13 , ⎫ ⎧ k ⎬ ⎨

 . (6.80) E exp ⎩ A N,j (β¯ i )|Y ε,0,j|⎭ < K47,i j =1

Relation (6.80) means that, for every ε ≤ ε13 , the condition D3 [β¯ ] (with the con­   stants K47,i , i = 1, . . . , k replacing the constants K14,i , i = 1, . . . , k) holds for the pro­  ε,n. cess Y

230 | 6 Convergence of option rewards – II Let us now take ε14 = min(ε11 , ε2 , ε13 ), where ε11 and ε2 were defined in relations (6.60) and (5.11). By the definition ε14 > 0. For every fixed 0 ≤ ε ≤ ε14 , the conditions C5 [β¯ ] and B1 [𝛾¯], and D3 [β¯ ] (with the   , K47,i , i = 1, . . . , k, and L19 , L20,i , i = 1, . . . , k replacing, respectively, constants K45,i   the constants K13,i , K14,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k) hold for the process ε,n . Y ε,n , for every 0 ≤ ε ≤ ε14 . Therefore, Lemma 4.3.16 can be applied to the process Y  This yields inequality (6.79). The constant M18 penetrating the inequalities given   in Lemma 4.3.16 is a function of the corresponding constants K13,i , K14,i , i = 1, . . . , k and L1 , L2,i , i = 1, . . . , k. The explicit expression for these function is given in Re­  mark 4.3.16. The same formula gives the expressions for the constant M42 as a function   of the corresponding constants K45,i , K47,i , i = 1, . . . , k, and L19 , L20,i , i = 1, . . . , k.  Remark 6.3.15. The constant M42 is given by the following formula, which follows from the corresponding formula given in Remark 4.3.16:  = L19 + M42



L19 L20,i +

i:𝛾i =0



 L19 L20,i (K45,i )

𝛾

N βi

i

𝛾i

 (K47,i ) βi .

(6.81)

i:𝛾i >0

Remark 6.3.16. The parameter ε14 = min(ε11 , ε2 , ε13 ), where ε11 and ε2 were defined in relations (6.60), (5.11), and (6.80). Note also that one can replace the following first-type condition of exponential mo­ ment boundedness D9 [β¯ ] by the following condition of exponential moment bound­ edness based on moment-generating functions for the initial value of log-price pro­  ε,0. It is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with cesses Y nonnegative components:   F6 [β¯ ]: (a) limε→0 E exp{ kj=1 ı j A N,j (β¯ i )Y ε,0,j} < K48,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i =  1, . . . , k for some 1 ≤ K48,i < ∞, i = 1, . . . , k; ε,0 ∈ Zε,0} = 1, for every ε ∈ [0, ε0 ], where Zε,0 is the set penetrating the (b) P{Z condition C10 [β¯ ] . Lemma 6.3.10. The condition F6 [β¯ ] implies that the condition D9 [β¯ ] holds. Proof. It follows from inequality (6.77).  Remark 6.3.17. If the condition F6 [β¯ ] holds with the constant K48,i , i = 1, . . . , k, then  ¯  the condition D17 [β ] holds with the constant K47,i , i = 1, . . . , k given by the following formulas:   K47,i = 2k K48,i , i = 1, . . . , k . (6.82)

Let us now assume that either conditions of Lemma 6.3.7 or Lemma 6.3.9 hold. In this (ε) case, we can define the optimal expected reward Φ ε = Φ(Mmax,N ) for the log-price

6.4 Multivariate modulated LPP with unbounded characteristics

|

231

Z ε,n , respectively, for every 0 ≤ ε ≤ ε14 , or, for every 0 ≤ ε ≤ ε14 process  (ε)

Φ ε = Φ(Mmax,N ) =

sup (ε) τ ε,0 ∈Mmax,N



Eg ε (τ ε,0 , e Y ε,τε,0 , X ε,τ ε,0 ) .

(6.83)

Also, as follows from Theorem 2.3.4, the following formula takes place, respec­ tively, for every 0 ≤ ε ≤ ε14 , or, for every 0 ≤ ε ≤ ε14 : ε,0 , X ε,0) . Φ ε = Eϕ ε,0 (Y

(6.84)

The following two theorems give asymptotically uniform upper bounds for the optimal expected rewards for American-type options. Theorem 6.3.3. Let the conditions B5 [𝛾¯], C10 [β¯ ], and D9 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the following inequality takes place, for 0 ≤ ε ≤ ε14 , |Φ ε | ≤ M42 .

(6.85)

Proof. It follows from the definition of the functional Φ ε and Lemma 6.3.7 that, for 0 ≤ ε ≤ ε14 , |Φ ε | ≤

max

(ε) τ ε,0 ∈Mmax,N



E|g ε (τ ε,0 , e Y τε,0 , X τ ε,0 )| 

≤ E max |g ε (n, e Y ε,n , X ε,n )| ≤ M42 . 0≤ n ≤ N

(6.86)

This inequality proves the theorem. Theorem 6.3.4. Let the conditions B5 [𝛾¯], C10 [β¯ ], and D9 [β¯ ] hold and 0 ≤ 𝛾i ≤ β i , i = 1, . . . , k. Then, the following inequality takes place, for 0 ≤ ε ≤ ε14 ,  |Φ ε | ≤ M42 .

(6.87)

Proof. It is analogous to the proof of Theorem 6.3.3. The only difference is that Lemma 6.3.8 should be used instead of Lemma 6.3.7. In this case, the inequality analogous to (6.86) can be written down for 0 ≤ ε ≤ ε14 .

6.4 Multivariate modulated LPP with unbounded characteristics In this section, we present results about convergence for log-rewards for multivariate modulated Markov log-price processes with unbounded characteristics.

232 | 6 Convergence of option rewards – II

6.4.1 Convergence of option rewards for multivariate modulated Markov log-price processes Let us now consider the log-reward functions ϕ ε,n ( y , x) introduced in Subsection 6.3.2 and give conditions for their convergence. Theorem 6.4.1. Let the conditions B5 [𝛾¯] and C10 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also conditions I1 and J1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn ∩ Z n: y ε , x ε ) → ϕ0,n ( y0 , x0 ) ϕ ε,n (

as

ε → 0.

(6.88)

Proof. It is analogous to the proof of Theorem 5.2.1. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. According to Theorem 6.3.1, the conditions B5 [𝛾¯] and C10 [β¯ ] imply that there exist ε12 > 0 and the constant M40 , M41,i , i = 1, . . . , k such that the following upper bounds y , x) for  z = ( y , x) ∈ Z, n = 0, 1, . . . , N, take place for the reward functions ϕ ε,n ( ε ≤ ε12 , ⎧⎛ ⎞ ⎫ k ⎨



𝛾⎬ |ϕ ε,n ( y , x)| ≤ M40 + M41,i + M41,i exp ⎩⎝ A N −n,j(β¯ i )|y j |⎠ i ⎭ . (6.89) βi i:𝛾i =0

i:𝛾i >0

j =1

z = ( y , x) ∈ Z, Using inequalities (6.89) and the condition C10 [β¯ ], we get for any  n = 0, 1, . . . , N − 1, ε ≤ ε12 , ⎛

α ε,n+1 , X ε,n+1)| ≤ ⎝M49 + M41,i Ez ,n |ϕ ε,n+1(Y i:𝛾i =0

⎧⎛ ⎞ ⎫⎞ α k ⎨

𝛾⎬ + M41,i exp ⎩⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j|⎠ i ⎭⎠ βi i:𝛾i >0 j =1 ⎛

α α ≤ Ez ,n (k + 1)α−1 ⎝M40 + M41,i i:𝛾i =0

⎧⎛ ⎫⎞ ⎞ k ⎨

⎬

𝛾 α i α ⎠ + M41,i exp ⎩⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j|⎠ βi ⎭ i:𝛾i >0 j =1 ⎛

α α −1 ⎝ α ≤ (k + 1) M40 + M41,i i:𝛾i =0

⎧⎛ ⎫ ⎞ k ⎨

⎬

𝛾 α α + M41,i exp ⎩⎝ A N −n−1,j(β¯ i )|y j |⎠ i ⎭ βi i:𝛾 >0 j =1 i

6.4 Multivariate modulated LPP with unbounded characteristics

| 233

⎧⎛ ⎫⎞ ⎞ k ⎨

⎬ 𝛾 α i ⎠ × Ez ,n exp ⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j − y j |⎠ ⎩ βi ⎭ j =1 ⎛

α α ≤ (k + 1)α−1 ⎝M40 + M41,i i:𝛾i =0

⎧⎛ ⎫ ⎞ k ⎨

⎬

𝛾 α α + M41,i exp ⎩⎝ A N −n−1,j(β¯ i )|y j |⎠ i ⎭ βi i:𝛾i >0 j =1 ⎫⎞ ⎧⎛ k ⎬ ⎨

× Ez ,n exp ⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j − Y ε,n,j| ⎠ ⎭ ⎩ j =1 ⎛

α α ≤ (k + 1)α−1 ⎝M40 + M41,i i:𝛾i =0

⎧⎛ ⎫ ⎞ k ⎨

⎬

𝛾 α i α + M41,i exp ⎩⎝ A N −n−1,j(β¯ i )|y j |⎠ βi ⎭ i:𝛾i >0 j =1 ⎫ ⎧ ⎞ k ⎬ ⎨

¯  N −n−1(β i ))|y j | K45,i ⎠ = Q A ( × exp A j (A y) < ∞ . ⎭ ⎩

(6.90)

j =1

The following proof repeats the proof of Theorem 5.2.1. The only difference is that relation (5.57) is replaced by the analogous relation ε,N ( lim E|ϕ ε,N (Y z ε ), X ε,N ( z ε ))|α ≤ Q A ( y0 ) < ∞ , ε →0

(6.91)

y ) defined in relation (5.42) is replaced by the function Q A (y) where the function Q( defined in relation (6.90). Remark 6.4.1. According to Lemma 6.3.5, the condition C10 [β¯ ] can be replaced in The­ orems 6.4.1 by the condition E14 [β¯ ]. Remark 6.4.2. As follows from the remarks made in Subsection 5.2.1, the follow­ ing pointwise convergence relation take place for the reward functions, for any  z = ( y , x) ∈ Zn ∩ Z n , n = 0, 1, . . . , N, y , x) → ϕ0,n ( y , x) ϕ ε,n (

as

ε → 0.

(6.92)

Also, the reward function ϕ0,n ( y , x) is continuous in points  z = ( y , x) ∈ Zn ∩ Z n , for every n = 0, 1, . . . , N. The following theorem gives alternative conditions for convergence of log-reward functions for American-type options in the above model. Theorem 6.4.2. Let the conditions B5 [𝛾¯] and C10 [β¯ ] hold with vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also conditions H1 , I1 and J1 hold. Then, for every n = 0, 1, . . . , N, the

234 | 6 Convergence of option rewards – II following relation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn ∩ Z n, ϕ ε,n ( y ε , x ε ) → ϕ0,n ( y0 , x0 )

as

ε → 0.

(6.93)

Proof. It is analogous to the proof of Theorem 5.2.2. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. According to Theorem 6.3.2, the conditions B5 [𝛾¯] and C10 [β¯ ] imply that there exist    , M41,i , i = 1, . . . , k such that the following upper bounds ε12 > 0 and the constant M40 take place for the reward functions ϕ ε,n ( y , x) for  z = ( y , x) ∈ Zε,n , n = 0, 1, . . . , N, ε ≤ ε12 ,  |ϕ ε,n ( y , x)| ≤ M40 +

i:𝛾i =0



 M41,i +

 M41,i exp{(

i:𝛾i >0

k

𝛾 A N −n,j(β¯ i )|y j |) i } . β i j =1

(6.94)

ε,n+1 ∈ Zε,n+1} = 1 for Recall that the condition C10 [β¯ ] (b) implies that Pz,n{Z z ∈ Zε,n, n = 0, . . . , N − 1. Using inequalities (6.94) and the condition C10 [β¯ ], we get for any  z = ( y , x) ∈ Zε,n , n = 0, 1, . . . , N − 1, ε ≤ ε12 , ⎛

α   ε,n+1 , X ε,n+1)| ≤ ⎝M40 Ez ,n |ϕ ε,n+1(Y + M41,i i:𝛾i =0

⎧⎛ ⎞ ⎫⎞ α k ⎨

𝛾⎬  ⎝ + M41,i exp ⎩ A N −n−1,j(β¯ i )|Y ε,n+1,j|⎠ i ⎭⎠ βi i:𝛾i >0 j =1 ⎛

  α ≤ Ez ,n (k + 1)α−1 ⎝(M40 ) + (M41,i )α i:𝛾i =0

⎧⎛ ⎫⎞ ⎞ k ⎨

⎬

𝛾 α  ⎠ + (M41,i )α exp ⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j|⎠ i ⎩ βi ⎭ i:𝛾i >0 j =1 ⎛

  α ) + (M41,i )α ≤ (k + 1)α−1 ⎝(M40 i:𝛾i =0

⎧⎛ ⎫ ⎞ k ⎨

⎬

𝛾 α  + (M41,i )α exp ⎝ A N −n−1,j(β¯ i )|y j |⎠ i ⎩ βi ⎭ i:𝛾i >0 j =1 ⎧⎛ ⎫⎞ ⎞ k ⎨

⎬ 𝛾 α ⎠ × Ez ,n exp ⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j − y j |⎠ i ⎩ βi ⎭ j =1 ⎛

 α −1 ⎝  α ≤ (k + 1) ) + (M41,i )α (M40 i:𝛾i =0

⎧⎛ ⎫ ⎞ k ⎨

⎬

𝛾 α i  α + (M41,i ) exp ⎝ A N −n−1,j(β¯ i )|y j |⎠ ⎩ βi ⎭ i:𝛾i >0

j =1

6.4 Multivariate modulated LPP with unbounded characteristics

| 235

⎫⎞ ⎧⎛ k ⎬ ⎨

× Ez ,n exp ⎝ A N −n−1,j(β¯ i )|Y ε,n+1,j − Y ε,n,j| ⎠ ⎭ ⎩ j =1 ⎛

 α  ≤ (k + 1)α−1 ⎝(M40 ) + (M41,i )α i:𝛾i =0

⎧⎛ ⎫ ⎞ k ⎨

⎬

𝛾 α  + (M41,i )α exp ⎝ A N −n−1,j(β¯ i )|y j |⎠ i ⎩ βi ⎭ i:𝛾i >0 j =1 ⎫ ⎧ ⎞ k ⎬ ⎨

 ⎠  N −n−1(β¯ i ))|y j | K45,i = QA ( A y) < ∞ . × exp j (A ⎭ ⎩

(6.95)

j =1

The following proof repeats the proof of Theorem 5.2.2. The only difference is that relation (5.64) should be slightly modified. As in the proof of Theorem 5.2.2, we choose an arbitrary function  z· =  zε , e ∈ zε →  z0 ∈ ZN −1 ∩ Z as ε → 0, and one should prove that the [0, ε0 ] such that  N −1 following relation, which replaces relation (5.64), holds: ε,N ( lim E|ϕ ε,N (Y z ε ), X ε,N ( z ε ))|α ≤ QA ( y0 ) < ∞ . ε →0

(6.96)

In this case, we should use the condition H1 , due to which the point  z0 = ( y0 , x0 ) ∈ Zε,N −1, for every ε ∈ [0, ε1 ( z0 )]. The parameter ε1 ( z0 ) ∈ (0, ε0 ] has been defined in relation (5.6) given in Remark 5.1.3. Since  zε →  z0 as ε → 0, this implies that there exists ε1 ( z (·)) ∈ (0, ε1 ( z0 )] such z ε = ( y ε , x ε ) ∈ Zε,N −1 for every ε ∈ [0, ε1 ( z (·))]. that  ε,N , X ε,N )|α given in relation (6.95) hold Therefore, upper bounds for Ez,N −1|ϕ ε,N (Y   for every ε ∈ [0, min(ε12 , ε1 ( z(·)))]. y ), relation (6.95) Due to the above remarks and continuity of the function QA ( implies that relation (6.96), replacing relation (5.64), holds. The remaining part of the proof repeats the corresponding part in the proof of Theorem 5.2.2.

6.4.2 Convergence of optimal expected rewards for multivariate modulated Markov log-price processes (ε)

Let us now consider the optimal expected rewards Φ ε = Φ(Mmax,N ) introduced in Subsection 6.3.2 and give conditions for their convergence. Theorem 6.4.3. Let the conditions B5 [𝛾¯], C10 [β¯ ], and D9 [β¯ ] hold with the vector pa­ rameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions I1 , J1 , and K1 hold. Then the following relation takes place: Φ ε → Φ0 as ε → 0 . (6.97)

236 | 6 Convergence of option rewards – II Proof. It is analogous to the proof of Theorem 5.2.3. Due to the above conditions imposed on the parameters 𝛾i , β i , i = 1, . . . , k, one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. According to Theorem 6.3.1, the conditions B5 [𝛾¯] and C10 [β¯ ] imply that there ex­ ist ε12 > 0 and the constants M40 , M41,i , i = 1, . . . , k such that the following upper bounds take place for the reward functions ϕ ε,0 ( y , x) for  z = ( y , x) ∈ Z and ε ≤ ε12 , ⎧⎛ ⎞ ⎫ k ⎨



𝛾⎬ |ϕ ε,0 ( y , x)| ≤ M40 + M41,i + M41,i exp ⎩⎝ A N,j (β¯ i )|y j |⎠ i ⎭ . (6.98) βi i:𝛾i =0

i:𝛾i >0

j =1

Also, according to relation (6.75), the condition D9 [β¯ ] implies that there exist 0 < ε13 ≤ ε0 and the constants K47,i , i = 1, . . . , k such that for 0 ≤ ε ≤ ε13 and i = 1, . . . , k, ⎧ ⎫ k ⎨

⎬  E exp ⎩ A N,j β¯ i |Y ε,0,j|⎭ < K47,i . (6.99) j =1

Using inequalities (6.98) and (6.99), we get for any ε ≤ ε14 = min(ε12 , ε13 ), ε,0 , X ε,0)|α E|ϕ ε,0 (Y ⎧⎛ ⎛ ⎞ ⎫⎞ α k ⎨



𝛾⎬ ≤ ⎝M40 + M41,i + M41,i exp ⎩⎝ A N,j (β¯ i )|Y ε,0,j|⎠ i ⎭⎠ βi i:𝛾i =0 i:𝛾i >0 j =1 ⎧⎛ ⎫⎞ ⎞ ⎛ k ⎨

⎬



𝛾 α α α α ≤ E(k + 1)α−1 ⎝M40 + M41,i + M41,i exp ⎩⎝ A N,j (β¯ i )|Y ε,0,j|⎠ i ⎭⎠ βi i:𝛾i =0 i:𝛾i >0 j =1 ⎫⎞ ⎧ ⎛ k ⎬ ⎨



α α α ≤ (k + 1)α−1 ⎝M40 + M41,i + M41,i E exp ⎩ A N,j (β¯ i )|Y ε,0,j|⎭⎠ ⎛ α ≤ (k + 1)α−1 ⎝M40 +

i:𝛾i =0

i:𝛾i =0

i:𝛾i >0

α M41,i +



⎞

j =1

α M41,i K47,i ⎠ = Q A < ∞ .

(6.100)

i:𝛾i >0

The following proof repeats the proof of Theorem 5.2.3. The only difference is that relation (6.100) should be used instead of relation (5.68) used in Theorem 5.2.3. Remark 6.4.3. According to Lemmas 6.3.5 and 6.3.8, the conditions C10 [β¯ ] and D9 [β¯ ] can be replaced in Theorem 6.4.3, respectively, by the conditions E14 [β¯ ] and F7 [β¯ ]. The following theorem gives alternative conditions of the convergence for optimal ex­ pected rewards. Theorem 6.4.4. Let the conditions B5 [𝛾¯], C10 [β¯ ], and D9 [β¯ ] hold with the vector pa­ rameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also conditions H1 , I1 , J1 , and K1 hold. Then the following relation takes place: Φ ε → Φ0 as ε → 0 . (6.101)

6.4 Multivariate modulated LPP with unbounded characteristics

|

237

Proof. It is analogous to the proof of Theorem 5.2.4. Due to the above conditions imposed on the parameters 𝛾i , β i , one can always find α > 1 such that 𝛾i α ≤ β i , i = 1, . . . , k. According to Theorem 6.3.2, the conditions B5 [𝛾¯] and C10 [β¯ ] imply that there ex­   ist ε12 > 0 and the constants M40 , M41,i , i = 1, . . . , k such that the following upper bounds take place for the reward functions ϕ ε,0( y , x) for  z = ( y , x) ∈ Zε,0, ε ≤ ε14 = min(ε12 , ε13 ): ⎧⎛ ⎞ ⎫ k ⎨



𝛾⎬    |ϕ ε,0( y , x)| ≤ M40 + M41,i + M41,i exp ⎩⎝ A N,j (β¯ i )|y j |⎠ i ⎭ . (6.102) βi i:𝛾 =0 i:𝛾 >0 j =1 i

i

Also, according to relation (6.80), the condition D9 [β¯ ] implies that there exist 0 <  ε13 ≤ ε0 and the constants K47,i , i = 1, . . . , k such that for 0 ≤ ε ≤ ε13 and i = 1, . . . , k, ⎫ ⎧ k ⎬ ⎨

 E exp ⎩ A N,j (β¯ i )|Y ε,0,j|⎭ < K47,i . (6.103) j =1

ε,0 ∈ Zε,0} = 1 for  z ∈ Zε,0. Recall that the condition D9 [β¯ ] (b) implies that Pz,n {Z Using this relation, inequalities (6.102) and the condition D9 [β¯ ], we get for any ε ≤ ε14 = min(ε12 , ε13 ), ε,0 , X ε,0)|α E|ϕ ε,0 (Y ⎧⎛ ⎛ ⎞ ⎫⎞ α k ⎨



𝛾⎬    ≤ ⎝M40 + M41,i + M41,i exp ⎩⎝ A N,j (β¯ i )|Y ε,0,j|⎠ i ⎭⎠ βi i:𝛾i =0 i:𝛾i >0 j =1 ⎛

  α ≤ E(k + 1)α−1 ⎝(M40 ) + (M41,i )α i:𝛾i =0

⎧⎛ ⎫⎞ ⎞ k ⎨

⎬

𝛾 α i  α ⎠ + (M41,i ) exp ⎝ A N,j (β¯ i )|Y ε,0,j|⎠ ⎩ βi ⎭ i:𝛾i >0 j =1 ⎫⎞ ⎧ ⎛ k ⎬ ⎨



  α −1 ⎝  α α α ≤ (k + 1) (M41,i ) + (M41,i ) E exp A N,j (β¯ i )|Y ε,0,j|⎭⎠ (M40 ) + ⎩ i:𝛾i =0 i:𝛾i >0 j =1 ⎛ ⎞



 α    ⎠ ≤ (k + 1)α−1 ⎝(M40 ) + (M41,i )α + (M41,i )α K47,i = QA < ∞. i:𝛾i =0

i:𝛾i >0

(6.104) The following part of the proof repeats the corresponding part of the proof of Theo­ rem 5.2.4, with the only difference that relation (6.104) should be used instead of re­ lation (5.82). Also, relation analogous to (5.75) does require reference to relation (6.93) given in Theorem 6.4.2 instead of relation (6.88) given in Theorem 6.4.1.

238 | 6 Convergence of option rewards – II

6.5 Conditions of convergence for Markov price processes In this section, we reformulate basic conditions and theorems given in Chapter 5 and in this chapter in terms of price processes.

6.5.1 Condition of asymptotic polynomial rate of growth for pay-off functions In what follows the simple fact is used. If e y = s then e| y| = s ∨ s−1 , for any y ∈ R1 . The condition B5 [𝛾¯], expressed in terms of argument ( s , x), takes the following equivalent form: | g ( n, s,x )| B˙ 5 [𝛾¯]: limε→0 max0≤n≤N supv=(s ,x)∈V 1+k εL (s ∨s −1)𝛾i < L9 , for some 0 < L9 < ∞ i =1 10,i i i and 0 ≤ L10,1 , . . . , L10,k < ∞ .

6.5.2 Conditions of asymptotically uniform moment compactness for price processes  ε,n = ( Recall that the price process V S ε,n , X ε,n ) = ((S ε,n,1, . . . , S ε,n,k), X ε,n), is con­ ε,n , X ε,n ) = ((Y ε,n,1, . . . , Z ε,n = (Y nected with the corresponding log-price process   Y ε,n,i Y ε,n,k), X ε,n) by the formulas  S ε,n,i = e , i = 1, . . . , k, n = 0, 1, . . . . Recall that we use the notations P˙ v ,n and E˙ v ,n for conditional probabilities and  ε,n =  v. expectations under the condition V

The modulus Δ β i (Y ε,· ,i , N ), expressed in terms of price processes, takes the fol­ lowing form: &

Δ˙ β (S ε,· ,i , N ) = max sup E˙ v ,n 0≤ n ≤ N − 1  v∈V

S ε,n+1,i S ε,n,i ∨ S ε,n,i S ε,n+1,i

'β

.

(6.105)

The condition C6 [β¯ ], expressed in terms of price process, takes the following equivalent form: C˙ 6 [β¯ ]: limε→0 Δ˙ β i (S ε,· ,i , c, T ) = K30,i < ∞, i = 1, . . . , k, for some 1 < K30,i < ∞, i = 1, . . . , k . The modulus Ξ± β ( Y ε, · ,i , N ), expressed in terms of price processes, takes the following equivalent form: ˙ v ,n ( S ε,n+1,i )±β . Ξ˙ ± β ( S ε, · ,i , N ) = max sup E 0≤ n ≤ N − 1  S ε,n,i v∈V

(6.106)

6.5 Conditions of convergence for Markov price processes

| 239

The condition E10 [β¯ ], expressed in terms of price processes, takes the following equivalent form: E˙ 10 [β¯ ]: limε→0 Ξ˙ ± β i ( S ε, · ,i , N ) < K 31,i , i = 1, . . . , k, for some 1 < K 31,i < ∞, i = 1, . . . , k . The condition E˙ 10 [β¯ ] implies that the condition C˙ 6 [β¯ ] holds. The condition D7 [β¯ ], expressed in terms of price processes, takes the following equivalent form: 1 βi ˙ 7 [β¯ ]: limε→0 E(S ε,0,i ∨ S− < K32,i , i = 1, . . . , k, for some 1 < K32,i < ∞, i = D ε,0,i ) 1, . . . , k . The condition F4 [β¯ ], expressed in terms of price processes, takes the following equiv­ alent form: F˙ 4 [β¯ ]: limε→0 E(S ε,0,i)±β i < K33,i , i = 1, . . . , k, for some 1 < K33,i < ∞, i = 1, . . . , k . ˙ 7 [β¯ ] holds. The condition F˙ 4 [β¯ ] implies that the condition D

6.5.3 Conditions of convergence for pay-off functions The condition I1 , expressed in terms of argument ( s , x), takes the following equivalent form: I˙1 : There exists measurable sets Vn ⊆ V, n = 0, . . . , N, such that the pay-off function g ε (n, s ε , x ε ) → g0 (n,  s0 , x0 ) as ε → 0 for any  v ε = ( sε , xε ) →  v0 = ( s0 , x0 ) ∈ Vn as ε → 0, and n = 0, . . . , N .

6.5.4 Conditions of convergence for price processes The condition J1 , expressed in terms of price processes, takes the following equivalent form: J˙1 : There exist measurable sets V n ⊆ V, n = 0, . . . , N such that: (a) P˙ ε,n ( v ε , ·) ⇒ P˙ 0,n ( v0 , ·) as ε → 0, for any  v ε = ( sε , xε ) →  v0 = ( s0 , x 0 ) ∈ V as ε → 0, and n = 1, . . . , N; n −1 (b) P˙ 0,n ( v0 , Vn ∩ V v0 ∈ Vn−1 ∩ V n −1 and n = 1, . . . , N, where n ) = 1, for every   V n , n = 1, . . . , N are sets introduced in the condition I˙1 . 



The condition J˙1 (b) holds, if (a) the sets Vn , Vn , n = 1, . . . , N are empty sets or (b) are at most finite or countable sets and measures P˙ 0,n ( v 0 , A ),  v0 ∈ Vn−1 ∩ V n −1 have   no atoms at points from the sets Vn , Vn , for every n = 1, . . . , N, or (c) measures P˙ 0,n ( v0 , A),  v0 ∈ Vn−1 ∩ V n −1 , n = 1, . . . , N are absolutely continuous with respect to   some σ-finite measure P˙ (A) on BV and P˙ (Vn ), P˙ (Vn ) = 0, n = 1, . . . , N.

240 | 6 Convergence of option rewards – II The condition K1 , expressed in terms of price processes, takes the following equiv­ alent form: ˙ 1 : (a) P˙ ε,0(·) ⇒ P˙ 0,0 (·) as ε → 0; K   (b) P˙ 0,0 (V0 ∩V 0 ) = 1, where V0 and V0 are the sets introduced in the conditions ˙I1 and J˙1 .

6.5.5 Examples of convergence results As examples, we reformulate in terms of price processes just two basic Theorems 5.2.1  ε,n . and 5.2.3 in terms of rewards for price processes V Theorem 6.5.1. Let the conditions B˙ 5 [𝛾¯] and C˙ 6 [β¯ ] hold with vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions I˙1 and J˙1 hold. Then, for every n = 0, 1, . . . , N, the v ε = ( sε , xε ) →  v0 = ( s0 , x0 ) ∈ Vn ∩ V following relation takes place for any  n: s ε , x ε ) → ϕ˙ 0,n ( s0 , x 0 ) ϕ˙ ε,n (

as

ε → 0.

(6.107)

˙ 7 [β¯ ] hold with vector parameters Theorem 6.5.2. Let conditions B˙ 5 [𝛾¯], C˙ 6 [β¯ ], and D ¯ 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 ˙ 1 hold. Then the following relation takes or β i = 𝛾i = 0, and also conditions I˙1 , J˙1 , and K place, ˙ 0 as ε → 0 . ˙ε→Φ (6.108) Φ The other conditions and theorems presented in Chapter 5 and in this chapter can be reformulated in terms of price processes in analogous way.

7 Space-skeleton reward approximations In this chapter, we present results about space-skeleton approximations for rewards of American type options for discrete time multivariate modulated Markov log-price processes. This model is based on approximations of rewards for a multivariate mod­ ulated Markov log-price process by the corresponding rewards for American type op­ tions for multivariate modulated log-price processes represented by atomic Markov chains, whose transition probabilities and initial distributions are concentrated at fi­ nite sets of skeleton points. The rewards for approximating atomic Markov chains can be effectively computed using backward recurrence relations presented in Chapter 3. The space-skeleton approximations do also require special fitting of transition prob­ abilities and initial distributions for approximating processes to the corresponding transition probabilities and initial distributions for approximated multivariate modu­ lated Markov chains. Convergence of the approximating rewards can be proven using the general convergence results presented in Chapters 5 and 6. In Section 7.1, we describe a general atomic approximation model for discrete time multivariate modulated Markov log-price processes, including the corresponding backward recurrence systems of linear equations for computing of reward functions for approximating atomic Markov log-price processes. In Section 7.2, we give results about convergence of space-skeleton approxima­ tions for rewards of American type options for univariate modulated Markov log-price processes with bounded characteristics. In Section 7.3, we give results about convergence of space-skeleton approxi­ mations for rewards of American type options for multivariate modulated Markov log-price processes with bounded characteristics. In Section 7.4, we give results about convergence of space-skeleton approxima­ tions for rewards of American type options for log-price processes represented by mul­ tivariate modulated random walks. In Section 7.5, we give results about convergence of space-skeleton approxi­ mations for rewards of American type options for multivariate modulated Markov log-price processes with unbounded characteristics. The space-skeleton approximation results presented in this chapter are based on the combination of backward recurrence algorithms for computing of rewards for atomic Markov chains presented in Chapter 3 and general convergence results for rewards of American type options for multivariate modulated Markov log-price processes given in Chapters 5 and 6. This combination yields an effective stochastic approximation method for approximative computing of rewards for American type options for multivariate modulated Markov log-price processes. The main convergence results given in the chapter are Theorems 7.4.1–7.4.6, and 7.5.1–7.5.4. These results are new.

242 | 7 Space-skeleton reward approximations

7.1 Atomic approximation models In this section, we describe general atomic approximation models for discrete time multivariate modulated Markov log-price processes. We also present the correspond­ ing backward recurrence systems of linear equations for computing of log-reward functions for approximating atomic Markov log-price processes. Also, the two most important atomic approximation models, a space-skeleton approximation Markov chain model with fixed skeleton structure and a space-skeleton approximation ran­ dom walk model with additive skeleton structure are described.

7.1.1 An atomic Markov chain approximation model ε,n , X ε,n ), n = 0, 1, . . . , Let us consider a model, where a log-price process  Z ε,n = (Y is a discrete time multivariate modulated Markov process with a phase space Z = Rk × X, an initial distribution P ε,0(A) = P{ Z ε,0 ∈ A}, and the transition probabil­   ities P ε,n ( z , A) = P{Z ε,n ∈ A/Z ε,n−1 =  z }, which depend on a perturbation parameter ε ∈ [0, ε0 ]. Z0,n (for ε = 0) can We additionally assume that the limiting log-price process  be a nonatomic Markov chain, while the prelimiting log-price process  Z ε,n (for every ε ∈ (0, ε0 ]) is an atomic Markov chain, i.e. its transition probabilities P ε,n ( z , A) and the initial distributions P ε,0(A) are discrete distributions concentrated at finite sets of points. Thus, it is assumed that, for every ε ∈ (0, ε0 ], the one-step transition probabil­ z , A) are concentrated at the finite sets Fε,z,n = {f ε,l ( z , n ), l = m − z , n ), ities P ε,n ( ε ( + . . . , m ε ( z , n)} and are connected with the corresponding one-point transition prob­ abilities p ε,l ( z , n ) = P{  Z ε,n = f ε,l ( z , n)/ Z ε,n−1 =  z }, l = m− z , n) , . . . , m + z , n), by ε ( ε ( z ∈ Z and n = 1, 2, . . . : the following relation, for every 

P ε,n ( z , A) = p ε,l ( z , n) , A ∈ BZ . (7.1) f ε,l ( z ,n )∈ A

Additionally, it is assumed that, for every ε ∈ (0, ε0 ], the initial distribution + z ε,0,l, l = m− P ε,0(A) is also concentrated at a finite set Fε,0 = { ε,0 , . . . , m ε,0 } and is con­ + ε,0 =  nected with the corresponding probabilities p ε,l = P{Z z ε,0,l }, l = m− ε,0 , . . . , m ε,0, by the following relation:

p ε,l , A ∈ BZ . (7.2) P ε,0(A) =  z ε,0,l ∈ A

The above assumption (7.1) implies that the r-step transition probabilities P ε (n,  z , n + r, A) = P{ Z ε,n+r ∈ A/ Z ε,n =  z } are also concentrated on the finite sets − Fε,z,n,n+r = {f ε,l ( z , n, n + r), l = m ε ( z , n, n + r), . . . , m+ z , n, n + r)}, with the ε (

7.1 Atomic approximation models | 243

z , n, n + r) = P{ Z ε,n+r = corresponding one-point transition probabilities p ε,l ( + f ε,l ( z , n, n + r)/ Z ε,n =  z }, l = m− ( z , n, n + r ) , . . . , m ( z , n, n + r ) , i.e. for every ε ε  z ∈ Z and n = 0, 1, . . . , and r = 1, 2, . . . ,

p ε,l ( z , n, n + r) , A ∈ BZ . (7.3) P ε (n, z , n + r, A) = f ε,l ( z ,n,n + r )∈A

z , n, n + r) and Also, the following recurrent relations link transition functions f ε,l ( the corresponding transition probabilities p ε,l ( z , n, n + r), for r = 1, 2, . . . , f ε,l ( z , n, n + r) ⎧ ⎪ f ε,l ( z , n + 1) if r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = f ε,j (f ε,i ( z , n, n + r − 1), n + r)if r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= 1, m− z , n + 1) ε ( ≤ l ≤ m+ z , n + 1) , ε ( > 1, (i, j) ∈ Fε,z,n,n+r,l ,

(7.4)

m− z , n, n + r) ε ( ≤ l ≤ m+ z , n, n + r) , ε (

and z , n, n + r) p ε,l ( ⎧ ⎪ p ε,l ( z , n + 1) if r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ = ⎪ (i,j)∈Fε,z,n,n+r,l p ε,i ( z , n, n + r − 1) ⎪ ⎪ ⎪ ⎪ ×p ε,j (f ε,i ( z , n, n + r − 1), n + r)if r ⎪ ⎪ ⎪ ⎪ ⎩

= 1, m− z , n + 1) ε ( ≤ l ≤ m+ z , n + 1) , ε (

(7.5)

> 1, m− z , n, n + r) ε ( ≤ l ≤ m+ z , n, n + r) , ε (

where  Fε,z,n,n+r,l = (i, j) : f ε,j (f ε,i ( z , n, n + r − 1), n + r)

z , n, n + r), m− z , n, n + r − 1) = f ε,l ( ε ( ≤ i ≤ m+ z , n, n + r − 1), m− z , n, n + r − 1), n + r) ε ( ε ( f ε,i (  + ≤ j ≤ m ε (f ε,i ( z , n, n + r − 1), n + r) .

(7.6)

Any node ( z , n) ∈ Z×N generates the tree of nodes T ε,(z,n). This tree includes one z , n), additional nodes (f ε,l ( z , n, n + 1), n + 1), l = m− z , n, n + 1), . . . , initial node ( ε ( + m ε ( z , n, n + 1), appearing in the tree after the first transition of the Markov chain, additional nodes (f ε,l ( z , n, n + 2), n + 2), l = m− z , n, n + 2), . . . , m+ z , n, n + 2), ε ( ε ( appearing in the tree after the second transition, etc.

244 | 7 Space-skeleton reward approximations According to the above remarks, the tree T ε,(z,n) is the following set of nodes: z , n, n + r), n + r), T ε,(z,n) = (f ε,l( l = m− z , n, n + r), . . . , m+ z , n, n + r) , ε ( ε (

r = 0, 1, . . .  ,

(7.7)

z , n, n) =  z, m± z , n, n) = 0. where f ε,0( ε ( The total number of nodes in the tree after r transitions is given by the following formula: z , n) = 1 + ( m + z , n, n + 1) − m− z , n, n + 1) + 1) L ε,r ( ε ( ε ( + · · · + (m+ z , n, n + r) − m− z , n, n + r) + 1) . ε ( ε (

(7.8)

We are specially interested in trees, where the number of nodes after r steps z , n) ≤ Lr h , r = 1, 2, . . . , where L, h > 0, i.e. models where the tree has not more L ε,r ( than polynomial rate of growth for L ε,r ( z , n) as function of r. The standard case is, where the parameters m± z , n, n + r) = ±m ε , 0 ≤ n < ε ( n + r < ∞, and m ε are nonnegative integer numbers. In this case, L ε,r ( z , n) = 1 + (2m ε + 1) + · · · + (2m ε + 1) = r(2m ε + 1) + 1 is a linear function of r. Another standard case is, where the parameters m± z , n, n + r) = ±rm ε , 0 ≤ ε ( n < n + r < ∞, and m ε are nonnegative integer numbers. In this case, L ε,r ( z , n) = 1 + (2m ε + 1) + · · · + (2rm ε + 1) = r + 1 + r(r + 1)m ε = r2 m ε + r(m ε + 1) + 1 is a quadratic function of r. In order to give a more convenient representation for the backward reward algo­ rithm for atomic modulated Markov log-price processes, we should introduce addi­ tional notations, explicitly separating log-price and index components for transition z , n, n + r) ∈ Z in the following form: points f ε,l (   z , n, n + r) = (f ε,l ( z , n, n + r), f ε,l ( z , n, n + r)) , f ε,l (

(7.9)

  ( z , n, n + r) ∈ Rk while f ε,l ( z , n, n + r) ∈ X. where f ε,l The condition A2 holds for every ε ∈ (0, ε0 ] since, in this case, the following z = ( y , x) ∈ Z, n = 0, 1, . . . : inequality takes place for any  

Ez ,n max |g ε (n + r, e Y ε,n+r , X ε,n+r)| 0≤ r ≤ N − n

≤

max

max

0≤ r ≤ N − n m − z ,n,n + r )≤ l≤m + z ,n,n + r ) ε ( ε (



|g ε (n, e f ε,l (z ,n,n+r),

 ( z , n, n + r))| < ∞ . f ε,l

(7.10)

The following lemma is a variant of Lemma 3.4.1. Lemma 7.1.1. Let, for every ε ∈ (0, ε0 ], the prelimit log-price processes  Z ε,n is an atomic Markov chain, with transition probabilities and initial distribution defined, z ) and respectively, in relations (7.1) and (7.2). Then, the log-reward functions ϕ ε,n ( + ϕ ε,n+r(f ε,l ( z , n, n + r)), for points f ε,l ( z , n, n + r), l = m− ( z , n, n + r ) , . . . , m ( z , n, n + ε ε

7.1 Atomic approximation models | 245

z ∈ Z, n = 0, . . . , N, the unique r), r = 1, . . . N − n given by formulas (7.4), are, for every  solution for the following finite recurrence system of linear equations: #  ⎧   ⎪ ϕ ε,N (f ε,l ( z , n, N )) = g ε N, e f ε,l (z ,n,N ), f ε,l ( z , n, N ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z , n, N ), . . . , m+ z , n, N ) , l = m− ⎪ ε ( ε ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r(f ε,l ( z , n, n + r)) ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ = max g (n + r, e f ε,l (z ,n,n+r), f  ( ⎨ ε ε,l z , n, n + r)) , (7.11) ⎪ ⎪ ' ⎪ m+ ( z ,n,n + r +1) ⎪ ε ⎪

⎪ ⎪ ⎪ ϕ ε,n+r+1(f ε,l (f ε,l ( z , n, n + r), n + r + 1)) ⎪ ⎪ ⎪ ⎪ ⎪ l =m− z ,n,n + r +1) ε ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ×p ε,l (f ε,l ( z , n, n + r), n + r + 1) , ⎪ ⎪ ⎪ ⎪ ⎩ l = m− ( z , n, n + r), . . . , m+ ( z , n, n + r), r = N − n − 1, . . . , 0 . ε

ε

z , n, n) =  z , m± z , n, n) = 0. where as above, f ε,0( ε ( (ε)

As far as the optimal expected reward Φ ε = Φ ε (Mmax,N ) is concerned, it can be found using the following summation formula: m+ ε,0

Φε =



p ε,l ϕ ε,0( z ε,0,l) .

(7.12)

l=m− ε,0

7.1.2 An atomic Markov chain approximation model with a fixed skeleton structure The atomic Markov chain approximation model with a fixed skeleton structure is an important particular case of the atomic Markov chain approximation model consid­ ered in Subsection 7.1.1. Let us assume that the sets of points Fε,n = { z ε,n,l = ( y ε,n,l, x ε,n,l), l = m− ε,n , . . . , } ∈ B are defined for n = 0, 1, . . . , m+ Z ε,n In the space-skeleton Markov chain model: (a) the skeleton points f ε,l ( z , n) = + z ε,n,l , m− z , n) = m − z , n) = m +  ε ( ε,n ≤ l ≤ m ε ( ε,n , n = 1, 2, . . . , and, in the se­ z ∈ Z, while quel, the skeleton sets Fε,z,n = Fε,n , n = 1, 2, . . . , do not depend on  (b) the one-point transition probabilities p ε,l ( z , n ) = P{  Z ε,n =  z ε,n,l/ Z ε,n−1 =  z }, − + l = m ε,n , . . . , m ε,n , n = 1, 2, . . . , do depend on  z ∈ Z. z , A) have the following form, for  z ∈ Z and n = In this case, measures P ε,n ( 1, 2, . . . :

P ε,n ( z , A) = p ε,l ( z , n) , A ∈ BZ . (7.13)  z ε,n,l ∈ A

As far as the initial distribution P ε,0(A) is concerned, it is defined, as in the gen­ eral model of atomic Markov chain model, i.e. by the following formula, analogous to

246 | 7 Space-skeleton reward approximations formula (7.2): P ε,0(A) =



p ε,l ,

A ∈ BZ .

(7.14)

 z ε,0,l ∈ A − The tree of nodes T ε,(z,n) includes one initial node ( z , n ), m + ε,n +1 − m ε,n +1 + 1 + additional nodes ( z ε,n+1,l, n + 1), l = m− ε,n +1 , . . . , m ε,n +1, after the first transition, + − + z ε,n+2,l , n + 2), l = m− m ε,n+2 − m ε,n+2 + 1 additional nodes ( ε,n +2 , . . . , m ε,n +2, after the second transition, etc. According to the above remarks, the tree T ε,(z,n) is the following set of nodes: , + + T ε,(z,n) = ( z , n), ( z ε,n+r,l, n + r), l = m− r = 1, 2, . . . . (7.15) ε,n + r , . . . , m ε,n + r ,

The total number of nodes in the tree after r transitions is − + − L ε,r (n) = 1 + (m+ ε,n +1 − m ε,n +1 + 1) + · · · + ( m ε,n + r − m ε,n + r + 1) .

(7.16)

The standard case is where the parameters m± ε,n + r = ±m ε , 0 ≤ n < n + r < ∞ where m ε are nonnegative integers. In this case, L ε,r (n) = 1 +(2m ε + 1)+· · ·+(2m ε + 1) = r(2m ε + 1) + 1 is a linear function of r. The following lemma is a variant of Lemma 3.4.2. Lemma 7.1.2. Let, for every ε ∈ (0, ε0 ], the prelimit log-price processes  Z ε,n is a spaceskeleton Markov chain, with transition probabilities and initial distribution defined, z) and respectively, in relations (7.13) and (7.14). Then, the log-reward functions ϕ ε,n ( − + ϕ ε,n+r( z ε,n+r,l), for points z ε,n+r,l, l = m ε,n+r , . . . , m ε,n+r, r = 1, . . . N − n, are, for every  z ∈ Z, n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equations: ⎧ + ⎪ ϕ ε,N ( z ε,N,l) = g ε (N, ey ε,N,l , x ε,N,l) , l = m− ⎪ ε,N , . . . , m ε,N , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ( z ) = max g ε (n + r, ey ε,n+r,l , x ε,n+r,l) , ε,n + r ε,n + r,l ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ m+ ε,n+r+1 ⎪

⎪ ⎨ ϕ ε,n+r+1( z n+r+1,l )p ε,l ( z ε,n+r,l, n + r + 1) , (7.17) ⎪ l=m− ⎪ ε,n+r+1 ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ l = m− r = N − n − 1, . . . , 1 , ⎪ ε,n + r , . . . , m ε,n + r , ⎪ ⎪ ⎪ ⎪ + & ' ⎪ m ε,n+1 ⎪

⎪ ⎪  y ⎪ ⎪ ( z ) = max g ( n, e , x ) , ϕ ( z ) p ( z , n + 1 ) . ϕ ε,n ε ε,n +1 ε,n +1,l ε,l ⎪ ⎪ ⎩ l=m− ε,n+1

7.1.3 A space-skeleton Markov chain approximation model The space-skeleton Markov chain approximation model is an important particular case of the atomic Markov chain approximation model considered in Subsection 7.1.2.

7.1 Atomic approximation models |

247

+ Let us choose some δ ε,n,i > 0, i = 1, . . . , k, n = 0, 1, . . . , integers m− ε,n,j ≤ m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . , and λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . . + Let us now define points y ε,n,i,l = lδ ε,n,i + λ ε,n,i, l = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, − + n = 0, 1, . . . and choose points x ε,n,l ∈ X, l = m ε,n,0, . . . , m ε,n,0, n = 0, 1, . . . . + Now, let us define the skeleton points for ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,n,j , . . . , m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . ,

 z ε,n,¯l = ( y ε,n,¯l , x ε,n,¯l) = ((y ε,n,1,l1 , . . . , y ε,n,k,l k ), x ε,n,l0) .

(7.18)

As in the general space-skeleton approximation model introduced above, the z , n ) = P{  Z ε,n =  z ε,n,¯l/ Z ε,n−1 =  z }, ¯l = one-point transition probabilities p ε,¯l ( − + (l0 , l1 , . . . , l k ), l j = m ε,n,j , . . . , m ε,n,j, j = 0, . . . , k, n = 1, 2, . . . , do depend on  z ∈ Z. z , A) have the following form, for In this case, the transition probabilities P ε,n (  z ∈ Z and n = 1, 2, . . . :

z , A) = p ε,¯l ( z , n) , A ∈ BZ . (7.19) P ε,n (  z ε,n,¯l ∈ A + ε,0 =  Also, let us p ε,¯l = P{Z z ε,0,¯l}, ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,0,j , . . . , m ε,0,j, j = 0, . . . , k be the corresponding one-point initial probabilities. In this case, the initial distribution P ε,0(A) takes the following form:

P ε,0 (A) = p ε,¯l , A ∈ BZ . (7.20)  z ε,0,¯l ∈ A

The difference in notations (between  z ε,n,l and  z ε,n,¯l and between p ε,l ( z , n) and − p ε,¯l ( z , n)) can be removed by a natural renumeration of indices, namely (m− ε,n,0 , m n,1, − − − + + − − . . . , m− ε,n,k) ↔ m ε,n , . . . , ( m ε,n,0 + 1, m ε,n,1 , . . . , m ε,n,k) ↔ m ε,n + 1, . . . , ( m ε,n,0, m ε,n,1, + + . . . , m ε,n,k) ↔ m ε,n, where − m+ ε,n − m ε,n + 1 =

k 

− (m+ ε,n,j − m ε,n,j + 1) ,

n = 0, 1, . . . .

(7.21)

j =0

The simplest variant is to choose integerst ±m± ε,n ≥ 0, n = 0, 1, . . . . − z , n ), m + The tree of nodes T ε,(z,n) includes one initial node ( ε,n +1 − m ε,n +1 + 1 ad­ + ditional nodes  z ε,n+1,¯l , ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,n +1,j , . . . , m ε,n +1,j, j = 0, . . . , k, ap­ + − pearing in the tree after the first jump, m ε,n+2 − m ε,n+2 + 1 additional nodes  z ε,n+2,¯l , ¯l = − + (l0 , l1 , . . . , l k ), l j = m ε,n+2,j, . . . , m ε,n+2,j, j = 0, . . . , k, appearing in the tree after the second jump, etc. In this case, the tree T ε,(z,n) is the following set of nodes: + z , n), ( z ε,n+r,¯l, n + r), ¯l = (l0 , l1 , . . . , l k ), T ε,(z,n) = ( , + l j = m− j = 0, . . . , k , r = 1, 2, . . . . (7.22) ε,n + r,j , . . . , m ε,n + r,j ,

248 | 7 Space-skeleton reward approximations The total number of nodes in the tree after r transitions is given by the following formula: − + − L ε,r (n) = 1 + (m+ ε,n +1 − m ε,n +1 + 1) + · · · + ( m ε,n + r − m ε,n + r + 1) .

(7.23)

The following lemma is a variant of Lemma 3.4.3. Z ε,n is a spaceLemma 7.1.3. Let, for every ε ∈ (0, ε0 ], the prelimit log-price processes  skeleton Markov chain with transition probabilities and initial distribution defined, respectively, in relations (7.19) and (7.20). Then, the log-reward functions ϕ ε,n ( z ) and + z n+r,¯l), for points  z n+r,¯l , ¯l = (l0 , l1 , . . . , l k ), l j = m− , . . . , m , j = 0, . . . , k, ϕ ε,n+r( ε,n,j ε,n,j r = 1, . . . , N − n, given by formulas (7.18), are, for every  z ∈ Z, n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equations: ⎧ ⎪ ϕ ε,N ( z ε,N,¯l) = g ε (N, ey ε,N,¯l , x ε,N,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ + ¯l = (l0 , l1 , . . . , l k ), l j = m− ⎪ ⎪ ε,N,j , . . . , m ε,N,j , j = 0, . . . , k , ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z ε,n+r,¯l) = max g ε (n + r, ey ε,n+r,¯l , x ε,n+r,¯l) , ⎪ ⎪ ϕ ε,n+r( ⎪ ⎪ ⎪ ' ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ϕ ε,n+r+1( z ε,n+r+1,¯l  )p ε,¯l  ( z ε,n+r,¯l, n + r + 1) , ⎪ ⎪ ⎪ ⎨  z ε,n+r+1,¯l  ∈Fε,n+r+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ l j = m− ε,n + r,j , . . . , m ε,n + r,j ,

¯l = (l0 , l1 , . . . , l k ) , r = N − n − 1, . . . , 1 , & ϕ ε,n ( z ) = max g ε (n, ey , x) ,



j = 0, . . . , k ,

ϕ ε,n+1( z ε,n+1,¯l  )

 z ε,n+1,¯l  ∈Fε,n+1

'

(7.24)

z , n + 1) . × p ε,¯l  ( (ε)

As far as the optimal expected reward Φ ε = Φ ε (Mmax,N ) is concerned, it is given by the following formula analogous of formula (3.102):

p ε,¯l ϕ ε,0 ( z ε,0,¯l) . (7.25) Φε =  z ε,0,¯l ∈ Fε,0

7.1.4 A space-skeleton Markov chain approximation model with an additive skeleton structure The space-skeleton Markov chain approximation model with an additive skeleton structure also is a particular case of the atomic Markov chain model, introduced in Subsection 7.1.1. It can also be referred to as a space-skeleton Markov random walk approximation model. ε,n , X ε,n ), n = Note, first of all, that the modulated Markov chain  Z ε,n = (Y 0, 1, . . . , can always be represented in the form of the modulated Markov random

7.1 Atomic approximation models

| 249

ε,n , X ε,n ) = (Y ε,n−1 + W  ε,n , X ε,n), n = 1, 2, . . . , with an initial state Z ε,n = (Y walk  ε,0 , X ε,0) and random jumps W  ε,n − Y  ε,n−1, n = 1, 2, . . . .   ε,n = Y Z ε,0 = (Y In this case, it is convenient to operate with the transition jump probabilities ˜ ε,n (  ε,n , X ε,n ) ∈ A/Y ε,n−1 =  P y , x, A) = P{(W y , X ε,n−1 = x}, which are connected with the transition probabilities of the Markov chain  Z ε,n by the following relations, for  z = ( y , x) ∈ Z, A ∈ BZ , n = 1, 2, . . . :

˜ ε,n (  ε,n , X ε,n ) ∈ A/Y ε,n−1 =  y , x, A) = P{(W y , X ε,n−1 = x} P ε,n −  ε,n−1 =  = P{(Y y , X ε,n ) ∈ A/Y y , X ε,n−1 = x}

(7.26)

= P ε,n ( z , A[y] ), 



where A[y] = { z = ( y , x) : ( y − y , x) ∈ A}. In this case, we also use the standard space-skeleton construction, but applied to jumps of a random walk. Let us choose, for every ε ∈ (0, ε0 ], some δ ε,i > 0, i = 1, . . . , k, integers m− ε,n,i ≤ + m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . , λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . , and define + points y ε,n,i,li = l i δ ε,i + λ ε,n,i ∈ R1 , l i = m− ε,n,i . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . . − + Then, let us choose some integers m ε,n,0 ≤ m ε,n,0, n = 0, 1, . . . , and define points + x ε,n,l0 ∈ X, l0 = m− ε,n,0 , . . . , m n,0, n = 0, 1, . . . . Finally let us define skeleton points  z ε,n,¯l = ( y ε,n,¯l , x ε,n,¯l), ¯l = (l0 , l1 , . . . , l k ), l j = + m− , . . . , m , j = 0, . . . , k, n = 0, 1, . . . , ε,n,j ε,n,j  z ε,n,¯l = ( y ε,n,¯l , x ε,n,¯l) = ((y ε,n,1,l1 , . . . , y ε,n,k,l k ), x ε,n,l0) .

(7.27)

z = ( y , x) ∈ Z and n = 1, 2, . . . , skeleton sets Let us now introduce, for  Fε,z,n = {f ε,¯l ( z , n) = ( y+ y ε,n,¯l , x ε,n,¯l) : ( y ε,n,¯l , x ε,n,¯l) ∈ Fε,n} ,

(7.28)

+ z ε,n,¯l : ¯l = (l0 , l1 , . . . , l k ), l j = m− where the sets Fε,n = { ε,n,j , . . . , m ε,n,j, j = 0, . . . , k }, n = 1, 2, . . . . ˜ ε,¯l ( y , x, n) are defined by the In this case, the one-point transition probabilities p + following formulas, for  z = ( y , x) ∈ Z, ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,n,j , . . . , m ε,n,j, j = 0, . . . , k, n = 1, 2, . . . :

 ε,n , X ε,n ) = ( ε,n−1 =  ˜ ε,¯l ( p y , x, n) = P{(W y ε,n,¯l , x ε,n,¯l)/Z z} ε,n = f ε,¯l ( = P{ Z z , n)/ Z ε,n−1 =  z}

(7.29)

= p ε,¯l ( z , n) .

˜ ε,n ( y , x, A) have the following form, for Sequentially, the transition probabilities P  z ∈ Z , A ∈ BZ , and n = 1, 2, . . . ,

˜ ε,n ( ˜ ε,¯l ( p y , x, A) = y , x, n), P ( y ε,n,¯l ,x ε,n,¯l )∈ A

=



p ε,¯l ( z , n)

( y+ y ε,n,¯l ,x ε,n,¯l )∈ A [y]

z , A[y] ) . = P ε,n (

(7.30)

250 | 7 Space-skeleton reward approximations + ε,0 =  Also, let p ε,¯l = P{Z z ε,0,¯l}, ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,0,j , . . . , m ε,0,j , j = 0, . . . , k be the corresponding one-point initial probabilities. In this case, the initial distribution P ε,0(A) takes the following form:

P ε,0(A) = p ε,¯l , A ∈ BZ . (7.31)  z ε,0,¯l ∈ A

The difference in notations (between f ε,l ( z , n) and f ε,¯l ( z , n) and between p l ( z , n) and p ε,¯l ( z , n) = p ε,¯l ( y , x, n)) can be removed, as above, by a natural renumeration of − − − − + − indices, namely (m− ε,n,0, m ε,n,1, . . . , m ε,n,k) ↔ m ε,n , . . . , ( m ε,n,0 +1, m ε,n,1 , . . . , m ε,n,k) + + − − ↔ m ε,n +1, . . . , (m ε,n, +, m ε,n,1, . . . , m ε,n,k) ↔ m ε,n , where − m+ ε,n − m ε,n + 1 =

k 

− (m+ ε,n,j − m ε,n,j + 1) ,

n = 0, 1, . . . .

(7.32)

j =0

The simplest variant is to choose integers ±m± ε,n ≥ 0, n = 0, 1, . . . . Let us denote  y ε,n,n+r,¯l = (δ ε,1 l1 + λ ε,n,n+r,1, . . . , δ ε,k l k + λ ε,n,n+r,k), x ε,n,n+r,¯l = + x ε,n+r,l0 , ¯l = (l0 , . . . , l k ), l j = m− ε,n,n +r,j, . . . , m ε,n,n + r,j, j = 0, . . . , k. n + r n+r ± ± ± Here, m ε,n,n+r,0 = m ε,n+r,0 and m ε,n,n+r,j = l=n+1 m± l = n +1 λ ε,l,j, ε,l,j, λ ε,n,n + r,j = j = 1, . . . , k, for 0 ≤ n < n + r < ∞. ε,n , X ε,n ) = ( If  Z ε,n = (Y y , x) for some ( y , x) ∈ Z and n = 0, 1, . . . , then, for   y + y ε,n,n+r,¯l, x ε,n+r,¯l), ¯l = r = 1, 2, . . . , possible states for Z ε,n+r = (Y ε,n+r , X ε,n+r) are ( − + (l0 , . . . , l k ), l j = m ε,n,n+r,j, . . . , m ε,n,n+r,j, j = 0, . . . , k. The total number of such states is given by the following formula, for 0 ≤ n < n + r < ∞: − m+ ε,n,n +r − m ε,n,n + r + 1 =

k  − (m+ ε,n,n + r,j − m ε,n,n +r,j + 1) .

(7.33)

j =0

The simplest variant is again to choose integers ±m± ε,n,n + r ≥ 0, 0 ≤ n < n + r < ∞. The tree of nodes T ε,((y,x),n) includes one initial node (( y , x ) , n ), m + ε,n,n +1 − − m ε,n,n+1 + 1 additional nodes (( y + y ε,n+1,¯l, x ε,n+1,¯l), n + 1), ¯l = (l0 , l1 , . . . , l k ), + + − l j = m− n,n +1,j, . . . , m n,n +1,j, j = 0, . . . , k after the first jump, m ε,n,n +2 − m ε,n,n +2 +1 additional nodes (( y + y ε,n,n+2,¯l, x ε,n+2,¯l), n + 2), ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,n,n +2,j , . . . , + m ε,n,n+2,j, j = 0, . . . , k after the second jump, etc. According to the above remarks, the tree T ε,((y,x),n) is the following set of nodes: y , x), n), (( y + y ε,n,n+r,¯l, x ε,n+r,¯l), n + r), T ε,((y,x),n) = (( + ¯l = (l0 , l1 , . . . , l k ), l j = m− ε,n,n + r,j , . . . , m ε,n,n +r,j , j = 0, . . . , k, r = 1, 2, . . .  .

(7.34)

The total number of nodes in the tree after r jumps is − + − L ε,r (n) = 1 + m+ ε,n,n +1 − m ε,n,n +1 + 1 + · · · + m ε,n,n + r − m ε,n,n + r + 1 .

The following lemma is a variant of Lemma 3.4.4.

(7.35)

7.2 Univariate Markov LPP with bounded characteristics

|

251

Z ε,n is a spaceLemma 7.1.4. Let, for every ε ∈ (0, ε0 ], the prelimit log-price processes  skeleton random walk with transition probabilities and initial distribution defined, re­ spectively, in relations (7.30) and (7.31). Then the log-reward functions ϕ ε,n ( y , x) and y+ y ε,n,n+r,¯l, x ε,n+r,¯l) for points ( y + y ε,n,n+r,¯l, x ε,n+r,¯l), ¯l = (l0 , l1 , . . . , l k ), l j = ϕ ε,n+r( + m− ε,n,n +r,j, . . . , m ε,n,n +r,j, j = 0, . . . , k, r = 1, . . . N − n, given by formulas (7.27), are, for every  z ∈ Z, n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equations: ⎧ ⎪ ϕ ε,N ( y+ y ε,n,N,¯l, x ε,N,¯l) = g(N, ey+y ε,n,N,¯l , x ε,N,¯l) , ⎪ ⎪ ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ j = 0, . . . , k , ⎪ ε,n,N,j , . . . , m ε,n,N,j , ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r( y+ y ε,n,n+r,¯l, x ε,n+r,¯l) ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = max g(n + r, ey+y ε,n,n+r,¯l , x ε,n+r,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ϕ ε,n+r+1( y+ y ε,n,n+r,¯l +  y ε,n+r+1,¯l  , x ε,n+r+1,¯l  ) ⎪ ⎪ ⎪ ⎪ ⎪ ( y ε,n+r+1,¯l  ,x ε,n+r+1,¯l  )∈Fε,n+r+1 ⎪ ⎪ ⎨ ' (7.36) ⎪ y+ y ε,n,n+r,¯l, x ε,n+r,¯l, n + r + 1) , × p ε,¯l  ( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ j = 0, . . . , k , ⎪ ε,n + r,j, . . . , m ε,n + r,j , ⎪ ⎪ ⎪ ⎪ ⎪ r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n ( y , x) = max g(n, ey , x) , ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ϕ ε,n+1( y+ y ε,n+1,¯l  , x ε,n+1,¯l  )p ε,¯l  ( y , x, n + 1) . ⎪ ⎪ ⎩ ( y ε,n+1,¯l  ,x ε,n+1,¯l  )∈Fε,n+1

(ε)

As far as the optimal expected reward Φ ε = Φ ε (Mmax,N ) is concerned, it is defined as in the general model of atomic modulated Markov log-price processes, i.e. by the following analogous of formula (3.102):

Φε = p ε,¯l ϕ ε,0( z ε,0,¯l) . (7.37)  z ε,0,¯l ∈ Fε,0

7.2 Univariate Markov LPP with bounded characteristics In this section, we present space-skeleton approximation models for discrete time uni­ variate Markov log-price processes with bounded characteristics.

252 | 7 Space-skeleton reward approximations

7.2.1 Space-skeleton approximations for log-rewards functions for univariate Markov log-price processes Let us assume that, for every ε ∈ [0, ε0 ], a modulated log-price process Y ε,n , n = 0, 1, . . . , is an inhomogeneous Markov chain, with a phase space R1 , an initial distri­ bution P ε,0(A), and the one-step transition probabilities P ε,n (y, A). We would like to approximate the log-price process Y0,n by the log-price processes Y ε,n . We additionally assume that the log-price process Y0,n (for ε = 0) can be a nonatomic Markov chain while the log-price processes Y ε,n (for ε ∈ (0, ε0 ]) are atomic Markov chains, i.e. their transition probabilities P ε,n (y, A) and initial distributions P ε,0(A) are concentrated at finite sets of points. In this case, the model without index component is considered. In order to be able to interpret the model as modulated, one can always introduce a “virtual” index component X ε,n with the one-point phase space X = {x0 }. Thus, we also assume that a pay-off function g(n, e y , x) = g(n, e y ) does not depend on argument x and is a realvalued Borel function of argument y ∈ R1 , for every n = 0, 1, . . . , N. Note also that, we consider the case where the pay-off function g(n, e y ) does not depend on the parameter ε. For every ε ∈ (0, ε0 ], let us choose nonempty Borel sets Aε,n,l ⊆ R1 , l = + m− ε,n , . . . , m ε,n , n = 0, 1, . . . such that, A ε,n,l ∩ A ε,n,l = ∅, l ≠ l ;

m+

R1 = ∪l=ε,nm−ε,n A ε,n,l ,

n = 0, 1, . . . .

(7.38)

and skeleton points, y ε,n,l ∈ Aε,n,l ,

+ l = m− ε,n , . . . , m ε,n ,

n = 0, 1, . . . .

(7.39)

Let us now assume that the following fitting condition holds: L1 : For every ε ∈ (0, ε0 ], transition probabilities P{Y ε,n = y ε,n,l/Y ε,n−1 = y} = + p ε,l (y, n) = P0,n (y ε,n−1,l, Aε,n,l), y ∈ Aε,n−1,l , l = m− ε,n −1, . . . , m ε,n −1, l = − + m ε,n , . . . , m ε,n , n = 1, 2, . . . . If we are also interested to fit initial distributions, the following fitting condition can be assumed: M1 : For every ε ∈ (0, ε0 ], initial probabilities P{Y ε,0 = y ε,0,l} = p ε,l = P0,0 (Aε,0,l), + l = m− ε,0 , . . . , m ε,0 . A standard way is to chose as the skeleton sets Aε,n,l and space-skeleton points y ε,n,l is the following. Let us choose, for every ε ∈ (0, ε0 ], some δ ε,n > 0, n = 0, 1, . . . , integers m− ε,n ≤ + m ε,n , n = 0, 1, . . . , and λ ε,n ∈ R1 , n = 0, 1, . . . .

7.2 Univariate Markov LPP with bounded characteristics

| 253

First, the intervals Iε,n,l, which play the role of the skeleton sets Aε,n,l, are con­ + structed for l = m− ε,n , . . . , m ε,n , n = 0, 1, . . . , ⎧ 1 − ⎪ ⎪ if l = m− ε,n , ⎪(−∞, δ ε,n (m ε,n + 2 )] + λ ε,n ⎨ 1 1 − (7.40) Iε,n,l = A ε,n,l = (δ ε,n(l − 2 ), δ ε,n (l + 2 )] + λ ε,n if m ε,n < l < m+ ε,n , ⎪ ⎪ ⎪ ⎩(δ (m+ − 1 ), ∞) + λ + if l = m . ε,n

ε,n

2

ε,n

ε,n

Then the skeleton points y ε,n,l ∈ Iε,n,l are chosen in the following way: y ε,n,l = lδ ε,n + λ ε,n ,

l = m− ε,n , . . . ,

m+ ε,n ,

n = 0, 1, . . . .

(7.41)

In this case, the fitting condition L1 takes the following form: L2 : For every ε ∈ (0, ε0 ], transition probabilities P{Y ε,n = y ε,n,l/Y ε,n−1 = y} = + − p ε,l (y, n) = P0,n (y ε,n−1,l , Iε,n,l), y ∈ Iε,n−1,l , l = m− ε,n −1 , . . . , m ε,n −1, l = m ε,n , . . . , m+ ε,n , n = 1, 2, . . . , The fitting condition M1 takes the following form: M2 : For every ε ∈ (0, ε0 ], initial probabilities P{Y ε,0 = y ε,0,l} = p ε,l = P0,0 (Iε,0,l), + l = m− ε,0 , . . . , m ε,0 . + In this case, the corresponding skeleton sets Fε,n = {y ε,n,l, l = m− ε,n , . . . , m ε,n }, n = 0, 1, . . . . Let us define, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , skeleton functions, ⎧ − ⎪ ⎪ ⎪ δ ε,n m ε,n + λ ε,n ⎪ ⎪ 1 ⎪ ⎪ if y ≤ δ ε,n (m− ⎪ ε,n + 2 ) + λ ε,n , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δ ε,n l + λ ε,n ⎪ ⎨ h ε,n (y) = ⎪ (7.42) if δ ε,n (l − 21 ) + λ ε,n < y ≤ δ ε,n (l + 21 ) + λ ε,n , ⎪ ⎪ ⎪ + ⎪ m− ⎪ ε,n < l < m ε,n , ⎪ ⎪ ⎪ ⎪ + ⎪ δ m + λ ε,n ⎪ ⎪ ⎪ ε,n ε,n ⎪ ⎩ 1 if y > δ ε,n (m+ ε,n − 2 ) + λ ε,n .

In this case, the transition probabilities P ε,n (y, A) = P{Y ε,n ∈ A/Y ε,n−1 = y}, y ∈ R1 , A ∈ B1 , n = 1, . . . , N take the following form, for every ε ∈ (0, ε0 ]:

P0,n (h ε,n−1(y), Iε,n,l) P ε,n (y, A) = y ε,n,l ∈ A

= P{h ε,n (Y 0,n ) ∈ A/Y 0,n−1 = h ε,n−1(y)} .

(7.43)

As far as the initial distribution P ε,0 (A) = P{Y ε,0 ∈ A}, A ∈ B1 is concerned, it takes the following form, for every ε ∈ (0, ε0 ]:

P0,0 (Iε,0,l) = P{h ε,0(Y0,0 ) ∈ A} . (7.44) P ε,0(A) = y ε,0,l ∈ A

254 | 7 Space-skeleton reward approximations The possibility to construct approximating Markov chains Y ε,n satisfying the fit­ ting conditions L2 and M2 follows from the fact that the quantity P ε,n (y, A), defined in relation (7.43), satisfies conditions that should satisfy transition probabilities of a Markov chain. It is a measurable function in y ∈ R1 , for every A ∈ B1 , and a proba­ bility measure as a function of A ∈ B1 , for every y ∈ R1 . Also, the quantity P ε,0 (A), defined in relation (7.44), is a probability measure. It is worth to note that the above described space-skeleton approximation proce­ dure realizes distributional fitting. The Markov chains Y ε,n can be defined for differ­ ent ε on different probability spaces. However, it is possible to realize the so-called strong space-skeleton approxima­ tion procedure, where Markov chains Y ε,n are defined for all ε on the same probability space. The phase space of the Markov chain Y0,n is R1 . In this case, we can assume, with­ out loss of generality, that this Markov chain is defined in the dynamic form ⎧ ⎨ Y 0,n = A0,n (Y 0,n−1 , U0,n ) , n = 1, 2, . . . , (7.45) ⎩ Y 0,0 = A0,0 (U0,0 ) where (a) U0,n , n = 0, 1, . . . , are i.i.d. random variables uniformly distributed in the interval [0, 1), (b) A0,n (y, u ), n = 1, 2, . . . , are Borel functions acting from R1 × [0, 1) to R1 , and (c) A0,0 (u ) is a Borel function acting from [0, 1) to R1 . More precisely, as was pointed out in Subsection 1.1.4, we can always construct a  given in the described above dynamic form, with the same phase Markov chain Y0,n space, transition probabilities and initial distribution as the Markov chain Y0,n . In this particular case, where the phase space is a real line, the corresponding construction is very simple one. Let us define A0,n (y, u ) = inf (v : P0,n (y, (−∞, v] > u ), u ∈ [0, 1), for every y ∈ R1 , n = 1, 2, . . . , and A0,0 (u ) = inf (v : P0,0 ((−∞, v] > u ), u ∈ [0, 1). Let also U n , n = 0, 1, . . . , is a sequence of i.i.d. random variables uniformly dis­   = A0,n (Y 0,n tributed in interval [0, 1). In this case, the random sequence Y0,n −1 , U n ),  n = 1, 2, . . . , Y0,0 = A0,0 (U0 ) is a Markov chain, which has the same phase space, transition probabilities and initial distribution as the Markov chain Y0,n . Thus, we can, indeed, assume that the Markov chain Y0,n is given in the dynamic form described above. In this case, the Markov chain Y ε,n can also be defined in the dynamic form, for every ε ∈ (0, ε0 ], according to the following relation: ⎧ ⎨ Y ε,n = h ε,n (A0,n (h ε,n−1(Y ε,n−1), U n )) , n = 1, 2, . . . , (7.46) ⎩ Y ε,0 = h ε,0(A0,0 (U0 )) . It is readily seen that the random sequence Y ε,n is a Markov chain, and its transi­ tion probabilities and initial distribution satisfy the conditions L2 and M2 and, subse­ quently, relations (7.43) and (7.44). It is worth to note that the conditions L2 and M2 imply that P{Y ε,n ∈ Fε,n, n = 0, 1, . . . } = 1. Since h ε,n (y) = y for y ∈ Fε,n , n = 0, 1, . . . , it seems that one can use

7.2 Univariate Markov LPP with bounded characteristics

| 255

simpler dynamic transition function h ε,n (A n (y, u )) in relation (7.46). However, this is not possible. The Markov chain Y ε,n has the phase space R1 . The conditions L2 and M2 require that the transition probabilities and the initial distribution of this Markov chain are concentrated at the above finite sets. That is why, the corresponding dynamic transi­ tion functions in relation (7.46) are defined as A ε,n (y, u ) = h ε,n (A n (h ε,n−1(y), u )) for (y, u ) ∈ R1 × [0, 1). Since the pay-off function g(n, e y ) does not depend on the index argument x, the log-reward functions ϕ ε,n (y), n = 0, 1, . . . , for the log-price processes Y ε,n are func­ tions of y, for every ε ∈ [0, ε0 ] . The following lemma is a variant of Lemma 7.1.3. Lemma 7.2.1. Let, for every ε ∈ (0, ε0 ], the prelimiting log-price process Y ε,n is a spaceskeleton Markov chain with transition probabilities and initial distribution defined, re­ spectively, in relations (7.43) and (7.44). Then, the log-reward functions ϕ ε,n (y) and + ϕ ε,n+r(y ε,n+r,l), for points y ε,n+r,l, l = m− ε,n + r , . . . , m ε,n + r , r = 1, . . . N − n, given by + formulas (7.41), are, for every y ∈ Iε,n,l, l = m− ε,n , . . . , m ε,n , n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equations: ⎧ + ⎪ ϕ ε,N (y ε,N,l) = g(N, e y ε,N,l ) , l = m− ⎪ ε,N , . . . , m ε,N , ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r(y ε,n+r,l) = max g(n + r, e y ε,n+r,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ m+ ε,n+r+1 ⎪

⎪ ⎪ ⎪ ⎪    ϕ ε,n+r+1(y ε,n+r+1,l )P0,n+r+1(y ε,n+r,l , Iε,n+r+1,l ) , ⎪ ⎪ ⎪ ⎨ l =m− ε,n+r+1 (7.47) + ⎪ ⎪ r = N − n − 1, . . . , 1 , l = m− ⎪ ε,n + r , . . . , m ε,n + r , ⎪ ⎪ ⎪ & ⎪ m+ ε,n+1 ⎪

⎪ ⎪ y ⎪ ⎪ ϕ ( y ) = max g ( n, e ) , ϕ ε,n+1(y ε,n+1,l) ε,n ⎪ ⎪ ⎪  =m− ⎪ l ⎪ ε,n+1 ⎪ ' ⎪ ⎪ ⎪ ⎪ ⎪ ⎪   P ( y , I ) . × ⎩ 0,n +1 ε,n,l ε,n +1,l

7.2.2 Convergence of space-skeleton approximations for logrewards functions for univariate Markov log-price processes Let us formulate conditions, which should be imposed on the log-price process Y0,n and the pay-off function g(n, e y ) in order the log-reward functions ϕ ε,n (y) for the approximation processes Y ε,n would converge to the limiting log-reward functions ϕ0,n (y) for the process Y0,n , as ε → 0. Let us introduce special shorten notations for the maximal and the minimal skele­ ton points, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , ± z± ε,n = δ ε,n m ε,n + λ ε,n .

(7.48)

256 | 7 Space-skeleton reward approximations We impose the following condition on parameters of the space-skeleton model: N1 : (a) δ ε,n → 0 as ε → 0, for n = 0, 1, . . . , N; (b) ±z± ε,n → ∞ as ε → 0, for n = 0, 1, . . . , N; (c) ±z± ε,n , n = 0, 1, . . . , N are nondecreasing sequences, for every ε ∈ (0, ε0 ] . − Note that the condition N1 implies that δ ε,n (m+ ε,n − m ε,n ) → ∞ as ε → 0, for n = − 0, 1, . . . , N, and m+ ε,n − m ε,n → ∞ as ε → 0, for n = 0, 1, . . . , N. Let us recall the first-type modulus of exponential moment compactness for the log-price process Y0,n , defined for β ≥ 0 and taken for the model without index com­ ponent the following form:

Δ β (Y0,· , N ) = max sup Ey,n e β| Y0,n+1 −Y0,n | . 0≤ n ≤ N −1 y ∈R1

(7.49)

We impose on the price process Y0,n the following first-type condition of exponen­ tial moment compactness, which is a one-dimensional variant of the condition C1 [β¯ ] and should be assumed to hold for some β ≥ 0: C11 [β ]: Δ β (Y0,· , N ) < K49 , for some 1 < K49 < ∞ . Let us also recall the second-type modulus of exponential moment compactness for the log-price process Y n , defined for β ≥ 0, ± β( Y n+1,i − Y n,i ) Ξ± . β ( Y · , N ) = max sup Ey,n e

0≤ n ≤ N −1 y ∈R1

(7.50)

As follows from Lemma 4.1.8, the following condition implies the condition C11 [β ] to hold: E15 [β ]: Ξ± β ( Y 0, · , N ) < K 50 , for some 1 < K 50 < ∞ . Let us introduce sequences of the sets Zε = Zε,0 , . . . , Zε,N , where the sets Zε,n are intervals defined, for every n = 0, . . . , N, by the following relation: ⎧ ⎨[z− , z+ ] for ε ∈ (0, ε0 ] , ε,n ε,n Zε,n = ⎩ (7.51) R for ε = 0 . 1

Note that the condition N1 (c) implies the following imbedding relation for the intervals Zε,n , for every ε ∈ (0, ε0 ]: Zε,0 ⊆ · · · ⊆ Zε,N .

(7.52)

Let us now recall the modified first-type modulus of exponential moment com­ pactness, for every ε ∈ [0, ε0 ] and β ≥ 0, Δ β,Zε (Y ε,· , N ) = max

sup Ey,n e β| Y ε,n+1 −Y ε,n | .

0≤ n ≤ N −1 y ∈Zε,n

(7.53)

7.2 Univariate Markov LPP with bounded characteristics

|

257

Note, that, by the definition, the modified modulus of exponential moment com­ pactness Δ β,Z0 (Y0,· , N ) coincides with the standard modulus of exponential moment compactness Δ β (Y0,· , N ). We shall now prove below that, in the skeleton approximation model described above, the conditions L2 , N1 , and C11 [β ] imply that the modified condition of exponen­ tial moment compactness C6 [β¯ ] (its one-dimensional variant) holds with the modified moduli of exponential moment compactness Δ β,Zε (Y ε,· , N ). While the standard con­ dition of exponential moment compactness C6 [β¯ ] (also its one-dimensional variant) does not hold for the log-price processes Y ε,n . We assume that the pay-off function g(n, e y ) satisfies the condition B4 [𝛾] holds for some 𝛾 ≥ 0. The condition I1 should be replaced by the following condition: I6 : There exist the Borel sets Yn ⊆ R1 , n = 0, . . . , N such that function g(n, e y ) is continuous in points y ∈ Yn , for every n = 0, . . . , N . The condition I6 obviously implies that g(n, e y ε ) → g(n, e y0 ) as ε → 0 for any y ε → y0 ∈ Yn as ε → 0 and n = 0, . . . , N. Thus, the condition I1 holds with the sets Zn = Yn × {x0 }, n = 0, . . . , N. The condition J1 should be replaced by the following condition: J9 : There exist the Borel sets Y n ⊆ R1 , n = 0, . . . , N such that: (a) P0,n (y ε , ·) ⇒ P0,n (y0 , ·) as ε → 0, for any y ε → y0 ∈ Y n −1 as ε → 0, and n = 1, . . . , N;   (b) P0,n (y0 , Yn ∩ Y n ) = 1, for every y 0 ∈ Y n −1 ∩ Y n −1 and n = 1, . . . , N, where  Yn , n = 0, . . . , N are sets introduced in the condition I6 . The condition J9 (a) means that the transition probabilities P0,n (y, ·) are weakly con­ tinuous at points y ∈ Y n −1 for every n = 1, . . . , N.   Note also that the condition J9 (b) automatically holds, if the sets Y n , Y n = ∅, n = 1, . . . , N, or these sets are finite or countable and the measures P0,n (y, A), y0 ∈ Yn−1 ∩   Y n −1 have no atoms in points of the sets Y n , Y n , for every n = 1, . . . , N, as well as in the case, where measures P0,n (y0 , ·), y0 ∈ Yn−1 ∩ Y n −1 , n = 1, . . . , N are absolutely continuous with respect to some σ-finite measure P(A) on the Borel σ-algebra B1 and   P(Yn ), P(Yn ) = 0, n = 1, . . . , N. We shall prove below that the conditions L2 , N1 and J9 imply that the condition J1  holds with the sets Zn = Yn × {x0 } and Z n = Y n × {x 0 }, n = 0, . . . , N. The following result can be obtained by application of Theorem 5.2.2 to the above skeleton model. Theorem 7.2.1. Let a Markov log-price process Y 0,n and the corresponding space-skele­ ton approximation Markov processes Y ε,n are defined as described above. Let also the conditions B4 [𝛾] and C11 [β ] hold with the parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L2 , N1 , I6 , and J9 hold. Then, for every n =

258 | 7 Space-skeleton reward approximations 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: ϕ ε,n (y ε ) → ϕ0,n (y0 )

as

ε → 0.

(7.54)

Proof. We are going to check that all conditions of Theorem 5.2.2 hold and to apply this theorem. The following inequality holds for ε ∈ (0, ε0 ] and n = 0, . . . , N, y ∈ R1 , − |h ε,n (y) − y| ≤ (z− ε,n − y ) I ( y ≤ z ε,n ) + + + + δ ε,n I (z− ε,n < y ≤ z ε,n ) + ( y − z ε,n ) I ( y > z ε,n ) .

(7.55)

Relation (7.43) and the condition L2 imply that the following inequality holds for every ε ∈ (0, ε0 ], y ∈ R1 , n = 0, . . . , N − 1, and β ≥ 0, Ey,n e β| Y ε,n+1 −Y ε,n | = Ey,n e β| Y ε,n+1 −y| = Eh ε,n (y),n e β| h ε,n+1 (Y0,n+1 )−y| .

(7.56)

Using relations (7.52), (7.55), and (7.56) we get the following inequalities for every ε ∈ (0, ε0 ], n = 0, . . . , N − 1, and β ≥ 0: sup Ey,n e β| Y ε,n+1 −Y ε,n | = sup Eh ε,n (y),n e β| h ε,n+1 (Y0,n+1 )−y|

y ∈Zε,n

y ∈Zε,n

≤ sup Eh ε,n (y),n e

β( y − Y 0,n+1 )

y ∈Zε,n

I (Y0,n+1 ≤ z− ε,n +1)

+ + e βδ ε,n+1 sup Eh ε,n (y) e β| Y0,n+1 −y| I (z− ε,n +1 < Y 0,n +1 ≤ z ε,n +1)

y ∈Zε,n

+ sup Eh ε,n (y),n e β(Y0,n+1 −y) I (Y 0,n+1 > z+ ε,n +1) y ∈Zε,n

≤e

βδ ε,n+1

sup Eh ε,n (y),n e β| Y0,n+1 −y|

y ∈Zε,n

≤ e βδ ε,n+1 e βδ ε,n sup Eh ε,n (y),n e β| Y0,n+1 −h ε,n (y) | y ∈Zε,n

≤e

β( δ ε,n+1 + δ ε,n )

sup Ey,n e β| Y0,n+1 −y| .

(7.57)

y ∈R1

Relation (7.57) implies the following inequality for moduli Δ β,Zε (Y ε,· , N ) and Δ β (Y0,· , N ) which holds for every ε ∈ (0, ε0 ] and β ≥ 0, Δ β,Zε (Y ε,· , N ) ≤ max e β(δ ε,n+1 +δ ε,n ) Δ β (Y0,· , N ) . 0≤ n ≤ N − 1

(7.58)

By the condition N1 , max0≤n≤N −1 e β(δ ε,n+1 +δ ε,n ) → 1 as ε → 0. This relation, inequal­ ity (7.58), and the condition C11 [β ] imply the following relation: lim Δ β,Zε (Y ε,· , N ) ≤ Δ β (Y0,· , N ) < K49 . ε →0

(7.59)

Therefore, the condition C6 [β¯ ] (its one-dimensional variant for the model without index component, and with the sets Zε,n given by relation (7.51)) holds for the modified moduli of exponential moment compactness Δ β,Zε (Y ε,· , N ).

7.2 Univariate Markov LPP with bounded characteristics

| 259

The condition N1 obviously implies that the condition H1 , with the sets Hε,n = Zε,n × {x0 }, n = 0, . . . , N, holds. As was mentioned above, the condition I6 implies that the condition I1 (its onedimensional variant without modulating component) holds. Let us choose an arbitrary 1 ≤ n ≤ N and y ε → y0 ∈ Yn−1 ∩ Y n −1 as ε → 0. It follows from relation (7.55) that, in this case, also h ε,n−1(y ε ) → y0

as

ε → 0.

(7.60)

Let us take arbitrary κ, ϱ > 0. One can find T κ > 0 such that points ±T κ are points of continuity for the distribution functions Py0 ,n−1{Y0,n ≤ u } = P0,n (y0 , (−∞, u ]), n = 1, . . . , N and, for n = 1, . . . , N, P0,n (y0 , (−∞, −T κ ]) + P0,n (y0 , [T κ , ∞)) ≤ κ .

(7.61)

Using the above remarks, the conditions N1 and J9 (a) and relations (7.55), (7.60), and (7.61), we get the following relation, for n = 1, . . . , N: lim Ph ε,n−1 (y ε ),n−1{|h ε,n (Y0,n ) − Y0,n | ≥ ϱ } # ≤ lim Ph ε,n−1 (y ε ),n−1{Y 0,n ≤ z− ε,n }

0< ε → 0

0< ε → 0

 + I (δ ε,n ≥ ϱ ) + Ph ε,n−1 (y ε ),n−1{Y 0,n > z+ ε,n } # ≤ lim Ph ε,n−1 (y ε ),n−1{Y 0,n ≤ −T κ } 0< ε → 0  + Ph ε,n−1 (y ε ),n−1{Y 0,n ≥ T κ }

= Py0 ,n−1{Y 0,n ≤ −T κ } + Py0 ,n−1 {Y 0,n ≥ T κ } ≤ κ .

(7.62)

Since an arbitrary choice of κ, ϱ > 0, relation (7.62) implies that for any for n = 1, . . . , N and ϱ > 0, lim Ph ε,n−1 (y ε ),n−1{|h ε,n (Y0,n ) − Y0,n | ≥ ϱ } = 0 .

0< ε → 0

(7.63)

Relation (7.63) implies that, for every n = 1, . . . , N, probability measures Ph ε,n−1 (y ε ),n−1{h ε,n (Y0,n ) ∈ ·} and Ph ε,n−1 (y ε ),n−1{Y0,n ∈ ·} have the same weak lim­ its as 0 < ε → 0. This proposition, relations (7.43), (7.60), and the condition J9 (a) imply that the following relation of weak convergence holds for any y ε → y0 ∈ Yn−1 ∩ Y n −1 as 0 < ε → 0, for every 1 ≤ n ≤ N, P ε,n (y ε , ·) = Ph ε,n−1 (y ε ),n−1{h ε,n (Y0,n ) ∈ ·} ⇒ P0,n (y0 , ·) = Py0 ,n−1{Y 0,n ∈ ·}

as 0 < ε → 0 .

(7.64)

The equality of the distributions on the right-hand side of relation (7.64) obvi­ ously implies that this asymptotic relation also holds as 0 ≤ ε → 0. Thus, the con­  dition J1 (a) holds with the sets Zn = Yn × {x0 } and Z n = Y n × {x 0 }, n = 0, . . . , N.

260 | 7 Space-skeleton reward approximations Since, the conditions J9 (b) and J1 (b) coincide, we can conclude that the condition J1 (its one-dimensional variant without modulating component) holds in this case, with sets Zn , Z n , n = 0, . . . , N. It follows from the above remarks that all conditions of Theorem 5.2.2 hold, and, therefore, Theorem 7.2.1 is a corollary of Theorem 5.2.2. Remark 7.2.1. It is not out of picture to note that Theorem 5.2.1, based on the standard condition of exponential moment compactness C6 [β¯ ] (its one-dimensional variant), cannot be applied in this case. Indeed, as follows from formula (7.56), the modulus Δ β (Y ε,· , N ) ≡ ∞, for every ε ∈ (0, ε0 ] and, thus, the condition C6 [β¯ ] (its one-dimen­ sional variant) does not holds.

7.2.3 Convergence of space-skeleton approximations for optimal expected rewards for univariate Markov log-price processes Let us now assume that we have found, using backward recurrence algorithm pre­ sented above in Subsection 7.2.1, the values of the reward functions ϕ ε,0(y ε,0,l) for the following initial points: y ε,0,l = δ ε,0 l + λ ε,0 ,

+ m− ε,0 ≤ l ≤ m ε,0 .

(7.65)

(ε)

The optimal expected reward functional Φ ε = Φ(Mmax,N ) for the approximation log-price processes Y ε,n can be found using the following formula: m+ ε,0

Φε =



ϕ ε,0 (y ε,0,l)p ε,0 ,

(7.66)

l=m− ε,0 + where p ε,l = P0,0 (Iε,0,l) = P{h ε,0(Y0,0 ) = y ε,0,l}, l = m− ε,0 , . . . , m ε,0 . Let us formulate conditions that should be imposed on the log-price process Y0,n and the pay-off function g(n, e y ) in order that the corresponding optimal expected rewards Φ ε would converge to the limiting optimal expected reward Φ0 as ε → 0. We impose (on the initial value of the log-price process Y0,0 ) the following first-­ type condition of exponential moment boundedness, which is a one-dimensional vari­ ant of the condition D1 [β¯ ] and should be assumed to hold for some β ≥ 0: D10 [β ]: Ee β| Y0,0 | < K51 , for some 1 < K51 < ∞ .

The condition K1 should be replaced by the following condition imposed on the initial distribution P0,0 (A):   K8 : P0,0 (Y0 ∩ Y 0 ) = 1, where Y0 and Y0 are the sets introduced, respectively, in the conditions I6 and J9 .

7.2 Univariate Markov LPP with bounded characteristics

| 261

The following theorem is a variant of Theorem 5.2.4 for the above space-skeleton ap­ proximation model. Theorem 7.2.2. Let a Markov log-price process Y 0,n and the corresponding space-skele­ ton approximation Markov processes Y ε,n are defined as described above. Also let con­ ditions B4 [𝛾], C11 [β ], and D10 [β ] hold with the parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L2 , M2 , N1 , I6 , J9 , and K8 hold. Then, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(7.67)

Proof. We are going to check that all conditions of Theorem 5.2.4 hold and to apply this theorem. In fact, we should only check that one-dimensional variants of the con­ ditions D7 [β¯ ] (its one-dimensional variant for the model without index component, − and the sets Zε,0 = [z− ε,0 , z ε,0 ]) and K1 hold. Indeed, this check was made for other conditions in the proof of Theorem 7.2.1. By the condition N1 , there exist ε15 ∈ (0, ε0 ] such that ±z± ε,0 ≥ 0, for ε ∈ (0, ε15 ]. The following inequality holds for ε ∈ (0, ε15 ]: + − + |h ε,0 (y)| ≤ |y|I (y ∉ [z− ε,0 , z ε,0 ]) + (|y | + δ ε,0) I ( y ∈ [ z ε,0 , z ε,0 ])

≤ (|y| + δ ε,0) .

(7.68)

Relation (7.44) and the condition M2 imply that the following relation holds for every ε ∈ (0, ε0 ] and β ≥ 0: Ee β| Y ε,0 | = Ee β| h ε,0(Y0,0 )| .

(7.69)

Using relations (7.68) and (7.69) we obtain the following inequality, for ε ∈ (0, ε15 ]: (7.70) Ee β| Y ε,0 | = Ee β| h ε,0 (Y0,0 )| ≤ e βδ ε,0 Ee β| Y0,0 | . Since, by the condition N1 , e βδ ε,0 → 1 as ε → 0, the condition D10 [β ] and inequal­ ity (7.70) imply the following relation: lim Ee β| Y ε,0 | ≤ lim e βδ ε,0 Ee β| Y0,0 | < K51 . ε →0

ε →0

(7.71)

Also, relation (7.44) implies that P{Y ε,0 ∈ Zε,0} = P{h ε,0 (Y0,0) ∈ Zε,0} = 1, for every ε ∈ (0, ε0 ]. Thus, the one-dimensional variant of the condition D7 [β¯ ] (its one-dimensional variant for the model without index component and with the sets Zε,0 given by relation (7.51)) holds. Let us take an arbitrary ϱ > 0. Using the condition N1 and inequality (7.55), we get the following relation: lim P{|h ε,0 (Y0,0 ) − Y0,0| ≥ ϱ } #  + ≤ lim P{Y 0,0 ≤ z− ε,0 } + I ( δ ε,0 ≥ ϱ ) + P{Y 0,0 > z ε,0 } = 0 .

0< ε → 0

0< ε → 0

(7.72)

262 | 7 Space-skeleton reward approximations Relation (7.72) implies the following relation: P ε (·) = P{h ε,0(Y0,0 ) ∈ ·} ⇒ P0 (·) = P{Y0,0 ∈ ·} as

ε → 0.

(7.73)

Relation (7.73) and the condition K8 imply that the condition K1 (its univariate  variant) holds with the sets Z0 = Y0 × {x0 } and Z 0 = Y0 × {x 0 }. It follows from the above remarks that all conditions of Theorem 5.2.4 hold, and, therefore, Theorem 7.2.2 is a corollary of Theorem 5.2.4.

7.3 Multivariate Markov LPP with bounded characteristics In this section, we present space-skeleton approximation models for discrete time multivariate modulated Markov log-price processes with bounded characteristics, de­ scribe the corresponding backward algorithms for computing reward functions and formulate convergence theorems for log-reward functions and optimal expected re­ wards for approximating space-skeleton Markov log-price processes.

7.3.1 Convergence of space-skeleton approximations for rewards functions for multivariate modulated Markov log-price processes ε,n , X ε,n ) is, for Let us now assume that the modulated log-price process  Z ε,n = (Y every ε ∈ [0, ε0 ], an inhomogeneous Markov chain with a phase space Z = Rk × X, an initial distribution P ε (A), and one-step transition probabilities P ε,n ( z , A ). We assume that the phase space of the index component X is a Polish metric space, i.e. complete, separable, metric space, with a metric dX (x , x ). We would like to approximate the log-price process  Z0,n by the log-price pro­ cesses  Z ε,n . We additionally assume that the log-price process  Z0,n (for ε = 0) can  be a nonatomic Markov chain, while the log-price processes Z ε,n (for ε ∈ (0, ε0 ]) are atomic Markov chains, i.e. their transition probabilities P ε,n ( z , A) and initial distri­ butions P ε (A) are distributions concentrated at finite sets of points. We also assume that the pay-off function g(n, ey , x) does not depend on the pa­ rameter ε. In general, space-skeleton approximations can be constructed in the following way. + Let us choose nonempty sets Aε,n,l ∈ BZ , l = m− ε,n , . . . , m ε,n , n = 0, 1, . . . , such that A ε,n,l ∩ A ε,n,l = ∅, l ≠ l ;

m+

Z = ∪l=ε,nm−ε,n A ε,n,l ,

n = 0, 1, . . . .

(7.74)

7.3 Multivariate Markov LPP with bounded characteristics

|

263

Let us also choose skeleton points, + l = m− ε,n , . . . , m ε,n ,

 z ε,n,l ∈ Aε,n,l ,

n = 0, 1, . . . .

(7.75)

Let us assume that the following fitting condition holds: ε,n =  z ε,n,l/ Z ε,n−1 =  z} = L3 : For every ε ∈ (0, ε0 ], transition probabilities P{Z + p ε,l ( z , n) = P0,n ( z ε,n−1,l, Aε,n,l),  z ∈ Aε,n−1,l , l = m− , . . . , m , ε,n −1 ε,n −1 l = − + m ε,n , . . . , m ε,n , n = 1, 2, . . . . If we are also interested to fit initial distributions, the following fitting condition should be assumed: ε,0 =  z ε,0,l} = p ε,l = P0,0 (Aε,0,l), M3 : For every ε ∈ (0, ε0 ], initial probabilities P{Z − + l = m ε,0 , . . . , m ε,0 . The most natural variant of the above space-skeleton approximation model is the fol­ lowing. Let us choose δ ε,n,i > 0, λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . . and integer num­ + bers m− ε,n,j ≤ m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . . + First, the intervals Iε,n,i,l should be constructed for l = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . , ⎧ 1 ⎪ ⎪ , δ ε,n,i(m− + )] + λ ε,n,i if l = m− ε,n,i , ⎪(−∞ ⎨ # #  ε,n,i 2# ! 1 1 − (7.76) Iε,n,i,l = δ l − 2 , δ ε,n,i l + 2 + λ ε,n,i if m ε,n,i < l < m+ ε,n,i , ⎪ # ε,n,i  ⎪ ⎪ 1 + + ⎩ δ (m − ), ∞ + λ if l = m . ε,n,i

ε,n,i

2

ε,n,i

ε,n,i

+ Then cubes Iε,n,l1,...,l k = Iε,n,1,l1 × · · · × I ε,n,k,l k , l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . should be defined. By the definition, the points y ε,n,i,li = l i δ ε,n,i + λ ε,n,i ∈ Iε,n,i,li and, thus, the vector + points (y ε,n,1,l1 , . . . , y ε,n,k,l k ) ∈ Iε,n,l1,...,l k , for l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . . + Second, nonempty sets Jε,n,l ⊂ BX , l = m− ε,n,0 , . . . , m ε,n,0, n = 0, 1, . . . , such that   (a) Jε,n,l ∩ Jε,n,l = ∅, l ≠ l , n = 0, 1, . . . ; (b) X = ∪m−ε,n,0 ≤l≤m+ε,n,0 Jε,n,l, n = 0, 1, . . . , should be constructed. Recall that one of our model assumption is that X is a Polish space, i.e. a com­ plete, separable, metric space. In this case, it is natural to assume that there ex­ ist “large” sets K ε,n, n = 0, 1, . . . , and “small” nonempty sets K ε,n,l ⊆ BX , l = +   m− ε,n,0, . . . , m ε,n,0, n = 0, 1, . . . , such that (c) K ε,n,l  ∩ K ε,n,l = ∅ , l ≠ l , n = 0, 1, . . . ; (d) ∪m−ε,n,0 ≤l≤m+ε,n,0 K ε,n,l = K ε,n, n = 0, 1, . . . . The exact sense of the epithets “large” and “small” used above is clarified in the condition N2 formulated below. Then, the sets Jε,n,l can be defined in the following way, for n = 0, 1, . . . : ⎧ + ⎨K ε,n,l if m− ε,n,0 ≤ l < m ε,n,0 , Jε,n,l = (7.77) + ⎩K + ∪ K ε,n if l = m .

ε,n,m ε,n,0

ε,n,0

264 | 7 Space-skeleton reward approximations + Finally, skeleton points x ε,n,l ∈ Jε,n,l, l = m− ε,n,0 , . . . , m ε,n,0, n = 0, 1, . . . , should be chosen. The particular important case is, where the space X = {1, . . . , m} is a finite set and metric dX (x, y) = I (x ≠ y). In this case, the standard choice is to take sets K ε,n = X, + n = 0, 1, . . . ; one-point sets Jε,n,l = K ε,n,l = {l}, 1 = m− ε,n,0 ≤ l ≤ m ε,n,0 = m, n = 0, 1, . . . ; and points x ε,n,l = l, l = 1, . . . , m, n = 0, . . . . If the space X = {1, 2, . . . } is a countable set and again metric dX (x, y) = I (x ≠ y), the standard choice is to take sets K ε,n = {l : l ≤ m ε }, n = 0, 1, . . . , and one-point + sets K ε,n,l = {l}, 1 = m− ε,n,0 ≤ l ≤ m ε,n,0 = m ε , n = 0, 1, . . . . In this case, the sets − + Jε,n,l = {l}, 1 = m ε,n,0 ≤ l < m ε,n,0 = m ε , n = 0, 1, . . . , while Jε,n,m ε = {l : l ≥ m ε }, and points x ε,n,l = l, l = 1, . . . , m ε , n = 0, 1, . . . . Another important case is, where X = Rp . In this case one can always choose as + K ε,n and K ε,n,l, m− ε,n,0 ≤ l ≤ m ε,n,0 cubes satisfying the conditions imposed above on these sets. Third, the sets Aε,n,¯l and skeleton points,  z ε,n,¯l ∈ Aε,n,¯l can be defined for ¯l = − + (l0 , l1 , . . . , l k ), l j = m ε,n,j, . . . , m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . , in the following way:

A ε,n,¯l = Iε,n,l1,...,l k × Jε,n,l0 ,

(7.78)

 z ε,n,¯l = ( y ε,n,¯l, x ε,n,¯l) = ((y ε,n,1,l1 , . . . , y ε,n,k,l k ), x ε,n,l0) .

(7.79)

and z ε,n,¯l : ¯l ∈ Lε,n }, where sets In this case, the corresponding skeleton sets Fε,n = { of vector indices Lε,n are defined in the following way, for n = 0, 1, . . . : + Lε,n = {¯l = (l0 , . . . , l k ), l j = m− ε,n,j , . . . , m ε,n,j ,

j = 0, . . . , k } .

(7.80)

In this case, the fitting condition L3 and M3 takes the following form: ε,n =  L4 : For every ε ∈ (0, ε0 ], transition probabilities P{Z z ε,n,¯l/ Z ε,n−1 =  z} = p ε,¯l ( z , n) = P0,n ( z ε,n−1,¯l, Aε,n,¯l),  z ∈ Aε,n−1,¯l , ¯l = (l0 , . . . , lk ) ∈ Lε,n−1, ¯l = (l0 , . . . , l k ) ∈ Lε,n, n = 1, 2, . . . . and ε,0 =  z ε,0,¯l} = p ε,¯l = P0,0 (Aε,0,¯l), ¯l = M4 : For every ε ∈ (0, ε0 ], initial probabilities P{Z (l0 , . . . , l k ) ∈ Lε,0 . z ε,n,l The difference in notations between the sets Aε,n,l and Aε,n,¯l and between points  and  z ε,n,¯l can be removed by a natural renumeration of indices, namely (m− , ε,n,0 . . . , − − − + − − m ε,n,k) ↔ m ε,n , . . . , (m ε,n,0 + 1, . . . , m ε,n,k) ↔ m ε,n + 1, . . . , (m ε,n,0, . . . , m+ ε,n,k) ↔ m+ , where ε,n − m+ ε,n − m ε,n + 1 =

k 

− (m+ ε,n,j − m ε,n,j + 1) ,

n = 0, 1, . . . .

j =0

The simplest variant is to choose integers ±m± ε,n ≥ 0, n = 0, 1, . . . .

(7.81)

7.3 Multivariate Markov LPP with bounded characteristics

|

265

Let us define the skeleton functions, h ε,n,i(y), y ∈ R1 , for ε ∈ (0, ε0 ] and i = 1, . . . , k, n = 0, 1, . . . , ⎧ 1 − ⎪ δ ε,n,i m− ⎪ ε,n,i + λ ε,n,i if y ≤ δ ε,n,j( m ε,n,i + 2 ) + λ ε,n,i , ⎪ ⎪ ⎪ ⎪ ⎨δ ε,n,i l + λ ε,n,i if δ ε,n,i(l − 21 ) + λ ε,n,i < y ≤ δ ε,n,i(l + 21 ) + λ ε,n,i , h ε,n,i(y) = ⎪ + ⎪ m− ⎪ ε,n,i < l < m ε,n,i , ⎪ ⎪ ⎪ 1 ⎩δ + + ε,n,i m ε,n,i + λ ε,n,i if y > δ ε,n,i( m ε,n,i − 2 ) + λ ε,n,i . (7.82) and the skeleton functions h ε,n,0(x), x ∈ X, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , h ε,n,0(x) = {x ε,n,l

if

+ x ∈ Jε,n,l, m− ε,n,0 ≤ l ≤ m ε,n,0 .

(7.83)

ˆ ε,n ( Finally, let us define the vector skeleton functions h y ),  y = (y1 , . . . , y k ) ∈ Y, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , ˆ ε,n ( h y) = (h ε,n,1(y1 ), . . . , h ε,n,k(y k )) .

(7.84)

and the vector skeleton functions h¯ ε,n ( z ),  z = ( y , x) ∈ Z, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , ˆ ε,n ( h¯ ε,n ( z) = (h y), h ε,n,0(x)) . (7.85) The transition probabilities P ε,n ( z , A ) = P{  Z ε,n ∈ A/ Z ε,n−1 =  z },  z ∈ Z, n = 1, 2, . . . , take the following form, for every ε ∈ (0, ε0 ]:

P ε,n ( z , A) = P0,n (h¯ ε,n−1( z), Aε,n,¯l)  z ε,n,¯l ∈ A

¯ ε,n ( = P{ h Z0,n ) ∈ A/ Z0,n−1 = h¯ ε,n−1( z)} ,

A ∈ BZ ,

(7.86)

ε,0 ∈ A} is concerned, it takes the As far as the initial distribution P ε,0(A) = P{Z following form, for every ε ∈ (0, ε0 ]:

P ε,0(A) = P0,0 (Aε,0,¯l) = P{h¯ ε,0 ( Z0,0) ∈ A} , A ∈ BZ . (7.87)  z ε,0,¯l ∈ A

z , A), defined in relation (7.86), is a measurable function in  z The quantity P ε,n ( and a probability measure as a function of A and, thus, it can serve as transition prob­ abilities for a Markov chain. Also the quantity P ε,0 (A), defined in relation (7.87), is a probability measure and, thus, it can serve as an initial distribution for a Markov chain. The role of approximating the log-price process  Z ε,n can play any Markov chain with the phase space Z, the initial distribution P ε,0(A) defined in relation (7.87), and the transition probabilities P ε,n ( z , A) defined in relation (7.86). z , A) and the initial distribution Obviously, the transition probabilities P ε,n ( P ε,0(A) of the Markov chain  Z ε,n satisfy the conditions L4 and M4 and, in sequel, relations (7.86) and (7.87).

266 | 7 Space-skeleton reward approximations As in the one-dimensional case, we can realize the strong fitting instead of the distributional one. Let us assume that the Markov chain  Z0,n is given in the dynamic form ⎧ ⎨ Z0,n = A0,n ( Z0,n−1 , U0,n ) , n = 1, 2, . . . , (7.88) ⎩ Z = A (U ) , 0,0

0,0

0,0

where (a) U0,n , n = 0, 1, . . . , are independent random variables taking values in a measurable space U and with distributions G n (u ) = P{U0,n ∈ A}, n = 0, 1, . . . , (b) A0,n ( z , u ), n = 0, 1, . . . , are the measurable functions acting from Z × U to Z, and (c) A0,0 (u ) is a measurable function acting from U to Z. Since Z is a Polish metric space, the above construction can be realized with the space U = [0, 1] and random variables U0,n , n = 0, 1, . . . , uniformly distributed in this interval. Z ε,n can be defined in the dynamic form The Markov chain  ⎧ ⎨ Z ε,n = h¯ ε,n (A0,n (h¯ ε,n−1( Z ε,n−1), U0,n )) , n = 1, 2, . . . , (7.89) ⎩ Z = h¯ (A (U )) . ε,0

ε,0

0

0,0

The following lemma is a variant of Lemma 7.1.3. Z ε,n is a space-skele­ Lemma 7.3.1. Let, for every ε ∈ (0, ε0 ], the log-price process  ton Markov chain with transition probabilities and initial distribution defined, respec­ y , x) and tively, in relations (7.86) and (7.87). Then, the log-reward functions ϕ ε,n ( ϕ n+r ( y ε,n,¯l, x ε,n,¯l), for points ( y ε,n,¯l, x ε,n,¯l), ¯l ∈ Lε,n+r, r = 1, . . . , N − n, given by formulas (7.79), are, for every  z = ( y , x) ∈ Aε,n,¯l , ¯l ∈ Lε,n , n = 0, . . . , N, the unique solution for the finite recurrence system of linear equations, ⎧ ⎪ ϕ ε,N ( y ε,N,¯l , x ε,N,¯l) = g(N, ey ε,N,l , x ε,N,¯l), ¯l ∈ Lε,N , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r( y ε,n+r,¯l, x ε,n+r,¯l) = max g(n + r, ey ε,n+r,¯l , x ε,n+r,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ϕ ε,n+r+1( y ε,n+r+1,¯l , x ε,n+r+1,¯l) ⎪ ⎪ ⎪ ⎪ ¯l  ∈Lε,n+r+1 ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z ε,n+r,¯l, Aε,n+r+1,¯l) , × P0,n+r+1( ⎪ ⎪ ⎨ (7.90) ¯l ∈ Lε,n+r , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n ( y , x) = max g(n, ey , x) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ϕ ε,n+1( y ε,n+1,¯l , x ε,n+1,¯l) ⎪ ⎪ ⎪ ⎪ ¯l  ∈Lε,n+1 ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ P ( z  , A  ) . × ¯ ¯ ⎩ 0,n +1 ε,n, l ε,n +1, l

7.3 Multivariate Markov LPP with bounded characteristics

| 267

7.3.2 Convergence of space-skeleton approximations for logrewards functions for multivariate modulated Markov log-price processes The backward algorithm for computing of the reward functions ϕ ε,n ( y , x) given in Lemma 7.3.1 should be supplemented by the proposition that would show that the logy , x) converge to the log-reward functions ϕ0,n ( y , x) as ε → 0, rewards functions ϕ ε,n ( and, thus, they can serve as approximations for the log-reward functions ϕ0,n ( y , x ). Let us introduce special shorten notations for the maximal and the minimal skele­ ton points, for j = 1, . . . , k, n = 0, . . . , N, and ε ∈ (0, ε0 ], ± z± ε,n,j = δ ε,n,j m ε,n,j + λ ε,n,j .

(7.91)

We impose the following on the parameters of the space-skeleton model defined in relations (7.74)–(7.87): N2 : (a) δ ε,n,j → 0 as ε → 0, for j = 1, . . . , k, n = 0, 1, . . . , N ; (b) ±z± ε,n,j → ∞ as ε → 0, for j = 1, . . . , k, n = 0, 1, . . . , N ; (c) ±z± ε,n,j , n = 0, 1, . . . , N are nondecreasing sequences, for every j = 1, . . . , k and ε ∈ (0, ε0 ] ; (d) for any x ∈ X and d > 0, there exists ε x,d ∈ (0, ε0 ] such that the ball R d (x) ⊆ K ε,n , n = 0, . . . , N, for ε ∈ (0, ε x,d ] ; (e) the sets K ε,n,l have diameters d ε,n,l = supx ,x∈Kε,n,l dX (x , x ) such that d ε,n = maxm−ε,n,0 ≤l≤m+ε,n,0 d ε,n,l → 0 as ε → 0, n = 0, . . . , N . − Note that the condition N2 implies that δ ε,n,j(m+ ε,n,j − m ε,n,j) → ∞ as ε → 0, for j = + − 1, . . . , k, n = 0, 1, . . . , N, and m ε,n,j−m ε,n,j →∞ as ε → 0, for j = 1, . . . , k, n = 0, 1, . . . , N. − Note, however, that the condition N2 does not require that m+ ε,n,0 − m ε,n,0 → ∞ as ε → 0. + Note also that the sets K ε,n and K ε,n,l, m− ε,n,0 ≤ l ≤ m ε,n,0 satisfying the conditions N2 (d) and (e) can be always constructed for the most important cases, where X is a discrete space or a Euclidian space or a product of such spaces. The role of the sets K ε,n,l can be played, respectively, by one-point sets, cubes, or products of one-point sets and cubes. We impose on the price process Y0,n the following first-type condition of expo­ nential moment compactness, which is a variant of the condition C1 [β¯ ], and should be assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: C12 [β¯ ]: Δ β i (Y0,· ,i , N ) < K52,i , i = 1, . . . , k, for some 1 < K52,i < ∞, i = 1, . . . , k .

Let us recall the second-type modulus of exponential moment compactness for the components of the log-price process, ± β( Y n+1,i − Y n,i ) . Ξ± z ,n e β ( Y · ,i , N ) = max sup E

0≤ n ≤ N − 1  z∈Z

(7.92)

268 | 7 Space-skeleton reward approximations As follows from Lemma 4.1.8, the following condition implies the condition C12 [β¯ ] to hold: E16 [β¯ ]: Ξ± β i ( Y 0, · ,i , N ) < K 53,i , i = 1, . . . , k, for some 1 < K 53,i < ∞, i = 1, . . . , k . Let us introduce sequences of the sets Zε = Zε,0 , . . . , Zε,N , where the sets Zε,n are defined, for every n = 0, . . . , N, by the following relation,: ⎧ ⎨[z− , z+ ] × · · · × [z− , z+ ] × X for ε ∈ (0, ε0 ] , ε,n,1 ε,n,1 ε,n,k ε,n,k Zε,n = (7.93) ⎩Z = R × X for ε = 0 . k Note that the condition N2 (c) implies the following imbedding relation for the sets Zε,n , for every ε ∈ (0, ε0 ]: Zε,0 ⊆ · · · ⊆ Zε,N . (7.94) Let us also recall the modified first-type modulus of exponential moment com­ pactness, defined for every ε ∈ [0, ε0 ] and i = 1, . . . , k, β ≥ 0, Δ β,Zε (Y ε,· ,i , N ) = max

sup Ez ,n e β| Y ε,n+1,i −Y ε,n,i | .

0≤ n ≤ N − 1  z∈Z

(7.95)

ε,n

Note, that, by the definition, the modified first-type modulus of exponential mo­ ment compactness Δ β,Z0 (Y0,· ,i , N ) coincides with the standard first-type modulus of exponential moment compactness Δ β (Y0,· ,i , N ). It is worth to note that the conditions L4 , N2 , and C12 [β¯ ] imply that the modi­ fied first-type condition of exponential moment compactness C6 [β¯ ] (with the modi­ fied moduli of exponential moment compactness Δ β i ,Zε (Y ε,· .i , N ), i = 1, . . . , k) holds, while the standard condition of exponential moment compactness C6 [β¯ ] (with the standard moduli of exponential moment compactness Δ β i (Y ε,· .i , N ), i = 1, . . . , k) does not. Let us recall the condition B1 [𝛾¯] for the pay-off function g(n, ey , x), which is as­ sumed to hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative compo­ nents. The condition I1 should be replaced by the following condition: I7 : There exist the sets Zn ∈ BZ , n = 0, . . . , N such that the function g(n, ey , x) is continuous in points  z = ( y , x) ∈ Zn , for n = 0, . . . , N . zε = The condition I7 means that g(n, ey ε , x ε ) → g(n, ey0 , x0 ) as ε → 0 for any   yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn as ε → 0, for n = 0, . . . , N. Thus, the condition I1 ( holds with the sets Zn , n = 0, . . . , N. The condition J1 should be replaced by the following condition: J10 : There exist the sets Z n ∈ BZ , n = 0, . . . , N such that: (a) P0,n ( z ε , ·) ⇒ P0,n ( z0 , ·) as ε → 0, for any  zε →  z0 ∈ Z n −1 as ε → 0, and n = 1, . . . , N;  (b) P0,n ( z0 , Zn ∩ Z z0 ∈ Zn−1 ∩ Z n −1 , n = 1, . . . , N, where Z n , n ) = 1, for every  n = 0, . . . , N are sets introduced in the condition I7 .

7.3 Multivariate Markov LPP with bounded characteristics

| 269

z , ·) are weakly con­ The condition J10 (a) means that the transition probabilities P0,n (  z ∈ Zn−1 for every n = 1, . . . , N. tinuous at points    Note also that the condition J10 (b) automatically holds if the sets Z n , Z n = ∅, n = 1, . . . , N, or these sets are finite or countable and measures P0,n ( z0 , ·),  z0 ∈ Zn−1 ∩    Zn−1 have no atoms in points of the sets Z n , Z n , for n = 1, . . . , N, as well as in the case, where measures P0,n (y0 , ·),  z0 ∈ Zn−1 ∩ Z n −1 , n = 1, . . . , N are absolutely continuous   with respect to some σ-finite measure P(A) on the σ-algebra BZ and P(Zn ), P(Zn ) = 0, for n = 1, . . . , N. We shall prove below that the conditions L4 , N2 , and J10 imply that the condition J1 holds with the sets Zn , Z n , n = 0, . . . , N. The following theorem is a variant of Theorem 5.2.2 for the space-skeleton approx­ imation model for multivariate modulated Markov log-price processes. Theorem 7.3.1. Let the multivariate modulated Markov log-price process  Z0,n and the corresponding space-skeleton approximation Markov processes  Z ε,n are defined as de­ scribed above. Also let the conditions B1 [𝛾¯] and C12 [β¯ ] hold with the vector parame­ ters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions L4 , N2 , I7 , and J10 hold. Then, z ε = ( yε , xε ) → for every n = 0, 1, . . . , N, the following relation takes place for any    z0 = ( y0 , x0 ) ∈ Zn ∩ Zn ,  y ε , x ε ) → ϕ0,n ( y0 , x0 ) ϕ ε,n (

as

ε → 0.

(7.96)

Proof. We are going to check that all conditions of Theorem 5.2.2 hold and then to apply this theorem. It follows from the definition of functions h ε,n,i(y), i = 1, . . . , k that the following inequality holds for ε ∈ (0, ε0 ] and i = 1, . . . , k, n = 0, . . . , N, y ∈ R1 , − |h ε,n,i(y) − y| ≤ (z− ε,n,i − y ) I ( y ≤ z ε,n,i) + + + + δ ε,n I (z− ε,n,i < y ≤ z ε,n,i) + ( y − z ε,n,i) I ( y > z ε,n,i) .

(7.97)

Relation (7.86) and the condition L4 imply that the following inequality holds for every ε ∈ (0, ε0 ] and  z = ( y , x) = ((y1 , . . . , y k ), x) ∈ Z, i = 1, . . . , k, n = 0, . . . , N −1, and β ≥ 0, Ez ,n e β| Y ε,n+1,i −Y ε,n,i | = Ez ,n e β| Y ε,n+1,i −y| = Eh¯ ε,n (z),n e β| h ε,n+1,i (Y0,n+1,i )−y| .

(7.98)

270 | 7 Space-skeleton reward approximations Using relations (7.94), (7.97), and (7.98) we get the following inequalities, for every ε ∈ (0, ε0 ] and i = 1, . . . , k, n = 0, . . . , N − 1, and β ≥ 0, sup  z=(( y 1 ,...,y k ) , x )∈Zε,n

Ez ,n e β| Y ε,n+1,i −Y ε,n,i | = sup Eh¯ ε,n (z),n e β| h ε,n+1,i (Y0,n+1,i )−y i |  z∈Zε,n

≤ sup Eh¯ ε,n (z),n e β(y i −Y0,n+1,i ) I (Y 0,n+1,i ≤ z− ε,n +1,i)  z∈Zε,n

+ e βδ ε,n+1,i sup Eh¯ ε,n (z) e β| Y0,n+1,i −y i |  z∈Zε,n

×

I ( z− ε,n +1,i

< Y 0,n+1,i ≤ z+ ε,n +1,i)

+ sup Eh¯ ε,n (z),n e β(Y0,n+1,i −y i ) I (Y 0,n+1,i > z+ ε,n +1,i)  z∈Zε,n

≤e

βδ ε,n+1,i

sup Eh¯ ε,n (z),n e β i | Y0,n+1,i −y i |

 z∈Zε,n

≤ e βδ ε,n+1,i e βδ ε,n,i sup Eh¯ ε,n (z),n e β| Y0,n+1 −h ε,n,i (y i )|  z∈Zε,n

≤e

β( δ ε,n+1,i + δ ε,n,i )

sup Ez ,n e β| Y0,n+1,i −y i | .

(7.99)

 z∈Z

This relation implies that the following inequality holds, for every ε ∈ (0, ε0 ] and i = 1, . . . , k: Δ β,Zε (Y ε,· ,i , N ) ≤ max e β(δ ε,n+1,i +δ ε,n,i ) Δ β (Y0,· ,i , N ) . 0≤ n ≤ N − 1

(7.100)

By the condition N2 , max0≤n≤N −1 e β(δ ε,n+1,i +δ ε,n,i ) → 1 as ε → 0, for i = 1, . . . , k and β ≥ 0. This relation, inequality (7.100) and the condition C12 [β¯ ] imply the following relation, for i = 1, . . . , k: lim Δ β i ,Zε (Y ε,· ,i , N ) ≤ Δ β i (Y0,· ,i , N ) < K52,i . ε →0

(7.101)

Therefore, the condition C6 [β¯ ] (with sets Zε,n given by relation (7.93)) holds. The condition N2 obviously implies that the condition H1 (with the sets Hε,n = Zε,n , n = 0, . . . , N) holds. As was mentioned above, the condition I6 implies that the condition I1 holds. Let us choose an arbitrary 1 ≤ n ≤ N and  z ε = ( y ε , x ε ) = ((y ε,1 , . . . , y ε,k ), x ε ) →  z0 = ( y0 , x0 ) = ((y0,1 , . . . , y0,k ), x0 ) ∈ Zn−1 ∩ Z as ε → 0. n −1 Inequality (7.97) and the condition N2 imply that, for i = 1, . . . , k, n = 1, . . . , N, h ε,n−1,i(y ε,i ) → y0,i

as

ε → 0.

(7.102)

It follows from the definition of functions h ε,n,0(x) that the following inequality also takes place ε ∈ (0, ε0 ] and n = 0, . . . , N, x ∈ X: dX (h ε,n,0(x) , x) ≤ dX (x ε,n,m+ε,n,0 , x)I (x ∈ K ε,n) + d ε,n I (x ∈ K ε,n) .

(7.103)

Inequality (7.103) and the condition N2 imply that, for n = 1, . . . , N, h ε,n−1,0(x ε ) → x0

as

ε → 0.

(7.104)

7.3 Multivariate Markov LPP with bounded characteristics

| 271

Relations (7.102) and (7.104) obviously imply that, for n = 1, . . . , N, zε ) →  z0 h¯ ε,n−1(

as

ε → 0.

(7.105)

Let us take arbitrary κ, ϱ > 0. One can find T κ > 0 such that points ±T κ are points of continuity for the distribution functions Pz0 ,n−1{Y0,n,i ≤ u }, i = 1, . . . , k, n = 1, . . . , N and, for i = 1, . . . , k, n = 1, . . . , N, Pz0 ,n−1{Y0,n,i ≤ −T κ } + Pz0 ,n−1{Y0,n,i ≥ T κ } ≤ κ .

(7.106)

Using the above remarks, the conditions N2 and J10 (a) and relations (7.97), (7.102), (7.104), and (7.106), we get the following relation, for i = 1, . . . , k, n = 1, . . . , N: lim Ph¯ ε,n−1 (z ε ),n−1{|h ε,n,i(Y0,n,i) − Y0,n,i| ≥ ϱ } # ≤ lim Ph¯ ε,n−1 (z ε ),n−1 {Y 0,n,i ≤ z− ε,n,i}

0< ε → 0

0< ε → 0

 +I (δ ε,n,i ≥ ϱ ) + Ph¯ ε,n−1 (z ε ),n−1 {Y 0,n,i > z+ ε,n,i} # ≤ lim Ph¯ ε,n−1 (z ε ),n−1 {Y 0,n,i ≤ −T κ } 0< ε → 0  +Ph¯ ε,n−1 (z ε ),n−1{Y 0,n,i > T κ }

= Pz0 ,n−1{Y 0,n,i ≤ −T κ } + Pz0 ,n−1{Y 0,n,i ≥ T κ } ≤ κ .

(7.107)

Since an arbitrary choice of κ, ϱ > 0, relation (7.107) implies that for any ϱ > 0 and i = 1, . . . , k, n = 1, . . . , N, lim Ph¯ ε,n−1 (z ε ),n−1{|h ε,n,i(Y0,n,i) − Y0,n,i| ≥ ϱ } = 0 .

0< ε → 0

(7.108)

Let us choose an arbitrary sequence ε r ∈ (0, ε0 ], r = 1, 2, . . . , such the ε r → 0 as r → ∞. The condition J10 and relation (7.105) imply that, for every n = 1, . . . , N, probabil­ ity measures, Ph¯ εr ,n−1 (z εr ),n−1{X0,n ∈ ·} ⇒ Pz0 ,n−1{X0,n ∈ ·}

as

r → ∞.

(7.109)

Let us again take an arbitrary κ > 0. Relation (7.109) implies by the Prokhorov theorem that there exists a compact C κ ⊆ X such that, for n = 1, . . . , N, max Ph¯ εr ,n−1 (z εr ),n−1{X0,n ∈ C κ } ≤ κ. r ≥1

(7.110)

Since X is a Polish metric space and C κ is a compact, there exists a ball R κ = R d κ (x κ ) such that C κ ⊆ R κ . By conditions N2 (d) there exists a ε x κ ,d κ ∈ (0, ε0 ] such that R κ ⊆ K ε,n , n = 0, . . . , N, for ε ∈ (0, ε x κ ,d κ ]. Thus, lim Ph¯ εr ,n−1 (z εr ),n−1{X0,n ∈ Kε r ,n }

r →∞

≤ lim Ph¯ εr ,n−1 (z εr ),n−1{X0,n ∈ R κ } r →∞

≤ lim Ph¯ εr ,n−1 (z εr ),n−1{X0,n ∈ C κ } ≤ κ . r →∞

(7.111)

272 | 7 Space-skeleton reward approximations Let us again take an arbitrary ϱ > 0. Using the condition N2 (e) and relations (7.103) and (7.111), we get the following relation, for n = 1, . . . , N: lim Ph¯ εr ,n−1 (z εr ),n−1{dX (h ε r ,n,0(X0,n ), X0,n ) ≥ ϱ }  # ≤ lim Ph¯ εr ,n−1 (z εr ),n−1{X0,n ∈ K ε r ,n } + I (d ε r ,n ≥ ϱ ) ≤ κ .

r →∞

r →∞

(7.112)

Since an arbitrary choice of κ > 0 and an arbitrary choice of sequence 0 < ε r → 0 as r → ∞, relation (7.112) implies that, for ϱ > 0 and n = 1, . . . , N, lim Ph¯ ε,n−1 (z ε ),n−1{dX (h ε,n,0(X0,n ), X0,n ) ≥ ϱ } = 0 .

0< ε → 0

(7.113)

Relations (7.108) and (7.113) imply that, for every n = 1, . . . , N, probability mea­ 0,n ∈ ·} have the same weak Z0,n ) ∈ ·} and Ph¯ ε,n−1 (z ε ),n−1{Z sures Ph¯ ε,n−1 (z ε ),n−1{h¯ ε,n ( limits as 0 < ε → 0. This proposition, relations (7.86), (7.105), and the condition J10 (a) imply that the zε →  z0 ∈ Zn−1 ∩ Z following relation of weak convergence holds, for any  n −1 as 0 < ε → 0, for every 1 ≤ n ≤ N, P ε,n ( z ε , ·) = Ph¯ ε,n−1 (z ε ),n−1{h¯ ε,n ( Z0,n ) ∈ ·} 0,n ∈ ·} ⇒ P0,n ( z0 , ·) = Pz0 ,n−1{Z

as

ε → 0.

(7.114)

Thus, the condition J1 (a) holds with the sets Zn , Z n , n = 0, . . . , N. Since, the conditions J10 (b) and J1 (b) coincide, we can conclude that the condition J1 holds in this case with the above sets Zn , Z n , n = 0, . . . , N. Therefore, all conditions of Theorem 5.2.2 hold, and, therefore, Theorem 7.2.3 is a corollary of Theorem 5.2.2.

7.3.3 Convergence of space-skeleton approximations for optimal expected rewards for multivariate modulated Markov log-price processes Let us now assume that we have found, using backward recurrence algorithm pre­ sented in Lemma 7.3.1, the values of log-reward functions ϕ ε,0( y ε,0,¯l, x ε,0,¯l), ¯l = (l0 , . . . , l k ) ∈ Lε,0 for the initial points,  z ε,0,¯l = ( y ε,0,¯l , x ε,0,¯l) = ((y ε,0,1,l1 , . . . , y ε,0,k,l k ), x ε,0,l0 ) = ((δ ε,0,1 l1 + λ ε,0,1 , . . . , δ ε,0,k l k + λ ε,0,k ), x ε,0,l0 ) . (ε)

(7.115)

Then, the optimal expected reward functional Φ ε = Φ(Mmax,N ) for the approxi­ mating log-price processes  Z ε,n can be found using the following formula:

Φε = ϕ ε,0( z ε,0,¯l)p ε,¯l , (7.116) ¯l∈Lε,0

where p ε,¯l = P0,0 (Aε,0,¯l) = P{h¯ ε,0 ( Z0,0) =  z ε,0,¯l}, ¯l = (l0 , . . . , l k ) ∈ Lε,0.

7.3 Multivariate Markov LPP with bounded characteristics

|

273

Let us formulate conditions that should be imposed on the log-price process  Z0,n and the pay-off function g(n, ey , x) in order the optimal expected rewards Φ ε = (ε) (0) Φ(Mmax,N ) would converge to the limiting optimal expected reward Φ0 = Φ(Mmax,N ) as ε → 0. We impose on the initial value of the log-price process  Z0,0 the following condi­ tion, which is a variant of the condition D1 [β¯ ], and should be assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: D11 [β¯ ]: Ee β i | Y0,0,i | < K54,i , i = 1, . . . , k, for some 1 < K54,i < ∞, i = 1, . . . , k . The condition K1 should be replaced by the following condition imposed on the initial distribution P0,0 (A):   K9 : P0,0 (Z0 ∩ Z 0 ) = 1, where Z0 and Z0 are the sets introduced, respectively, in the conditions I7 and J10 . The following theorem is a variant of Theorem 5.2.4 for the multivariate space-skeleton approximation model. Z0,n and the corresponding space spaceTheorem 7.3.2. Let a Markov log-price process  Z ε,n are defined as described above. Let also skeleton approximation Markov processes  the conditions B1 [𝛾¯], C12 [β¯ ], and D11 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions L4 , M4 , N2 , I7 , J10 , and K9 hold. Then, the following relation takes place: Φ ε → Φ0 as ε → 0 . (7.117) Proof. We are going to check that all conditions of Theorem 5.2.4 hold and to apply this theorem. In fact, we should only check that the condition D1 [β¯ ], with the sets Zε,0 defined in relation (7.93), and the condition K1 hold. Indeed, this check was made for other conditions of Theorem 5.2.4 in the proof of Theorem 7.2.3. By the condition N2 , there exists ε16 ∈ (0, ε0 ] such that ±z± ε,0,i ≥ 0, i = 1, . . . , k, for ε ∈ (0, ε16 ]. The following inequality holds for i = 1, . . . , k and ε ∈ (0, ε16 ]: + |h ε,0,i(y) | ≤ |y|I (y ∉ [z− ε,0,i , z ε,0,i ]) + + (|y| + δ ε,0,i)I (y ∈ [z− ε,0,i , z ε,0,i ]) ≤ (|y | + δ ε,0,i) .

(7.118)

Relation (7.87) and the condition M2 imply that the following relation holds for ε ∈ (0, ε0 ], i = 1, . . . , k, and β ≥ 0: Ee β| Y ε,0,i | = Ee β| h ε,0,i (Y0,0,i )| .

(7.119)

Using relations (7.118) and (7.119) we get the following inequality, for i = 1, . . . , k, ε ∈ (0, ε16 ], and β ≥ 0: Ee β| Y ε,0,i | = Ee β| h ε,0,i (Y0,0,i )| ≤ e βδ ε,0,i Ee β| Y0,0,i | .

(7.120)

274 | 7 Space-skeleton reward approximations Since, by the condition N2 , e βδ ε,0,i → 1 as ε → 0, for i = 1, . . . , k and β ≥ 0, the condition D11 [β¯ ] and inequality (7.120) imply that, for i = 1, . . . , k, lim Ee β i | Y ε,0,i | ≤ Ee β i | Y0,0,i | < K54,i .

(7.121)

ε →0

Also, relation (7.87) implies that, for every ε ∈ (0, ε0 ]. ¯ ε,0( ε,0 ∈ Zε,0} = P{h Z0,0 ) ∈ Zε,0} = 1 . P{ Z

(7.122)

Thus, the condition D7 [β¯ ] (with sets Zε,0 given by relation (7.93)) holds. Let us take an arbitrary ϱ > 0. Using the condition N2 and inequality (7.97), we get the following relations, for i = 1, . . . , k: lim P{|h ε,0,i(Y0,0,i ) − Y0,0,i| ≥ ϱ }

0< ε → 0

≤ lim

#

0< ε → 0

P{Y0,0,i ≤ z− ε,0,i }

 + I (δ ε,0,i ≥ ϱ ) + P{Y 0,0,i > z+ ε,0,i } = 0 .

(7.123)

Let us take arbitrary κ, ϱ > 0. There exists a compact C κ ⊆ X such that P{X0,0 ∈ C κ } ≤ κ. Since X is a Polish metric space and C κ is a compact, there exists a ball R κ = R d κ (x κ ) such that C κ ⊆ R κ . By the conditions N2 (d) there exists a ε x κ ,d κ ∈ (0, ε0 ] such that R κ ⊆ K ε,0, for ε ∈ (0, ε x κ ,d κ ]. The above remarks and relation (7.103) imply that, lim P{dX (h ε,0,0(X0,0 ), X0,0) ≥ ϱ }

0< ε → 0

≤ lim

0< ε → 0

 P{X0,0 ∈ K ε,0} + I (d ε,0 ≥ ϱ ) ≤ κ .

(7.124)

Due to an arbitrary choice of κ, ϱ > 0, relations (7.123) and 7.124) imply the follow­ ing relation: Z0,0 ) ∈ ·} ⇒ P0,0 (·) = P{ Z0,0 ∈ ·} P ε,0(·) = P{h¯ ε,0(

as

ε → 0.

(7.125)

Relation (7.125) and the condition K9 imply that the condition K1 holds with the sets Z0 and Z 0. It follows from the above remarks that all conditions of Theorem 5.2.4 hold, and, therefore, Theorem 7.1.4 is a corollary of Theorem 5.2.4.

7.4 LPP represented by multivariate modulated random walks |

275

7.4 LPP represented by multivariate modulated random walks In this section, we present space-skeleton approximation models for discrete time log-price processes represented by random walks. We describe the corresponding backward algorithms for computing log-reward functions and present convergence theorems, which give conditions of convergence for log-reward functions and optimal expected rewards for approximating space-skeleton approximation Markov log-price processes.

7.4.1 Space-skeleton approximations for log-price processes represented by univariate standard random walks Let us consider the model where, for every ε ∈ [0, ε0 ], the log-price process Y ε,n , n = 0, 1, . . . , is a one-dimensional random walk, Y ε,n = Y ε,n−1 + W ε,n ,

n = 1, 2, . . . ,

(7.126)

where (a) Y ε,0 is a real-valued random variable with distribution P ε,0(A), (b) W ε,n , n = 1, 2, . . . , is a sequence of independent real-valued random variables with distributions ˜ ε,n (A) for n = 1, 2, . . . , (c) the random variable Y ε,0 and the random sequence W ε,n , P n = 1, 2, . . . , are independent. We would like to approximate the log-price process Y0,n by the log-price processes Y ε,n. We additionally assume that distributions of jumps and an initial distribution for the random walk Y0,n (for ε = 0) can be nondiscrete, while distributions of jumps and an initial distribution for the random walk Y ε,n (for every ε ∈ (0, ε0 ]) should be discrete and finite, i.e. concentrated at finite sets of points. In this case, the model without index component is considered. In order to be able to interpret the model as modulated, one can always introduce a “virtual” index component X n with a one-point phase space X = {x0 }. Thus, we also assume that the pay-off function g(n, e y , x) = g(n, e y ) does not depend on argument x and is a real-valued Borel function of argument y ∈ R1 for every n = 0, 1, . . . , N. Let us choose the space-skeleton parameters δ ε > 0, λ ε,n ∈ R1 , n = 0, 1, . . . , and + integers m− ε,n ≤ m ε,n , n = 0, 1, . . . , as they have been defined in Subsection 7.1.4. + The intervals Iε,n,l should be constructed, for l = m− ε,n , . . . , m ε,n , n = 0, 1, . . . , using the following relation: ⎧ 1 ⎪ ⎪ (−∞, δ ε (m− if l = m− ε,n + 2 )] + λ ε,n ε,n , ⎪ ⎨ 1 1 (7.127) Iε,n,l = (δ ε (l − 2 ), δ ε (l + 2 )] + λ ε,n if m− < l < m+ ε,n ε,n , ⎪ ⎪ ⎪ ⎩(δ (m+ − 1 ), ∞) + λ if l = m+ . ε

ε,n

2

ε,n

ε,n

276 | 7 Space-skeleton reward approximations and the skeleton points y ε,n,l ∈ Iε,n,l should be defined according to the following formulas: y ε,n,l = lδ ε + λ ε,n ,

+ l = m− ε,n , . . . , m ε,n ,

n = 0, 1, . . . .

(7.128)

Let us now assume that the following fitting condition holds: ˜ ε,l (n) = P{W0,n ∈ Iε,n,l} L5 : For every ε ∈ (0, ε0 ], probabilities P{W ε,n = y ε,n,l} = p + ˜ 0,n (Iε,n,l), l = m− =P ε,n , . . . , m ε,n , n = 1, 2, . . . . If we also are interested to fit initial distributions, then the following fitting condition should also be assumed: M5 : For every ε ∈ (0, ε0 ], initial probabilities P{Y ε,0 = y ε,0,l} = p ε,l = P0,0 (Iε,0,l), + l = m− ε,0 , . . . , m ε,0 . Let us define, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , the skeleton functions, ⎧ 1 − ⎪ δ ε m− ⎪ ε,n + λ ε,n if y ≤ δ ε ( m ε,n + 2 ) + λ ε,n , ⎪ ⎪ ⎪ ⎪ 1 ⎨δ ε l + λ ε,n if δ ε (l − 2 ) + λ ε,n < y ≤ δ ε (l + 21 ) + λ ε,n , h ε,n (y) = ⎪ + ⎪ m− ⎪ ε,n < l < m ε,n , ⎪ ⎪ ⎪ 1 ⎩δ m + + λ + ε ε,n ε,n if y > δ ε ( m ε,n − 2 ) + λ ε,n .

(7.129)

˜ ε,n (A) = P{W ε,n ∈ A}, n = 1, . . . , take the In this case, the jump distributions P following form, for every ε ∈ (0, ε0 ]:

˜ 0,n (Iε,n,l) ˜ ε,n (A) = P P y ε,n,l ∈ A (7.130) = P{h ε,n (W0,n ) ∈ A} , A ∈ B1 . As far as the initial distribution P ε,0(A) = P{Y ε,0 ∈ A} is concerned, it takes the following form, for every ε ∈ (0, ε0 ]:

P0,0 (Iε,0,l) P ε,0(A) = y ε,0,l ∈ A (7.131) = P{h ε,n (Y 0,0) ∈ A} , A ∈ B1 . In order to realize the distributional fitting, we can use any standard random walk Y ε,n = Y ε,n−1 + W ε,n , n = 1, 2, . . . , defined by relation (7.126), with the jump distribu­ tions and the initial distribution defined, respectively, by relations (7.130) and (7.131). Any such random walk obviously will satisfy the fitting conditions L5 and M5 . Moreover, we can easily realize the strong fitting defining the random walk Y ε,n in the following dynamic form: Y ε,n = Y ε,n−1 + h ε,n (W0,n ) ,

n = 1, 2, . . . ,

Y ε,0 = h ε,0(Y0,0 ) .

(7.132)

In this case, relations (7.130) and (7.131) and, subsequently, the conditions L5 and M5 hold.

7.4 LPP represented by multivariate modulated random walks |

277

Let us denote y ε,n,n+r,l = δ ε l + λ ε,n,n+r ,

+ l = m− ε,n,n + r , . . . , m ε,n,n +r ,

(7.133)

where, for 0 ≤ n < n + r < ∞, m± ε,n,n + r = and λ ε,n,n+r =

n

+r

m± ε,l

(7.134)

λ ε,l .

(7.135)

l = n +1 n

+r l = n +1

If Y ε,n = y, for some y ∈ R1 , and n = 0, 1, . . . , then possible states for Y ε,n+r are + y + y ε,n,n+r,l, l = m− ε,n,n + r , . . . , m ε,n,n + r, r = 1, 2, . . . . Since the pay-off function g(n, e y ) does not depend on the index argument x, the reward functions ϕ ε,n (y), n = 0, 1, . . . , for the log-price processes Y ε,n are functions of y, for every ε ∈ [0, ε0 ]. The following lemma is a variant of Lemma 7.1.4. Lemma 7.4.1. Let, for every ε ∈ (0, ε0 ], the log-price process Y ε,n is a space-skeleton random walk with distributions of jumps and the initial distribution defined, respectively, in relations (7.130) and (7.131). Then, the log-reward functions ϕ ε,n (y) and ϕ ε,n+r(y + + y ε,n,n+r,l), for points y + y ε,n,n+r,l, l = m− ε,n,n +r , . . . , m ε,n,n +r, r = 1, . . . N − n given by formulas (7.128), are, for every y ∈ R1 , n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equations: ⎧ ϕ ε,N (y + y ε,n,N,l) = g(N, e y+y ε,n,N,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ l = m− ε,n,N , . . . , m ε,n,N , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r(y + y ε,n,n+r,l) = max g(n + r, e y+y ε,n,n+r,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m+ ⎪ ε,n+r+1 ⎪

⎪ ⎪ ⎪ ϕ ε,n+r+1(y + y ε,n,n+r,l + y ε,n+r+1,l) ⎪ ⎪ ⎪ ⎪  =m− ⎪ l ε,n+r+1 ⎪ ⎨ ' ⎪ ˜ ⎪ × P0,n+r+1(Iε,n+r+1,l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ r = N − n − 1, . . . , 1 , l = m− ⎪ ε,n,n +r , . . . , m ε,n,n +r , ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n (y) = max g(n, e y ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ m+ ε,n+1 ⎪

⎪ ⎪ ⎪ ˜ ⎪ ϕ ε,n+1(y + y ε,n+1,l)P0,n+1(Iε,n+1,l) . ⎪ ⎩ l =m− ε,n+1

(7.136)

278 | 7 Space-skeleton reward approximations

7.4.2 Convergence of space-skeleton approximations for log-price processes represented by univariate standard random walks The above backward algorithm for computing the log-reward functions ϕ ε,n(y) should be supplemented by the proposition that would prove that the log-rewards functions ϕ ε,n (y) converge to the log-reward functions ϕ0,n (y) as ε → 0. We impose the following condition on the parameters of the space-skeleton model defined in relations (7.126)–(7.131): N3 : (a) δ ε → 0 as ε → 0; ± (b) z± ε,n = δ ε m ε,n + λ ε,n → ±∞ as ε → 0, for n = 0, 1, . . . , N . − Note that the condition N3 implies that δ ε (m+ ε,n − m ε,n ) → ∞ as ε → 0, for n = − 0, 1, . . . , N, and m+ ε,n − m ε,n → ∞ as ε → 0, for n = 0, 1, . . . , N. In this case, we are going to apply Theorem 5.3.1, which give conditions of conver­ gence for reward functions for log-price processes represented by random walks. Let us recall the first-type modulus of exponential moment compactness for the log-price process Y0,n , defined for β ≥ 0,

Δβ (Y0,· , N ) = max Ee β| W0,n+1 | . 0≤ n ≤ N − 1

(7.137)

We impose on the price process Y0,n the following first-type condition of exponen­ tial moment compactness, which is a one-dimensional variant of the condition C7 [β¯ ] and should be assumed to hold for some β ≥ 0: C13 [β ]: Δβ (Y0,· , N ) < K55 , for some 1 < K55 < ∞ . Let us recall the second-type modulus of exponential moment compactness for the components of the log-price process Y0,n , takes the following form for β ≥ 0: ± βW 0,n+1 . Ξ± β ( Y 0, · , N ) = max Ee

0≤ n ≤ N − 1

(7.138)

As follows from Lemma 4.1.8, the condition C13 [β ] is implied by the following sec­ ond-type condition of exponential moment compactness, which should be assumed to hold for the same parameter β ≥ 0: E17 [β ]: Ξ± β ( Y 0, · , N ) < K 56 , for some 1 < K 56 < ∞ . We shall prove below that the conditions L5 , N3 , and C13 [β ] imply that the first-type condition of exponential moment compactness C7 [β¯ ] (its one-dimensional variant) holds for the log-price processes Y ε,n. As in Subsection 7.2.1, we impose the conditions B4 [𝛾] and I6 on the pay-off func­ tion g(n, e y ). As was pointed out in Subsection 7.2.1, the condition I6 implies that g(n, e y ε ) → g(n, e y0 ) as ε → 0 for any y ε → y0 ∈ Yn as ε → 0 and n = 0, . . . , N. Thus, the condition I3 holds with the sets Zn = Yn × {x0 }, n = 0, . . . , N.

7.4 LPP represented by multivariate modulated random walks | 279

The condition J4 should be replaced by the following condition: ˜ 0,n (Yn − y) = P{W0,n + y ∈ Yn } = 1, y ∈ Yn−1 , for n = 1, . . . , N, where Yn , J11 : P n = 0, . . . , N are the sets introduced in conditions I6 . We shall prove below that the conditions L5 and N3 imply that probability measures ˜ ε,n (·) ⇒ P ˜ 0,n (·) as ε → 0, for n = 1, . . . , N, and, thus, the condition J4 (a) holds. P Also, the condition J11 coincides with the condition J4 (b) . Thus the conditions L5 , N3 , and J11 imply that the condition J4 holds with the sets Zn = Yn × {x0 } and Z n = R1 × {x 0 }, n = 0, . . . , N.  Note also that the condition J11 (b) automatically holds if the sets Y n = ∅, n = 1, . . . , N, or if these sets are finite or countable and the measures P0,n (A − y), y ∈  Yn−1 have no atoms in points of the set Y n , for every n = 1, . . . , N, as well as in the case, where measures P0,n (A − y), y ∈ Yn−1 , n = 1, . . . , N are absolutely continuous  with respect to some σ-finite measure P(A) on the Borel σ-algebra B1 and P(Yn ) = 0, n = 1, . . . , N. The following result can be obtained by application of Theorem 5.3.1 to the above space-skeleton approximation model. Theorem 7.4.1. Let a log-price process Y 0,n and the corresponding space skeleton ap­ proximation processes Y ε,n represented by random walks are defined as described above. Let also the conditions B4 [𝛾] and C13 [β ] hold with the parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L5 , N3 , I6 , and J11 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn : ϕ ε,n (y ε ) → ϕ0,n (y0 ) as

ε → 0.

(7.139)

Proof. By the condition N3 , there exists ε17 ∈ (0, ε0 ] such that ±z± ε,n ≥ 0, n = 0, . . . , N, for ε ∈ (0, ε17 ]. The following inequality holds for every ε ∈ (0, ε17 ] and y ∈ R1 , n = 0, . . . , N: + − + |h ε,n (y) | ≤ |y|I (y ∉ [z− ε,n , z ε,n ]) + (|y | + δ ε ) I ( y ∈ [ z ε,n , z ε,n ])

≤ (|y| + δ ε ) .

(7.140)

Relation (7.130) and the condition L5 imply that the following inequality holds for every n = 1, . . . , N, ε ∈ (0, ε17 ], and β ≥ 0: Ee β| W ε,n | = Ee β| h ε,n (W0,n )| .

(7.141)

Using relations (7.140) and (7.141) we get the following inequality, for ε ∈ (0, ε17 ], and β ≥ 0: Δβ (Y ε,· , N ) = max Ee β| W ε,n+1 | = max Ee β| h ε,n+1 (W0,n+1 )| 0≤ n ≤ N − 1

≤ e βδ ε

0≤ n ≤ N − 1

max Ee β| W0,n+1 | = e βδ ε Δβ (Y0,· , N ) .

0≤ n ≤ N − 1

(7.142)

280 | 7 Space-skeleton reward approximations Since, by the condition N3 , e βδ ε → 1 as ε → 0, the condition C13 [β ] and inequality (7.142) imply the following relation: lim Δβ (Y ε,· , N ) ≤ Δβ (Y0,· , N ) < K55 .

(7.143)

ε →0

Thus, the condition C7 [β¯ ] (its one-dimensional variant) holds. Also, the following inequality holds for every ε ∈ (0, ε0 ] and n = 0, . . . , N, y ∈ R1 : − |h ε,n (y) − y| ≤ (z− ε,n − y ) I ( y ≤ z ε,n ) + + + + δ ε I ( z− ε,n < y ≤ z ε,n ) + ( y − z ε,n ) I ( y > z ε,n ) .

(7.144)

The condition N3 and relation (7.144) imply that, for every y ∈ R1 and n = 0, 1, . . . , N, h ε,n (y) → y as ε → 0 . (7.145) Relation (7.145) obviously implies that, for every y ∈ R1 and n = 0, 1, . . . , N, a.s.

h ε,n (W0,n ) −→ W0,n

as

ε → 0.

(7.146)

Since a.s. convergence of random variables implies their weak convergence, rela­ tion (7.146) implies that, for every y ∈ R1 and n = 0, 1, . . . , N, ˜ ε,n (·) = P{h ε,n (W0,n ) ∈ ·} P ˜ 0,n (·) = P{W0,n ∈ ·} as 0 < ε → 0 . ⇒P

(7.147)

As follows from the remarks made above, other conditions of Theorem 5.3.1 also holds. Therefore, Theorem 7.4.1 is a corollary of Theorem 5.3.1.

7.4.3 Convergence of space-skeleton approximations for optimal expected rewards of log-price processes represented by univariate standard random walks Let us now assume that we have found, using backward recurrence algorithm pre­ sented above in Subsection 7.3.1, the values of the reward functions ϕ ε,0 (y ε,0,l), for the following initial points: y ε,0,l = δ ε,0 l + λ ε,0 ,

+ m− ε,0 ≤ l ≤ m ε,0 .

(7.148) (ε)

Then the optimal expected reward functional Φ ε = Φ(Mmax,N ) for the approxi­ mation log-price processes Y ε,n can be found using the following formula: m+ ε,0

Φε =



ϕ ε,0 (y ε,0,l)p ε,0 ,

l=m− ε,0 + where p ε,0 = P0,0 (Iε,0,l) = P{h ε,0(Y0,0 ) = y ε,0,l}, l = m− ε,0 , . . . , m ε,0 .

(7.149)

7.4 LPP represented by multivariate modulated random walks | 281

We impose on the initial value of the price process Y0,n the condition D10 [β ] and the following variant of the condition K6 : K10 : P0,0 (Y0 ) = 1, where Y0 is the set introduced in the condition I6 . The following theorem is a corollary of Theorem 5.3.2. Theorem 7.4.2. Let a log-price process Y 0,n and the corresponding space skeleton ap­ proximation processes Y ε,n represented by random walks are defined as described above. Also let the conditions B4 [𝛾], C13 [β ], and D10 [β ] hold with the parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L5 , M5 , N3 , I6 , J11 , and K10 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(7.150)

7.4.4 Space-skeleton approximations for option rewards of log-price processes represented by Lévy random walks Let us consider the case, where the Y0,n is a Lévy random walk. This means that the random jumps W0,n , n = 1, 2, . . . , have infinitely divisible distributions with charac­ teristic functions, ψ0,n (s) = E exp{isW0,n } (

1 (eisy − 1 − isy)Π n (dy) = exp iμ n s − σ 2n s2 + 2 | y | 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L5 , N3 , I6 , and J11 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn : ϕ ε,n (y ε ) → ϕ0,n (y0 )

as

ε → 0.

(7.152)

The conditions D10 [β ] and K10 do not change. Theorem 7.3.2 takes the following form. Theorem 7.4.4. Let a log-price process Y 0,n represented by a Lévy random walk and the corresponding space skeleton approximation processes Y ε,n are defined as described above. Let also the conditions B4 [𝛾], C13 [β ], and D10 [β ] hold with the parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L5 , M5 , N3 , I6 , J11 , and K10 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(7.153)

7.4.5 Space-skeleton approximations for log-price processes represented by multivariate modulated random walks ε,n , X ε,n ) can be introduced, for In this model, a modulated price process  Z ε,n = (Y every ε ∈ [0, ε0 ], via the following stochastic transition dynamic relation: ⎧ ε,n−1 + W  ε,n , X ε,n = C ε,n(X ε,n−1 , U ε,n ) , ε,n = Y ⎪ Y ⎪ ⎪ ⎨ ˜ ε,n (X ε,n−1 , U ε,n ), n = 1, 2, . . . ,  ε,n = B where W (7.154) ⎪ ⎪ ⎪ ⎩  = B (U ) , X = C (U ) . Y ε,0

ε,0

ε,0

ε,0

ε,0

ε,0

where (a) U ε,n , n = 0, 1, . . . , is a sequence of independent random variables tak­ ing value in some measurable space U with a σ-algebra of measurable subsets BU , ˜ ε,n ( ˜ ε,n,1(y1 , u ), . . . , B ˜ ε,n,k(y k , u )), n = 1, 2, . . . , are vector measur­ (b) B y , u) = (B able functions acting from the space Rk × U to Rk , (c) C ε,n (x, u ), n = 1, 2, . . . , are measurable functions acting from the space X × U to the space X, (d) B ε,0(u ) = (B ε,0,1(u ), . . . , B ε,0,k(u )) is a vector measurable function acting from the space U to Rk , (e) C ε,0(u ) is a measurable function acting from the space U to the space  ε,n = (W ε,n,1 , . . . , W ε,n,k), n = 1, 2, . . . , where X, and (f) the sequence of jumps W ˜ W ε,n,i = B ε,n,i(X ε,n−1, U ε,n ), i = 1, . . . , k, n = 1, 2, . . . . In this case, the transition probabilities P ε,n ( z , A ) = P{  Z ε,n ∈ A/ Z ε,n−1 =  z } of ε,n , X ε,n ) can be expressed via the transition probabilities ε,n = (Y the Markov process Z ˜ ε,n (x, A) = P{(W ˜ ε,n (x, U ε,n ), C ε,n (x, U ε,n )) ∈ A}  ε,n , X ε,n) ∈ A/X ε,n−1 = x} = P{(B P by the following formulas, for  z = ( y , x) ∈ Z = Rk × X, A ∈ BZ , n = 1, 2, . . . :  ε,n , X ε,n) ∈ A/Y ε,n−1 =  P ε,n ( z , A) = P{( y+W y , X ε,n−1 = x} ˜ ε,n (x, U ε,n ), C ε,n (x, U ε,n )) ∈ A} y+B = P{(

˜ ε,n (x, A[y] ) , =P

(7.155)

7.4 LPP represented by multivariate modulated random walks

| 283

y  , x) : ( y + y , x) ∈ A}. where A[y] = A[y] = {( ε,0 ∈ A} is concerned, it has the As far as the initial distribution P ε,0(A) = P{Z following form, for A ∈ BZ : ε,0 , X ε,0) ∈ A} = P{(B ε,0(U ε,0), C ε,0(U ε,0)) ∈ A} . P ε,0(A) = P{(Y

(7.156)

A standard way for constructing skeleton sets and space-skeleton approximation Z0,n is the following. for the modulated random walk  Let us choose δ ε,i > 0 and λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . , and integers + m− ε,n,j ≤ m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . . + First, the intervals Iε,n,i,l should be constructed for l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . , ⎧ 1 ⎪ ⎪ (−∞, δ ε,i (m− λ if l = m− ε,n,i + 2 )] + ε,n,i , ⎪ ⎨# ! ε,n,i 1 1 − + Iε,n,i,l = (7.157) δ ( l − ) , δ ( l + ) + λ if m ε,i ε,i ε,n,i ε,n,i < l < m ε,n,i , 2 2 ⎪ ⎪ ⎪ + ⎩(δ (m+ − 1 ), ∞) + λ if l = m . ε,i

ε,n,i

2

ε,n,i

ε,n,i

+ Then the cubes Iε,n,l1,...,l k = Iε,n,1,l1 × · · · × I ε,n,k,l k , l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . , should be defined. By the definition, the points y ε,n,i,l i = l i δ ε,i + λ ε,n,i ∈ Iε,n,i,l i and, thus, the vector + points (y ε,n,1,l1 , . . . , y ε,n,k,l k ) ∈ Iε,n,l1,...,l k , for l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . . + Second, nonempty sets Jε,n,l ∈ BX , l = m− ε,n,0 , . . . , m ε,n,0, n = 0, 1, . . . , such that (a) Jε,n,l ∩ Jε,n,l = ∅, l ≠ l , n = 0, 1, . . . ; (b) X = ∪m−ε,n,0 ≤l≤m+ε,n,0 Jε,n,l, n = 0, 1, . . . , should be constructed. Recall that one of our model assumption is the X is a Polish space, i.e. a com­ plete, separable, metric space. In this case, it is natural to assume that there ex­ ist some “large” sets K ε,n , n = 0, 1, . . . , “small” nonempty sets K ε,n,l ⊆ BX , l = +   m− ε,n,0, . . . , m ε,n,0, n = 0, 1, . . . , such that (c) K ε,n,l  ∩ K ε,n,l = ∅ , l ≠ l , n = 0, 1, . . . ; (d) ∪m−ε,n,0 ≤l≤m+ε,n,0 K ε,n,l = K ε,n, n = 0, 1, . . . . Then, the sets Jε,n,l can be defined in the following way, for n = 0, 1, . . . : ⎧ + ⎨K ε,n,l if m− ε,n,0 ≤ l < m ε,n,0 , Jε,n,l = (7.158) ⎩K + ∪ K ε,n if l = m+ .

ε,n,m ε,n,0

ε,n,0

+ Then skeleton points x ε,n,l ∈ K ε,n,l, l = m− ε,n,0 , . . . , m ε,n,0, n = 0, 1, . . . , should be chosen. A particular important case is, where the space X = {1, . . . , m} is a finite set and metric dX (x, y) = I (x ≠ y). In this case, the standard choice is to take K ε,n = X, + n = 0, 1, . . . , one-point sets Jε,n,l = K ε,n,l = {l}, 1 = m− ε,n,0 ≤ l ≤ m ε,n,0 = m, n = 0, 1, . . . , N, and points x ε,n,l = l, l = 1, . . . , m, n = 0, . . . . If the space X = {1, 2, . . . } is a countable set and again metric dX (x, y) = I (x ≠ + y), the standard choice is to take one-point sets K ε,n,l = {l}, 1 = m− ε,n,0 ≤ l ≤ m ε,n,0 =

284 | 7 Space-skeleton reward approximations m ε , n = 0, 1, . . . , sets K ε,n = {l : l ≤ m ε }, n = 0, 1, . . . . In this case, the sets Jε,n,l = + {l}, 1 = m − ε,n,0 ≤ l < m ε,n,0 = m ε , n = 0, 1, . . . , while J ε,n,m ε = {l : l ≥ m ε }, and points x ε,n,l = l, l = 1, . . . , m ε ,, n = 0, 1, . . . . z ε,n,¯l ∈ Aε,n,¯l can be defined for ¯l = Third, the sets Aε,n,¯l and skeleton points,  − + (l0 , l1 , . . . , l k ), l j = m ε,n,j, . . . , m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . , in the following way: A ε,n,¯l = Iε,n,l1,...,l k × Jε,n,l0 ,

(7.159)

 z ε,n,¯l = ( y ε,n,¯l, x ε,n,¯l) = ((y ε,n,1,l1 , . . . , y ε,n,k,l k ), x ε,n,l0) .

(7.160)

and Let us introduce sets of vector indices Lε,n which are defined in the following way, for n = 0, 1, . . . : + Lε,n = {¯l = (l0 , . . . , l k ), l j = m− ε,n,j , . . . , m ε,n,j , j = 0, . . . , k } .

(7.161)

Let us now assume that the following fitting condition holds:  ε,n =  y ε,n,¯l , X ε,n = x ε,n,¯l/ L6 : For every ε ∈ (0, ε0 ], transition probabilities P{W ˜ 0,n (x ε,n−1,l ,  0,n , X0,n ) ∈ A ε,n,¯l/X0,n−1 = x} = P ˜ ε,¯l (x, n) = P{(W X ε,n−1 = x} = p 0  − + ¯  A ε,n,¯l), x ∈ Jε,n−1,l0 , l0 = m ε,n−1,0, . . . , m ε,n−1,0, l = (l0 , . . . , l k ) ∈ Lε,n , n = 1, . . . . If we are also interested to fit initial distributions, the following fitting condition should be assumed: ε,0 =  M6 : For every ε ∈ (0, ε0 ], initial probabilities P{Y y ε,0,¯l , X ε,0 = x ε,0,¯l} = p ε,¯l = ¯  P{(Y0,0 , X0,0 ) ∈ Aε,0,¯l)} = P0,0 (Aε,0,¯l), l = (l0 , . . . , l k ) ∈ Lε,0 . Let us define the skeleton functions, h ε,n,i(y), y ∈ R1 , for ε ∈ (0, ε0 ] and i = 1, . . . , k, n = 0, 1, . . . , ⎧ 1 − ⎪ δ ε,i m− ⎪ ε,n,i + λ ε,n,i if y ≤ δ ε,i ( m ε,n,i + 2 ) + λ ε,n,i , ⎪ ⎪ ⎪ ⎪ ⎨ δ ε,i l + λ ε,n,i if δ ε,i (l − 21 ) + λ ε,n,i < y ≤ δ ε,i (l + 21 ) + λ ε,n,i , h ε,n,i(y) = ⎪ + ⎪ m− ⎪ ε,n,i < l < m ε,n,i , ⎪ ⎪ ⎪ 1 ⎩δ m+ + λ + (7.162) ε,i ε,n,i ε,n,i if y > δ ε,i ( m ε,n,i − 2 ) + λ ε,n,i . ˆ ε,n ( y),  y = (y1 , . . . , y k ) ∈ Rk , for ε ∈ (0, ε0 ] and and the vector skeleton functions h n = 0, 1, . . . , ˆ ε,n ( h y ) = (h ε,n,1(y1 ), . . . , h ε,n,k(y k )) .

(7.163)

Let us also define skeleton functions h ε,n,0(x), x ∈ X, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , + h ε,n,0(x) = { x ε,n,l if x ∈ Jε,n,l , m− (7.164) ε,n,0 ≤ l ≤ m ε,n,0 .

7.4 LPP represented by multivariate modulated random walks

| 285

˜ ε,n (x, A) = P{(W  ε,n , X ε,n ) ∈ A /X ε,n−1 In this case, the transition probabilities P = x}, x ∈ X, A ∈ BZ , n = 1, . . . , take the following form, for every ε ∈ (0, ε0 ]:

˜ ε,n (x, A) = ˜ 0,n (h ε,n−1,0(x), Aε,n,¯l) P P ( y ε,n,¯l ,x ε,n,¯l )∈ A

  ˆ ε,n(W  0,n ), h ε,n,0(X0,n )) ∈ A/X0,n−1 = h ε,n−1,0(x) . = P (h

(7.165)

ε,0 , X ε,0) ∈ A}, A ∈ BZ is con­ As far as the initial distribution P ε,0(A) = P{(Y cerned, it takes the following form, for every ε ∈ (0, ε0 ]:

P0,0 (Aε,0,¯l) P ε,0(A) = ( y ε,0,¯l ,x ε,0,¯l )∈ A

  ˆ ε,0(Y 0,0), h ε,0,0(X0,0 )) ∈ A . = P (h

(7.166)

In this case, the dynamic representation (7.154) for the modulated log-price pro­ Z ε,n takes, for every ε ∈ (0, ε0 ], the following form: cess  ⎧ ⎪ ε,n = Y ε,n−1 + W  ε,n , Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆ ε,n (B ˜ 0,n (h ε,n−1,0(X ε,n−1), U0,n )) ,  ε,n = h ⎪ where W ⎪ ⎪ ⎨ X ε,n = h ε,n,0(C0,n (h ε,n−1,0(X ε,n−1), U0,n )) , n = 1, 2, . . . , (7.167) ⎪ ⎪ ⎪ ⎪ ⎪ ˆ ε,0(B0,0 (U0,0 )) ,  ε,0 = h ⎪ Y ⎪ ⎪ ⎪ ⎪ ⎩ X ε,0 = h ε,0,0(C0,0 (U0,0 )) . ε,n given by the dynamic relations (7.167) is a Obviously, the random sequence Z modulated random walk with transition probabilities and initial distribution satisfy­ ing relations (7.165) and (7.166) and, therefore, satisfying fitting conditions L6 and M6 . Let us denote  y ε,n,n+r,¯l = (δ ε,1 l1 + λ ε,n,n+r,1, . . . , δ ε,k l k + λ ε,n,n+r,k), x ε,n,n+r,¯l = + x ε,n+r,l0 , ¯l = (l0 , . . . , l k ), l j = m− ε,n,n + r,j , . . . , m ε,n,n +r,j, j = 0, . . . , k. Here, for i = 1, . . . , k, 0 ≤ n < n + r < ∞, ± m± ε,n,n +r,0 = m ε,n + r,0 ,

and λ ε,n,n+r,i =

m± ε,n,n + r,i = n

+r

λ ε,l,i .

n

+r

m± ε,l,i ,

(7.168)

l = n +1

(7.169)

l = n +1

ε,n , X ε,n ) = ( ε,n = (Y If Z y , x) for some ( y , x) ∈ Z and n = 0, 1, . . . , then, for   y + y ε,n,n+r,¯l, x ε,n,n+r,¯l), ¯l = r = 1, 2, . . . , possible states for Z ε,n+r = (Y ε,n+r , X ε,n+r) are ( − + (l0 , . . . , l k ), l j = m ε,n,n+r,j, . . . , m ε,n,n+r,j, j = 0, . . . , k. The following lemma is a variant of Lemma 7.1.4.

Z ε,n is a space-skeleton Lemma 7.4.2. Let, for every ε ∈ (0, ε0 ], the log-price process  random walk with jump distributions probabilities and the initial distribution defined,

286 | 7 Space-skeleton reward approximations respectively, in relations (7.165) and (7.166). Then, the log-reward functions ϕ ε,n ( y , x) and ϕ ε,n+r ( y+ y ε,n,n+r,¯l, x ε,n+r,¯l) for points  z = ( y , x) = ((y1 , . . . , y k ), x) and ( y+ − + ¯ y ε,n,n+r,¯l, x ε,n+r,¯l), l = (l0 , l1 , . . . , l k ), l j = m ε,n,n+r,j, . . . , m ε,n,n+r,j, j = 0, . . . , k, r =  − + 1, . . . N − n, are, for every  z = ( y , x) ∈ Rk × Jε,n,l0 , l 0 = m ε,n,0, . . . , m ε,n,0, n = 0, . . . , N, the unique solution for the following finite recurrence system of linear equa­ tions: ⎧ ⎪ ϕ ε,N ( y+ y ε,n,N,¯l, x ε,N,¯l) = g(N, ey+y ε,n,N,¯l , x ε,N,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ + ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ ⎪ ε,n,N,j, . . . , m ε,n,N,j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j = 0, . . . , k , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r( y+ y ε,n,n+r,¯l, x ε,n+r,¯l) = max g(n + r, ey+y ε,n,n+r,¯l , x ε,n+r,¯l) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ #

⎪ ⎪ ⎪ ⎪ ϕ ε,n+r+1  y+ y ε,n,n+r,¯l ⎪ ⎪ ⎪ ⎪ ( y ε,n+r+1,¯l  ,x ε,n+r+1,¯l  )∈Fε,n+r+1 ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ ⎪ + y ε,n+r+1,¯l  , x ε,n+r+1,¯l  ⎪ ⎪ ⎪ ⎪ ' ⎨ ˜ × P0,n+r+1(x ε,n+r,¯l, A ε,n+r+1,¯l , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ ⎪ ε,n,n +r,j, . . . , m ε,n,n + r,j , ⎪ ⎪ ⎪ ⎪ ⎪ j = 0, . . . , k , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n ( y , x) = max g(n, ey , x), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ # 

⎪ ⎪ ⎪ ⎪ ϕ ε,n+1  y+ y ε,n+1,¯l  , x ε,n+1,¯l  ⎪ ⎪ ⎪ ⎪ ( y ε,n+1,¯l  ,x ε,n+1,¯l  )∈Fε,n+1 ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ⎪  × P0,n+1(x ε,n,l0 , A ε,n+1,¯l . ⎩ (7.170)

7.4.6 Convergence of space-skeleton approximations for log-price processes represented by multivariate modulated random walks The above backward algorithm for computing of the log-reward functions ϕ ε,n ( y , x) given in Lemma 7.4.2 should be supplemented by the proposition that would prove that the log-rewards functions ϕ ε,n ( y , x) converge to the log-reward functions ϕ0,n ( y , x) as ε → 0, and, thus, they can serve as approximations for the log-reward functions ϕ0,n ( y , x ).

7.4 LPP represented by multivariate modulated random walks

| 287

Let us introduce special shorten notations for the maximal and the minimal skele­ ton points, for i = 1, . . . , k, n = 0, . . . , N and ε ∈ (0, ε0 ], ± z± ε,n,i = δ ε,i m ε,n,i + λ ε,n,i .

(7.171)

We impose the following condition on the parameters of the space-skeleton model defined in relations (7.154)–(7.166): N4 : (a) δ ε,i → 0 as ε → 0, for i = 1, . . . , k; (b) ±z± ε,n,i → ∞ as ε → 0, for i = 1, . . . , k, n = 0, 1, . . . , N; (c) for any x ∈ X and d > 0, there exists ε x,d ∈ (0, ε0 ] such that the ball R d (x) ⊆ K ε,n , n = 0, . . . , N, for ε ∈ (0, ε x,d ] ; (d) the sets K ε,n,l have diameters d ε,n,l = supx ,x∈Kε,n,l dX (x , x ) such that d ε,n = maxm−ε,n,0 ≤l≤m+ε,n,0 d ε,n,l → 0 as ε → 0, n = 0, . . . , N . − Note that the condition N4 implies that δ ε,i (m+ ε,n,i − m ε,n,i) → ∞ as ε → 0, for i = + − 1, . . . , k, n = 0, 1, . . . , N, and m ε,n,i − m ε,n,i → ∞ as ε → 0, for i = 1, . . . , k, n = 0, 1, . . . , N. − Note, however, that the condition N4 does not require that m+ ε,n,0 − m ε,n,0 → ∞ as ε → 0. Let us recall the first-type modulus of exponential moment compactness for the components log-price process Y0,n,i, for i = 1, . . . , k, defined for β ≥ 0,

β| Y 0,n+1,i − Y 0,n,i | Δ , β ( Y 0, · ,i , N ) = max sup Ex,n e 0≤ n ≤ N −1 x ∈X

(7.172)

where Ex,n and Px,n denote, respectively, conditional expectation and conditional probability under condition that X n = x. 0,n , X0,n ) the following first-type condi­ We impose on the price process  Z0,n = (Y tion of exponential moment compactness, which is a variant of the condition C2 [β¯ ], and should be assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with non­ negative components: C14 [β¯ ]: Δ β i ( Y 0, · ,i , N ) < K 58,i , i = 1, . . . , k for some 1 < K 58,i < ∞, i = 1, . . . , k . Let us recall the second-type modulus of exponential moment compactness for com­ Z0,n , defined for β ≥ 0, ponents of the log-price process  ± βW 0,n+1 Ξ± . β ( Y 0, · ,i , N ) = max sup Ex,n e

0≤ n ≤ N −1 x ∈X

(7.173)

As follows from Lemma 4.2.1, the condition C14 [β¯ ] is implied by the follow­ ing condition, which should be assumed to hold for the same vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: E18 [β¯ ]: Ξ± β i ( Y 0, · ,i , N ) < K 59,i , i = 1, . . . , k, for some 1 < K 59,i < ∞, i = 1, . . . , k .

288 | 7 Space-skeleton reward approximations We are going to apply Theorem 5.3.5, which give conditions of convergence for re­ ward functions for log-price processes represented by multivariate modulated random walks. We shall prove below that, in the skeleton approximation model described above, the conditions L6 , N4 , and C14 [β¯ ] imply that the first-type condition of exponential moment compactness C8 [β¯ ] holds for the log-price processes  Z ε,n .  y In this case, we impose on pay-off function g(n, e , x) the condition B1 [𝛾¯] as­ sumed to hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative compo­ nents. Also, the condition I1 should be replaced by the condition I7 . As was pointed out in Subsection 7.2.3, the condition I7 means that g(n, ey ε , x ε ) → g(n, ey0 , x0 ) as ε → 0 for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn as ε → 0 and n = 0, . . . , N. Thus, the condition I1 holds with the sets Zn , n = 0, . . . , N. The condition J7 should be replaced by the following condition: J12 : There exist measurable sets Xn ⊆ X, n = 0, . . . , N such that: ˜ 0,n (x0 , ·) as ε → 0, for any x ε → x0 ∈ Xn−1 as ε → 0, and ˜ 0,n (x ε , ·) ⇒ P (a) P n = 1, . . . , N; ˜ 0,n (x0 , (Zn ∩ Z (b) P z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Z y 0 ] ) = 1, for every  n −1 , n = n )[  1, . . . , N, where Zn , n = 0, . . . , N are sets introduced in the condition I7 and Z n = Rk × X n , n = 0, . . . , N . ˜ 0,n (x, ·) are weakly con­ The condition J12 (a) means that the transition probabilities P tinuous at points x0 ∈ Xn−1 for every n = 1, . . . , N.  A typical example is where the sets Zn , X n , n = 1, . . . , N are empty sets. Then the condition J12 (b) obviously holds.  Another typical example is where the sets Zn , Xn , n = 1, . . . , N are at most finite ˜ 0,n (x0 , A), A ∈ BZ and or countable sets. Then, the assumption that the measures P ˜ P0,n (x0 , Rk × B), B ∈ BX have no atoms, for every x0 ∈ Xn−1 , n = 1, . . . , N, implies that the condition J12 (b) holds. ˜ 0,n (x0 , A), x0 ∈ Xn−1 , n = 1, . . . , N One more example is, where the measures P are absolutely continuous with respect to some σ-finite measure P(A) on BZ and  P(Zn ), P(Rk × X n ) = 0, n = 1, . . . , N. This assumption implies that the condition J12 (b) holds. We shall prove below that the conditions L6 , N4 , and J12 imply that the condition J12 holds with the sets Zn , Z n , n = 0, . . . , N. The modulated random walk  Z0,n was introduced in the dynamic form. This makes it possible to replace the condition J12 by the following sufficient condition, which, by Lemma 5.2.1, implies the condition J12 to hold: J12 : There exist measurable sets Xn ⊆ X, n = 0, . . . , N and Un ∈ U, n = 1, . . . , N such that: ˜ 0,n (x0 , u ) and C0,n (x ε , u ) → C0,n (x0 , u ) as ε → 0, for any ˜ 0,n (x ε , u ) → B (a) B x ε → x0 ∈ Xn−1 as ε → 0, u ∈ Un , n = 1, . . . , N;

7.4 LPP represented by multivariate modulated random walks

| 289

(b) P{U0,n ∈ Un } = 1, n = 1, . . . , N; ˜ 0,n (x0 , (Zn ∩ Z (c) P z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Z n −1 , n = n )[ y 0 ] ) = 1, for every   1, . . . , N, where Zn , n = 0, . . . , N are sets introduced in the condition I7 and Z n = Rk × X n , n = 0, . . . , N . The following result can be obtained by application of Theorem 5.3.5. This theorem should be applied to the above skeleton model log-price processes represented by mul­ tivariate modulated random walks. Z0,n and the corresponding space skeleton ap­ Theorem 7.4.5. Let a log-price process  proximation processes  Z ε,n represented by multivariate modulated random walks are defined as described above. Let also the conditions B1 [𝛾¯] and C14 [β¯ ] hold with the vec­ tor parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ), β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions L6 , N4 , I7 , and J12 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Z , n −1 ϕ ε,n (y ε , x ε ) → ϕ0,n (y0 , x0 )

as

ε → 0.

(7.174)

Proof. By the condition N4 , there exists ε18 ∈ (0, ε0 ] such that ±z± ε,n,i ≥ 0, i = 1, . . . , k, n = 0, . . . , N, for ε ∈ (0, ε18 ]. The following inequality holds for every ε ∈ (0, ε18 ] and y ∈ R1 , i = 1, . . . , k, n = 0, . . . , N, |h ε,n,i(y)| + − + ≤ |y|I ( y ∉ [ z− ε,n,i , z ε,n,i]) + (|y | + δ ε,i ) I ( y ∈ [ z ε,n,i , z ε,n,i]) ≤ (|y | + δ ε,i ) .

(7.175)

Relation (7.165) and the condition L6 imply that the following inequality holds for every x ∈ X, i = 1, . . . , k, n = 0, . . . , N − 1, ε ∈ (0, ε18 ], and β ≥ 0: Ex,n e β| W ε,n+1,i | = Eh ε,n,0 (x),n e β| h ε,n+1,i (W0,n+1,i )| .

(7.176)

Using relations (7.175) and (7.176) we get the following inequality, for i = 1, . . . , k, e ∈ (0, ε18 ], and β ≥ 0: β| W ε,n+1,i | Δ β ( Y ε, · ,i , N ) = max sup Ex,n e 0≤ n ≤ N −1 x ∈X

=

max sup Eh ε,n,0 (x),n e β| h ε,n+1,i (W0,n+1,i )|

0≤ n ≤ N −1 x ∈X

≤ e βδ ε,i ≤ e βδ ε,i

max sup Eh ε,n,0 (x),n e β| W0,n+1,i |

0≤ n ≤ N −1 x ∈X

max sup Ex,n e β i | W0,n+1,i |

0≤ n ≤ N −1 x ∈X

= e βδ ε,i Δ β ( Y 0, · ,i , N ) .

(7.177)

Since, by the condition N4 , e βδ ε,i → 1 as ε → 0, for i = 1, . . . , k and β ≥ 0, the condition C14 [β¯ ] and inequality (7.177) imply that, for i = 1, . . . , k,  lim Δ β i ( Y ε, · ,i , N ) ≤ Δ β i ( Y 0, · ,i , N ) < K 58,i .

ε →0

(7.178)

290 | 7 Space-skeleton reward approximations Thus, the condition C8 [β¯ ] holds. Let us choose an arbitrary 1 ≤ n ≤ N and x ε → x0 ∈ Xn−1 as ε → 0. It follows from the definition of functions h ε,n,0(x) that the following inequality also takes place ε ∈ (0, ε0 ] and x ∈ X, n = 0, . . . , N: dX (h ε,n,0(x), x) ≤ dX (x ε,n,m+ε,n,0 , x)I (x ∈ Kε,n ) + d ε,n I (x ∈ K ε,n) .

(7.179)

Inequality (7.179) and the condition N4 imply that, for n = 1, . . . , N, h ε,n−1,0(x ε ) → x0

as

ε → 0.

(7.180)

The following inequality holds for every ε ∈ (0, ε0 ] and y ∈ R1 , i = 1, . . . , k, n = 0, . . . , N, − |h ε,n,i(y) − y| ≤ (z− ε,n,i − y ) I ( y ≤ z ε,n,i) + + + + δ ε,i I (z− ε,n,i < y ≤ z ε,n,i) + ( y − z ε,n,i) I ( y > z ε,n,i) .

(7.181)

Let us take arbitrary κ, ϱ > 0. One can find T κ > 0 such that points ±T κ are points of continuity for the distribution functions Px0 ,n−1{W0,n,i ≤ u }, i = 1, . . . , k, n = 1, . . . , N and, for i = 1, . . . , k, n = 1, . . . , N, Px0 ,n−1{W0,n,i ≤ −T κ } + Px0 ,n−1{W0,n,i ≥ T κ } ≤ κ .

(7.182)

Using the above remarks, the conditions N4 and J12 (a) and relations (7.180) and (7.181), we get the following relation, for i = 1, . . . , k, n = 1, . . . , N: lim Ph ε,n−1,0 (x ε ),n−1{|h ε,n,i(W0,n,i) − W0,n,i | ≥ ϱ } # ≤ lim Ph ε,n−1,0 (x ε ),n−1 {W0,n,i ≤ z− ε,n,i}

0< ε → 0

0< ε → 0

 + I (δ ε,i ≥ ϱ ) + Ph ε,n−1,0 (x ε ),n−1{W0,n,i > z+ ε,n,i} # ≤ lim Ph ε,n−1,0 (x ε ),n−1 {W0,n,i ≤ −T κ } 0< ε → 0  + Ph ε,n−1,0 (x ε ),n−1{W0,n,i > T κ }

= Px 0 ,n−1{W0,n,i ≤ −T κ } + Px 0 ,n−1{W0,n,i ≥ T κ } ≤ κ .

(7.183)

Since an arbitrary choice of κ, ϱ > 0, relation (7.183) implies that for any ϱ > 0 and i = 1, . . . , k, n = 1, . . . , N, lim Ph ε,n−1,0 (x ε ),n−1{|h ε,n,i(W0,n,i) − W0,n,i | ≥ ϱ } = 0 .

0< ε → 0

(7.184)

Let us choose an arbitrary sequence ε r ∈ (0, ε0 ], r = 1, 2, . . . , such the ε r → 0 as r → ∞. The condition J12 and relation (7.180) imply that, for every n = 1, . . . , N, probabil­ ity measures, Ph εr ,n−1,0 (x εr ),n−1{X0,n ∈ ·} ⇒ Px0 ,n−1{X0,n ∈ ·} as

r → ∞.

(7.185)

7.4 LPP represented by multivariate modulated random walks |

291

Let us take again an arbitrary κ > 0. Relation (7.185) implies by the Prokhorov theorem that there exists a compact C κ ⊆ X such that, for n = 1, . . . , N, max Ph εr ,n−1,0 (x εr ),n−1{X0,n ∈ C κ } ≤ κ . r ≥1

(7.186)

Since X is a Polish metric space and C κ is a compact, there exists a ball R κ = R d κ (x κ ) such that C κ ⊆ R κ . By conditions N4 (c), there exists a ε x κ ,d κ ∈ (0, ε0 ] such that R κ ⊆ K ε,n , n = 0, . . . , N, for ε ∈ (0, ε x κ ,d κ ]. Thus, lim Ph εr ,n−1,0 (x εr ),n−1{X0,n ∈ Kε r ,n }

r →∞

≤ lim Ph εr ,n−1,0 (x εr ),n−1{X0,n ∈ R κ } r →∞

≤ lim Ph εr ,n−1,0 (x εr ),n−1{X0,n ∈ C κ } ≤ κ . r →∞

(7.187)

Let us again take an arbitrary ϱ > 0. Using the condition N4 (d) and relations (7.179) and (7.187), we get the following relation, for n = 1, . . . , N: lim Ph εr ,n−1,0 (x εr ),n−1{dX (h ε r ,n,0(X0,n ), X0,n ) ≥ ϱ }  # ≤ lim Ph εr ,n−1,0 (x εr ),n−1 {X0,n ∈ Kε r ,n } + I (d ε r ,n ≥ ϱ ) ≤ κ .

r →∞

r →∞

(7.188)

Since an arbitrary choice of κ > 0 and an arbitrary choice of sequence 0 < ε r → 0 as r → ∞, relation (7.188) implies that, for ϱ > 0 and n = 1, . . . , N, lim Ph ε,n−1,0 (x ε ),n−1{dX (h ε,n,0(X0,n ), X0,n ) ≥ ϱ } = 0 .

0< ε → 0

(7.189)

Relations (7.184) and (7.189) obviously imply that, for every n = 1, . . . , N, the ˆ ε,n (W  0,n ), h ε,n,0(X0,n )) ∈ ·} and Ph ε,n−1,0 (x ε ),n−1 probability measures Ph ε,n−1,0 (x ε ),n−1{(h  0,n , X0,n ) ∈ ·} have the same weak limits as 0 < ε → 0. {(W This proposition, relations (7.165), (7.180), and the condition J12 (a) imply that the following relation of weak convergence holds, for any x ε → x0 ∈ Xn−1 as 0 < ε → 0, for every n = 1, . . . , N: ˆ ε,n (W ˜ ε,n (x ε , ·) = Ph ε,n−1,0 (x ε ),n−1{(h  0,n ), h ε,n,0(X0,n )) ∈ ·} P ˜ 0,n (x0 , ·) = Px0 ,n−1{(W  0,n , X0,n ) ∈ ·} ⇒P

as

ε → 0.

(7.190)

Thus, the condition J7 (a) holds with the sets Zn , Z n , n = 0, . . . , N. Since, the conditions J12 (b) and J7 (b) coincide, we can conclude that the condition J7 holds, in this case, with the above sets Zn , Z n , n = 0, . . . , N. As follows from the remarks made above, other conditions of Theorem 5.3.5 also holds. Therefore, Theorem 7.3.5 is a corollary of Theorem 5.3.5.

292 | 7 Space-skeleton reward approximations

7.4.7 Convergence of space-skeleton approximations for optimal expected rewards of log-price processes represented by multivariate modulated random walks Let us now assume that we have found, using backward recurrence algorithm pre­ sented in Lemma 7.4.2, the values of log-reward functions ϕ ε,0( y ε,0,¯l , x ε,0,¯l), ¯l = (l0 , . . . , l k ) ∈ Lε,0 for the initial points,  z ε,0,¯l = ( y ε,0,¯l , x ε,0,¯l) = ((y ε,0,1,l1 , . . . , y ε,0,k,l k ), x ε,0,l0 ) = ((δ ε,0,1 l1 + λ ε,0,1 , . . . , δ ε,0,k l k + λ ε,0,k ), x ε,0,l0 ) .

(7.191)

Then, the optimal expected reward functional Φ ε can be found using the follow­ ing formula:

ϕ ε,0( z ε,0,¯l)p ε,¯l , (7.192) Φε = ¯l∈Lε,0

ˆ ε,0(Y 0,0 ) = Y ε,0,¯l, h ε,0,0(X0,0 ) = x ε,0,¯l}, ¯l = (l0 , . . . , where p ε,¯l = P0,0 (Aε,0,¯l) = P{h l k ) ∈ Lε,0. Let us formulate conditions of convergence for optimal expected rewards. We impose on the initial value of the log-price process  Z0,0 the condition D11 [β¯ ], which should be assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components. Also, the following variant of the condition K5 is assumed to hold:   K11 : x P0,0 (Z0 ∩ Z 0 ) = 1, where Z0 and Z0 are the sets introduced, respectively, in the conditions I7 and J12 . The following result can be obtained by application of Theorem 5.3.6. Theorem 7.4.6. Let a Markov log-price process  Z0,n and the corresponding space spaceskeleton approximating Markov processes  Z ε,n represented by multivariate modulated random walks are defined as described above. Let also the conditions B1 [𝛾¯], C14 [β¯ ], and D11 [β¯ ] hold with the vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾, k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions L6 , M6 , N4 , I7 , J12 , and K11 hold. Then, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(7.193)

Proof. We are going to check that all conditions of Theorem 5.3.6 hold and then to apply this theorem. In fact, we should only check that the condition D7 [β¯ ] and the condition K6 hold. Indeed, this check was made for other conditions of Theorem 5.3.6 in the proof of Theorem 7.3.5. Relation (7.166) and the condition M6 imply that the following relation holds for i = 1, . . . , k and ε ∈ (0, ε0 ]: Ee β| Y ε,0,i | = Ee β| h ε,0,i (Y0,0,i )| .

(7.194)

7.4 LPP represented by multivariate modulated random walks

| 293

Using inequality (7.175) and relation (7.194) we get the following inequality, for i = 1, . . . , k, ε ∈ (0, ε18 ], and β ≥ 0: Ee β| Y ε,0,i | = Ee β| h ε,0,i (Y0,0,i )| ≤ e βδ ε,i Ee β| Y0,0,i | .

(7.195)

Since, by the condition N4 , e βδ ε,i → 1 as ε → 0, for i = 1, . . . , k, the condition D11 [β¯ ] and inequality (7.195) imply that, for i = 1, . . . , k, lim Ee β i | Y ε,0,i | ≤ Ee β i | Y0,0,i | < K54,i . ε →0

(7.196)

Thus, the condition D7 [β¯ ] holds. Let us take an arbitrary ϱ > 0. Using the condition N4 and inequality (7.181), we get the following relations, for i = 1, . . . , k:   lim P |h ε,0,i(Y0,0,i) − Y0,0,i | ≥ ϱ

0< ε → 0

≤ lim

0< ε → 0

#

 + P{Y0,0,i ≤ z− ε,0,i } + I ( δ ε,i ≥ ϱ ) + P{Y 0,0,i > z ε,0,i } = 0 .

(7.197)

Let us take arbitrary κ, ϱ > 0. There exists a compact C κ ⊆ X such that P{X0,0 ∈ C κ } ≤ κ. Since X is a Polish metric space and C κ is a compact, there exists a ball R κ = R d κ (x κ ) such that C κ ⊆ R κ . By the conditions N2 (d) there exists a ε x κ ,d κ ∈ (0, ε0 ] such that R κ ⊆ K ε,0, for ε ∈ (0, ε x κ ,d κ ]. The above remarks and relation (7.103) imply that, lim P{dX (h ε,0,0(X0,0 ), X0,0) ≥ ϱ }

0< ε → 0

≤ lim

0< ε → 0

 P{X0,0 ∈ K ε,0} + I (d ε,0 ≥ ϱ ) ≤ κ .

(7.198)

Due to an arbitrary choice of κ, ϱ > 0, relations (7.197) and (7.198) imply the fol­ lowing relation: ˆ ε,0 (Y 0,0), h ε,0,0(X0,0)) ∈ ·} P ε,0(·) = P{(h 0,0 , X0,0 ) ∈ ·} ⇒ P0,0 (·) = P{(Y

as

ε → 0.

(7.199)

Relation (7.199) and the condition K11 imply that the condition K6 holds with the sets Z0 and Z 0. It follows from the above remarks that all conditions of Theorem 5.3.6 hold, and, therefore, Theorem 7.3.6 is a corollary of Theorem 5.3.6.

294 | 7 Space-skeleton reward approximations

7.5 Multivariate Markov LPP with unbounded characteristics In this section, we present space-skeleton approximation models for discrete time multivariate modulated Markov log-price processes with unbounded characteristics.

7.5.1 Space-skeleton approximations for univariate Markov log-price processes with unbounded characteristics In this subsection, we consider the same model of log-price processes Y ε,n , n = 0, 1, . . . , and the same skeleton approximation model as in Subsection 7.2.1. The only difference is that we assume that the corresponding characteristics of the process Y0,n can be unbounded. In this case, we are going to use Theorems 6.2.2 and 6.2.4, which give conditions of convergence for, respectively, log-reward functions and optimal expected rewards for univariate modulated Markov log-price processes with unbounded characteristics. In fact, we shall use reduced variants of these theorems for the model without index component. Let us recall the class A of measurable, nonnegative, and nondecreasing func­ tions A(β ) defined for β ≥ 0 and such that A(0) = 0. Any function A(β ) ∈ A generates a nondecreasing sequence of functions A0 (β ) ≤ A1 (β ) ≤ A2 (β ) ≤ · · · from the class A that are defined by the follow­ ing recurrence relation, for every β ≥ 0: ⎧ ⎨β for n = 0 , A n (β) = ⎩ (7.200) A n−1(β ) + A(A n−1 (β )) for n = 1, 2, . . . . Let us also recall the first-type A-modulus of exponential moment compactness for the log-price process Y0,n , defined for β ≥ 0, Δ β,A (Y0,· , N ) = max sup

0≤ n ≤ N −1 y ∈R1

Ey,n e A N −n−1 (β)| Y0,n+1 −Y0,n | . e A(A N −n−1(β))| y|

(7.201)

Recall that we use the notations Py,n and Ey,n for conditional probabilities and expectations under the condition Y n = y. Note that modulus Δ β,A (Y0,· , N ) coincides with modulus Δ β (Y0,· , N ), if function A(β ) = 0. Indeed, in this case all functions A(A n (β )) ≡ 0 and A n (β ) ≡ β for n = 0, 1, . . . . Instead of the condition C11 [β ] based on the modulus Δ β (Y0,· , N ), we use the fol­ lowing first-type condition of exponential moment compactness for the log-price pro­ cess Y 0,n , which is assumed to hold for some β ≥ 0:

7.5 Multivariate Markov LPP with unbounded characteristics

| 295

C15 [β ]: Δ β,A (Y0,· , N ) < K60 , for some 1 < K60 < ∞ . Let us also recall the second-type A-modulus of exponential moment compactness for the log-price process Y0,n , for β ≥ 0, Ξ± β,A ( Y 0, · , N ) = max sup

0≤ n ≤ N −1 y ∈R1

Ey,n e±A N −n−1 (β)(Y0,n+1 −Y0,n ) . e A(A N −n−1 (β))| y|

(7.202)

The following second-type condition of exponential moment compactness is as­ sumed to hold for some β ≥ 0: E19 [β ]: Ξ± β,A ( Y 0, · , N ) < K 61 , for some 1 < K 61 < ∞ . The following lemma is a corollary of Lemma 4.3.7. Lemma 7.5.1. The condition E19 [β ] implies the condition C15 [β ] to hold. Let us also recall the sequences of the intervals Zε = Zε,0 , . . . , Zε,N > introduced in relation (7.51). Instead of the modulus of exponential moment compactness Δ β,Zε (Y ε,· , N ) we re­ call the modified first-type A-modulus of exponential moment compactness, defined for every ε ∈ [0, ε0 ] and β ≥ 0 by the following formula: Δ β,A,Zε (Y ε,· , N ) = max

sup

0≤ n ≤ N −1 y ∈Zε,n

Ey,n e A N −n−1 (β)| Y ε,n+1 −Y ε,n | . e A(A N −n−1(β))| y|

(7.203)

Theorem 7.5.1. Let a Markov log-price process Y 0,n and the corresponding space-skele­ ton approximation Markov processes Y ε,n be defined as described in Subsection 7.2.1. Also let the conditions B4 [𝛾] and C15 [β ] hold with the parameters 𝛾 and β such that ei­ ther β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L2 , N1 , I6 , and J9 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n, ϕ ε,n (y ε ) → ϕ0,n (y0 ) as

ε → 0.

(7.204)

Proof. We are going to check that all conditions of Theorem 6.2.2 hold and to apply this theorem. The first part of the proof is similar with the corresponding part in the proof of Theorem 7.2.1. The only difference that we use the modulus Δ β,A,Zε (Y ε,· , N ) instead on the modulus Δ β,Zε (Y ε,· , N ), Using relations (7.52), (7.55), and (7.56) we get the following inequalities (replac­ ing inequalities (7.57) given in the proof of Theorem 7.2.1), for every ε ∈ (0, ε0 ], n = 0, . . . , N − 1, and β ≥ 0, sup y ∈Zε,n

Ey,n e A N −n−1 (β)| Y ε,n+1 −Y ε,n | e A(A N −n−1 (β))| y|

= sup Eh ε,n (y),n y ∈Zε,n

e A N −n−1 (β)| h ε,n+1 (Y0,n+1 )−y| e A(A N −n−1 (β))| y|

296 | 7 Space-skeleton reward approximations & = sup Eh ε,n (y),n y ∈Zε,n

+ e A N −n−1 (β)δ ε,n+1 +

e A N −n−1 (β)(y−Y0,n+1 ) − I e A(A N −n−1 (β))| y| ε,n e A N −n−1 (β)| Y0,n+1 −y| I ε,n e A(A N −'n−1 (β))| y|

e A N −n−1 (β)(Y0,n+1 −y) + I e A(A N −n−1(β))| y| ε,n

≤ e A N −n−1 (β)δ ε,n+1 sup Eh ε,n (y),n y ∈Zε,n

≤e

A N −n−1 ( β) δ ε,n+1 A N −n−1 ( β) δ ε,n

e

× sup Eh ε,n (y),n y ∈Zε,n

≤e

e A N −n−1 (β)| Y0,n+1 −h ε,n (y)| e A(A N −n−1(β))| y|

A N −n−1 ( β) δ ε,n+1 A N −n−1 ( β) δ ε,n A ( A N −n−1 ( β)) δ ε,n

e

× sup Eh ε,n (y),n y ∈R1

≤e

e A N −n−1 (β)| Y0,n+1 −y| e A(A N −n−1 (β))| y|

e

e A N −n−1 (β)| Y0,n+1 −h ε,n (y)| e A(A N −n−1(β))| h ε,n (y)|

A N −n−1 ( β) δ ε,n+1 + A N −n−1 ( β) δ ε,n + A ( A N −n−1 ( β)) δ ε,n

× sup

y ∈R1

Ey,n e A N −n−1 (β)| Y0,n+1 −y| , e A(A N −n−1 (β))| y|

(7.205)

where, for n = 0, . . . , N − 1, − I− ε,n = I ( Y 0,n +1 ≤ z ε,n +1) , + I ε,n = I (z− ε,n +1 < Y 0,n +1 ≤ z ε,n +1) ,

I+ ε,n

= I (Y 0,n+1 >

(7.206)

z+ ε,n +1) .

This relation implies the following inequality, for the moduli Δ β,A,Zε (Y ε,· , N ) and Δ β,A (Y0,· , N ), which holds for every ε ∈ (0, ε0 ] and β ≥ 0, Δ β,A,Zε (Y ε,· , N ) ≤

max e A N −n−1 (β)δ ε,n+1 +A N −n−1 (β)δ ε,n +A(A N −n−1(β))δ ε,n × Δ β,A (Y0,· , N ) .

0≤ n ≤ N − 1

(7.207)

By the condition N1 , max e A N −n−1 (β)δ ε,n+1 +A N −n−1 (β)δ ε,n +A(A N −n−1(β))δ ε,n → 1 as

0≤ n ≤ N − 1

ε → 0.

(7.208)

Inequality (7.207), relation (7.208), and the condition C15 [β ] imply the following relation: lim Δ β,A,Zε (Y ε,· , N ) ≤ Δ β,A (Y0,· , N ) < K60 . (7.209) ε →0

Therefore, the condition C9 [β ] (its variant for the model without modulating in­ dex component) holds for the modified moduli of exponential moment compactness Δ β,A,Zε (Y ε,· , N ).

7.5 Multivariate Markov LPP with unbounded characteristics

| 297

The following part of the proof repeats the corresponding part of the proof for Theorem 7.2.1. In particular, it was shown in the proof of Theorem 7.2.1 that the condition I6 implies that the condition I1 (its univariate variant without modulating component) holds. But, this reduced variant of the condition I1 just coincides with the condition I5 . Analogously, it was shown in the proof of Theorem 7.2.1 that the condition J9 implies that the condition J1 (its univariate variant without modulating component) holds. But, this reduced variant of the condition J1 just coincides with the condition J8 . It follows from the above remarks that all conditions of Theorem 6.2.2 hold, and, therefore, Theorem 7.4.1 is a corollary of Theorem 6.2.2.

7.5.2 Convergence of space-skeleton approximations for optimal expected rewards of univariate Markov log-price processes with unbounded characteristics We continue consider the model of one-dimensional Markov log-price processes intro­ duced in Subsection 7.2.1. In this case, the condition D10 [β ] should be replaced by the following condition assumed to hold for some β ≥ 0: D12 [β ]: E exp{A N (β )|Y0,0 |} < K62 , for some 1 < K62 < ∞ . The following result can be obtained by application Theorem 6.2.4. Theorem 7.5.2. Let a Markov log-price process Y 0,n and the corresponding space-skele­ ton approximation Markov processes Y ε,n are defined in Subsection 7.2.1. Let also the conditions B4 [𝛾], C15 [β ], and D12 [β ] hold with the parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions L2 , M2 , N1 , I6 , J9 , and K8 hold. Then, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(7.210)

Proof. It repeats the proof of Theorem 7.2.2. The only difference is that the condition D12 [β ] should be used instead of the condition D10 [β ] that let one get the inequality analogous to (7.71), lim Ee A N (β)| Y ε,0 | ≤ lim e A N (β)δ ε,0 Ee A N (β)| Y0,0 | < K62 . ε →0

ε →0

(7.211)

In the above relation, we used the relation e A N (β)δ ε,0 → 1 as ε → 0 implied by the condition N1 . As was pointed out in the proof of Theorem 7.2.2, relation (7.44) implies that, for every ε ∈ (0, ε0 ], P{Y ε,0 ∈ Zε,0} = P{h ε,0(Y0,0 ) ∈ Zε,0} = 1 .

(7.212)

298 | 7 Space-skeleton reward approximations Relations (7.211) and (7.212) imply that the condition D8 [β ] (its variant for the model without modulating index component) holds. The remaining part of the proof repeats the corresponding part of the proof for Theorem 7.2.2.

7.5.3 Space-skeleton approximations for multivariate modulated Markov log-price processes with unbounded characteristics ε,n , In this subsection, we consider the same model of log-price processes  Z ε,n = (Y X ε,n ), n = 0, 1, . . . , and the same skeleton approximation model as those considered in Subsection 7.3.1. The only difference is that we assume that the corresponding char­ acteristics of the process  Z ε,n can be unbounded. In this case, we are going to use Theorem 6.4.2 instead of Theorem 6.2.2.  (β¯ ) = (A1 (β¯ ), Let us recall the class Ak of measurable, vector functions A ¯ ¯ . . . , A k (β )) defined for β = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k such that func­ tions A i (β¯ ), i = 1, . . . , k are nondecreasing in every argument β i , i = 1, . . . , k and ¯ ) = 0, i = 1, . . . , k. A i (0  (β¯ ) ∈ Ak generates a sequence of functions A  n (β¯ ) = (A n,1(β¯ ), Any function A ¯ . . . , A n,k (β )), n = 0, 1, . . . from the class Ak that are defined by the following recur­ rence relation, for any β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k, ⎧ ⎨ β¯ for n = 0 ,  n (β¯ ) = A (7.213) ⎩A  (A  n−1 (β¯ )) for n = 1, 2, . . . .  n−1 (β¯ ) + A

Note that, by the definition, 0 = A0,j (β¯ ) ≤ A1,j (β¯ ) ≤ · · · for every β¯ = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k and j = 1, . . . , k.  -modulus of exponential moment compactness for the Let us recall the first-type A ¯  log-price process Y0,n , for β = (β 1 , . . . , β k ), β i ≥ 0, i = 1, . . . , k,  Δ β, ¯ A  ( Y 0, · , N ) =

max

sup

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Z

Ez ,n e

k

j =1

e

k

A N −n−1,j ( β¯ )| Y 0,n+1,j − Y 0,n,j |

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(7.214)

Recall that we use the notations Pz ,n and Ez ,n for conditional probabilities and Zn =  z. expectations under the condition  0, · , N ), we use the fol­ Instead of the condition C12 [β¯ ] based on the modulus Δ β (Y lowing first-type condition of exponential moment compactness for the log-price pro­ 0,n , which is assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with cess Y nonnegative components and vectors β¯ i = (0, . . . , 0, β i , 0, . . . , 0), i = 1, . . . , k: 0, · , N ) < K63,i , i = 1, . . . , k, for some 1 < K63,i < ∞, i = 1, . . . , k . C16 [β¯ ]: Δ β¯ i , A (Y

7.5 Multivariate Markov LPP with unbounded characteristics

| 299

 -modulus of exponential moment compactness for the Let us recall the second-type A 0,n , defined for vector β¯ = (β 1 , . . . , β k ) with components of the log-price process Y nonnegative components and vectors ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , where Ik = {¯ı k : ı j = +, −, j = 1, . . . , k }. ¯ı k  Ξ β, ¯ A  ( Y 0, · ,

N ) = max

sup

Ez ,n e

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Z

k

e

¯

j =1 ı j A N −n−1,j ( β)( Y n+1,j − Y n,j )

k

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(7.215)

The following second-type condition of exponential moment compactness is as­ sumed to hold for some vector parameter β¯ with nonnegative components: ¯ı k 0, · , N ) < K64,i , ¯ı k = (ı1 , . . . , ı k ) ∈ Ik , i = 1, . . . , k, for some 1 < K64,i < E20 [β¯ ]: Ξ β¯ , A (Y i ∞, i = 1, . . . , k . The following lemma is a corollary of Lemma 4.3.17. Lemma 7.5.2. The condition E20 [β ] implies the condition C16 [β ] to hold. Let us recall the sequences of the sets Zε = Zε,0 , . . . , Zε,N  introduced in relation (7.93).  -modulus of exponential moment com­ Let us also recall the modified first-type A pactness, defined for every ε ∈ [0, ε0 ] and a vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components by the following formula:  Δ β, ¯ A  , Zε ( Y ε, · , N ) =

max

sup

Ez ,n e

0≤ n ≤ N − 1  z=(( y 1 ,...,y k ) , x )∈Zε,n

k

j =1

e

k

A N −n−1,j ( β¯ )| Y ε,n+1,j − Y ε,n,j |

j =1

 N −n−1 ( β¯ ))| y j | A j (A

.

(7.216)

Z0,n and the Theorem 7.5.3. Let a Markov multivariate modulated log-price process  corresponding space-skeleton approximation Markov processes  Z ε,n be defined as in Subsection 7.3.1. Let also the conditions B1 [𝛾¯] and C16 [β¯ ] hold with the vector param­ eters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions L4 , N2 , I7 , and J10 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn ∩ Z , n ϕ ε,n ( y ε , x ε ) → ϕ0,n ( y0 , x0 )

as

ε → 0.

(7.217)

Proof. We are going to check that all conditions of Theorem 6.4.2 hold and then to apply this theorem. The first part of the proof is similar with the corresponding part in the proof of  Theorem 7.3.2. The only difference is that we use the modulus Δ β, ¯ A  , Zε ( Y ε, · , N ) instead  on the modulus Δ β, ¯ Z ( Y ε, · , N ). ε

300 | 7 Space-skeleton reward approximations Using relations (7.94), (7.97), and (7.98), we get the following inequalities for every ε ∈ (0, ε0 ] and i = 1, . . . , k, n = 0, . . . , N − 1, sup

Ez ,n e

k

j =1

e

 z =(( y 1 ,...,y k ) , x )∈Zε,n

k

A N −n−1,j ( β¯ )| Y ε,n+1,j − Y ε,n,j |

j =1

 N −n−1 ( β¯ ))| y j | A j (A

Eh¯ ε,n (z),n e

= sup

 z ∈Zε,n

e

k

j =1

+ ≤

k 

k

j =1

k #  j =1

A N −n−1,j ( β¯ ) δ ε,n+1,j A N −n−1,j ( β¯ )| Y 0,n+1,j − y j |

e

¯ e A N −n−1,j (β)(Y0,n+1,j −y j ) I + ε,n,j ¯

e A N −n−1,j (β)δ ε,n+1,j sup  z∈Zε,n

× Eh¯ ε,n (z),n ≤

¯

e A N −n−1,j (β)(y j −Y0,n+1,j ) I − ε,n,j

j =1

k 

 N −n−1 ( β¯ ))| y j | A j (A

 N −n−1 ( β¯ ))| y j | A j (A

× Eh¯ ε,n (z),n +e

A N −n−1,j ( β¯ )| h ε,n+1,j ( Y 0,n+1,j )− y j |

j =1

e 1

 z ∈Zε,n

= sup

k

k 

e



k

I ε,n,j

1

¯  j =1 A j ( A N −n−1 ( β ))| y j |

¯

e A N −n−1,j (β)| Y0,n+1,j −y j |

j =1

¯

¯



¯

e A N −n−1,j (β)δ ε,n+1,j e A N −n−1,j (β)δ ε,n,j e A j (A N −n−1 (β))δ ε,n,j

j =1

× sup

 z∈Zε,n

e

≤

 N −n−1 ( β¯ ))| h ε,n,j ( y j )| A j (A

j =1

× Eh¯ ε,n (z),n k 

1

k

k 

¯

e A N −n−1,j (β)| Y0,n+1,j −h ε,n,j (y j )|

j =1

¯

¯



¯

e A N −n−1,j (β)δ ε,n+1,j e A N −n−1,j (β)δ ε,n,j e A j (A N −n−1 (β))δ ε,n,j

j =1

× sup

Ez ,n e

 z∈Zε,n

k

e

j =1

k

j =1

A N −n−1,j ( β¯ )| Y 0,n+1,j − y j |  N −n−1 ( β¯ ))| y j | A j (A

,

(7.218)

where, for i = 1, . . . , k, n = 0, . . . , N − 1, − I− ε,n,i = I ( Y 0,n +1,i ≤ z ε,n +1,i) , + I ε,n,i = I (z− ε,n +1,i < Y 0,n +1,i ≤ z ε,n +1,i) ,

I+ ε,n,i

= I (Y 0,n+1,i >

z+ ε,n +1,i) .

(7.219)

7.5 Multivariate Markov LPP with unbounded characteristics

| 301

Let us denote δ ε (β¯ ) = max (e 0≤ n ≤ N − 1 k

×e

j =1

k

j =1

A N −n−1,j ( β¯ ) δ ε,n+1,j

A N −n−1,j ( β¯ ) δ ε,n,j

e

k

j =1

 N −n−1 ( β¯ )) δ ε,n,j A j (A

).

(7.220)

Relation (7.218) implies the following inequality, which holds for every ε ∈ (0, ε0 ] and vectors β¯ with nonnegative components, ¯ ¯  (Y  0, · , N ) . Δ β, ¯ A  , Zε ( Y ε, · , N ) ≤ δ ε ( β ) Δ β, A

(7.221)

Since δ ε (β¯ i ) → 0 as ε → 0, for i = 1, . . . , k, by the condition N4 , inequality (7.221) and the condition C16 [β¯ ] imply the following relation, for i = 1, . . . , k: ε, · , N ) ≤ Δ ¯  (Y 0, · , N ) < K63,i . lim Δ β¯ i , A ,Zε (Y βi , A ε →0

(7.222)

Therefore, the condition C10 [β¯ ] holds for the modified modula of exponential mo­ ment compactness Δ β¯ i , A ,Zε (Y ε,· , N ), i = 1, . . . , k. The following part of the proof repeats the corresponding part of the proof for Theorem 7.2.3. In particular, it was shown in the proof of Theorem 7.2.3 that the condition I7 im­ plies that the condition I1 holds. Analogously, it was shown in the proof of Theorem 7.2.3 that the condition J10 im­ plies that the condition J1 . It follows from the above remarks that all conditions of Theorem 6.4.2 hold, and, therefore, Theorem 7.4.1 is a corollary of Theorem 6.4.2.

7.5.4 Convergence of space-skeleton approximations for optimal expected rewards for multivariate modulated Markov log-price processes with unbounded characteristics We continue consider the model of multidimensional modulated Markov log-price pro­ cesses introduced in Subsection 7.3.1. In this case, the condition D11 [β¯ ] should be replaced by the following condition assumed to hold for some vector β¯ = (β 1 , . . . , β k ) with nonnegative components and vectors β¯ i = (0, . . . , 0, β i , 0, . . . , 0) = (β i,1 , . . . , β i,k ), i = 1, . . . , k: k ¯ D13 [β¯ ] Ee j=1 A N,j (β i )| Y0,0,j | < K65,i , i = 1, . . . , k, for some 1 < K65,i < ∞, i = 1, . . . , k . Z0,n and the cor­ Theorem 7.5.4. Let a Markov multivariate modulated log-price process  Z ε,n are defined as in Sub­ responding space-skeleton approximation Markov processes  section 7.3.1. Let also the conditions B1 [𝛾¯] C16 [β¯ ] and D13 [β¯ ] hold with the vector pa­ rameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either

302 | 7 Space-skeleton reward approximations β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions L4 , M4 , N2 , I7 , J10 , and K9 hold. Then, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(7.223)

Proof. This theorem is a corollary of Theorem 6.4.4. The proof is analogous to the proof of Theorem 7.3.2. The only difference is that the condition D13 [β¯ ] should be used in­ stead of the condition D11 [β¯ ] that let one get the inequality analogous to (7.120), for i = 1, . . . , k, lim Ee ε →0

k

j =1

A N,j ( β¯ i )| Y ε,0,j |

≤ lim e ε →0

k

j =1

A N,j ( β¯ i ) δ ε,0,j

Ee

k

k

j =1

A N,j ( β¯ i )| Y 0,0,j |

< K65,i .

(7.224)

¯

In the above relation, we used the relation e j=1 A N,j (β i )δ ε,0,j → 1 as ε → 0 implied by the condition N2 . As was pointed out in the proof of Theorem 7.3.2, relation (7.87) implies that, for every ε ∈ (0, ε0 ], ¯ ε,0( ε,0 ∈ Zε,0} = P{h P{ Z Z0,0 ) ∈ Zε,0} = 1 .

(7.225)

Relations (7.224) and (7.225) imply that the condition D9 [β¯ ] holds. The remaining part of the proof repeats the corresponding part of the proof for Theorem 7.3.2.

8 Convergence of rewards for Markov Gaussian LPP In this chapter, we present results about convergence of rewards for American-type options for discrete time modulated Markov Gaussian log-price processes. These pro­ cesses play, in the class of discrete time Markov log-price processes, the role, analo­ gous to those played by diffusion processes in the class of continuous time Markov log-price processes. They can also be effectively used for constructing of stochastic approximations for rewards of diffusion-type log-price processes. In Section 8.1, we discuss results about convergence of rewards for perturbed, univariate modulated Markov Gaussian log-price processes with bounded and un­ bounded (linear bounded) drift and volatility coefficients. These results are applied to Markov Gaussian log-price processes with estimated parameters, in Section 8.3. In Section 8.2, we present results about convergence of rewards for perturbed, multivariate modulated Markov Gaussian log-price processes with bounded and un­ bounded (linear bounded) drift and volatility coefficients. In Section 8.3, we consider Markov Gaussian log-price processes with random drift and volatility coefficients given in the form of consistent statistical estimates and present results about consistency of the corresponding estimates for option rewards. In Section 8.4, we present results about convergence of space-skeleton reward ap­ proximations for multivariate modulated Markov Gaussian log-price processes with bounded and unbounded (linear bounded) drift and volatility coefficients. In Section 8.5, we present results about convergence of space-skeleton reward ap­ proximations for log-price processes represented by multivariate modulated Markov Gaussian random walks. The main results are given in Theorems 8.3.1–8.3.8 and 8.4.1–8.4.8. These results are new.

8.1 Univariate Markov Gaussian LPP In this section, we presents general results about convergence of rewards for Ameri­ can-type options for Markov Gaussian log-price processes.

8.1.1 Asymptotically uniform upper bounds for option rewards for univariate Markov Gaussian log-price processes with bounded characteristics Let us assume that a modulated log-price process Y ε,n , n = 0, 1, . . . is given, for every ε ∈ [0, ε0 ], by the following stochastic transition dynamic relation Y ε,n = Y ε,n−1 + μ ε,n (Y ε,n−1) + σ ε,n (Y ε,n−1)W ε,n ,

n = 1, 2, . . . ,

(8.1)

304 | 8 Convergence of rewards for Markov Gaussian LPP where (a) Y ε,0 is a real-valued random variable, (b) W ε,n , n = 1, 2, . . . is a sequence of standard i.i.d. normal random variables with mean 0 and variance 1, (c) the random variable Y ε,0 and the sequence of random variables W ε,n , n = 1, 2, . . . are indepen­ dent, (d) μ ε,n (y) are real-valued Borel functions defined on R1 , for n = 1, 2, . . . , and (e) σ ε,n(y) are real-valued Borel functions defined on R1 , for n = 1, 2, . . . . This is a model without an index component, and, therefore, the pay-off function g ε (n, e y ) is also assumed to be, for every ε ∈ [0, ε0 ], a function of the argument y. In sequel, the log-reward functions ϕ ε,n (y) are also functions argument y. Let us give upper bounds for option rewards for the model, where the following condition holds: G5 : limε→0 max1≤n≤N supy∈R1 (|μ ε,n (y)| + σ 2ε,n (y)) < K66 , for some 0 < K66 < ∞. In this case, there exist the moment-generating functions Ey,n e β(Y ε,n+1 −Y n ) defined, for every ε ∈ [0, ε0 ] and y ∈ R1 , n = 0, 1, . . . , N − 1, Ey,n e β(Y ε,n+1 −Y ε,n ) = Ee β(μ ε,n+1 (y) +σ ε,n+1 (y) W ε,n+1 ) 1

= e βμ ε,n+1 (y) + 2 β

2 2 σ ε,n+1 ( y )

β ∈ R1 .

,

(8.2)

The second-type modulus of exponential moment compactness Ξ± β ( Y ε, · , N ) takes, according to formula (8.2), the following form, for β ≥ 0: 1

± βμ ε,n+1 ( y ) + 2 β Ξ± β ( Y ε, · , N ) = max sup e

2 2 σ ε,n+1 ( y )

0≤ n ≤ N −1 y ∈R1

.

(8.3)

We are interested in the following condition, assumed to hold for some β ≥ 0: E21 [β ]: limε→0 Ξ± β ( Y ε, · , N ) < K 67 , for some 1 < K 67 < ∞. Lemma 8.1.1. The condition G5 implies that the condition E21 [β ] holds for any β ≥ 0. Proof. It is analogous to the proof of Lemma 4.4.1. The condition G5 and relations (8.2) and (8.3) imply that for any β ≥ 0, 1

± βμ ε,n+1 ( y ) + 2 β lim Ξ± β ( Y ε, · , N ) = lim max sup e

ε →0

2 2 σ ε,n+1 ( y )

ε →0 0≤ n ≤ N −1 y ∈R1

< K67 (β ) =

1 2 1 + e K 66 (β+ 2 β ) < ∞ . 2

(8.4)

Thus, the condition G5 implies that the condition E21 [β ] holds for any β ≥ 0. Remark 8.1.1. The constant K67 = K67 (β ) is determined by the constant K66 via for­ mula (8.4). We assume that the condition B7 [𝛾] holds for pay-off functions.

8.1 Univariate Markov Gaussian LPP |

305

Theorem 8.1.1. Let the condition G5 holds and, also, the condition B7 [𝛾] holds for some 𝛾 ≥ 0. Then, there exist 0 < ε19 ≤ ε0 and the constants 0 ≤ M43 , M44 < ∞ such that the log-reward functions ϕ ε,n (y) satisfy the following inequality, for y ∈ R1 , 0 ≤ n ≤ N and 0 ≤ ε ≤ ε19 : |ϕ ε,n (y) | ≤ M43 + M44 e𝛾| y| . (8.5) Proof. This theorem is a corollary of Theorem 5.1.1. Indeed, according to Lemma 8.1.1, the condition G5 implies that the condi­ tion E21 [β ] holds for any β ≥ 0. This condition is a variant of the condition E10 [β¯ ] (its univariate variant for the model without index component). According to Lemma 5.1.6, this condition implies that a reduced variant of the con­ dition C6 [β¯ ] (its univariate variant for the model without index component) holds. Moreover, the condition G5 implies that there exists 0 < ε20 ≤ ε0 such that for 0 ≤ ε ≤ ε20 , max sup (|μ ε,n (y)| + σ 2ε,n (y)) < K66 . (8.6) 1≤ n ≤ N y ∈R1

By Lemmas 4.1.8, Remark 4.1.7, and Lemma 8.1.1, the following inequality holds for any β ≥ 0 and 0 ≤ ε ≤ ε20 : 1

Δ β (Y ε,· ,i(·), N ) < K68 = 2K67 (β ) = 1 + 2e K 66 (β+ 2 β ) . 2

(8.7)

Also, the condition B7 [𝛾] is a reduced variant of the condition B5 [𝛾¯] (its univariate variant for the model without index component). This condition implies that there exists 0 < ε21 ≤ ε0 such that for 0 ≤ ε ≤ ε21 , max sup

0≤ n ≤ N y ∈R1

|g ε (n, e y )| < L13 . 1 + L14 e𝛾| y|

(8.8)

Note that we can always choose β ≥ 𝛾 that is required in Theorem 5.1.1. Thus, Theorem 5.1.1 can be applied that yields inequalities (8.5). Remark 8.1.2. The explicit formulas for constants M43 , M44 can be obtained by using formulas given in Remark 5.1.6, for constants M32 and M33,1, in the univariate case k = 1. They take the following form: M43 = L13 ,

𝛾

M44 = L13 L14 I (𝛾 = 0) + L13 L14 (2K67 (β ))N β I (𝛾 > 0) .

(8.9)

One can choose any β ≥ 𝛾 in formula (8.9). Remark 8.1.3. The parameter ε19 can be obtained by using formulas given in Re­ mark 5.1.7 for the parameter ε3 . It takes the following form: ε19 = min(ε20 , ε21 ), where the parameters ε20 and ε21 replace the parameters ε1 and ε2 penetrating Remark 5.1.7 and are determined, respectively, by relations (8.6) and (8.8).

306 | 8 Convergence of rewards for Markov Gaussian LPP Let us now impose the following condition on the distribution of random variables Y ε,0, assumed to hold for some β ≥ 0: D14 [β ]: limε→0 Ee β| Y ε,0 | < K69 , for some 1 < K69 < ∞. The following theorem gives asymptotically uniform upper bounds for optimal ex­ pected rewards Φ ε . Theorem 8.1.2. Let the condition G5 holds and, also, the conditions B7 [𝛾] and D14 [β ] hold for some β ≥ 𝛾 ≥ 0. Then, there exist 0 < ε22 ≤ ε0 and constants 0 ≤ M45 < ∞ such that the following inequality holds for 0 ≤ ε ≤ ε22 : |Φ ε | ≤ M45 .

(8.10)

Proof. This theorem follows from Theorem 5.1.3. Indeed, the condition D14 [β ] is a reduced variant of the condition D7 [β¯ ] (its univariate variant for the model without index component). Moreover, the condi­ tion D14 [β ] implies that there exists 0 < ε23 ≤ ε0 such that for 0 ≤ ε ≤ ε23 , Ee β| Y ε,0,i | < K69 .

(8.11)

Other conditions of this theorem also holds that was pointed out in the proof of Theorem 8.1.1. Thus, Theorem 5.1.3 can be applied that yields inequality (8.10). Remark 8.1.4. The explicit formulas for the constant M45 can be obtained with the use of formula given in Remark 5.1.13, for the constant M34 , in the univariate case where k = 1. It takes the following form: 𝛾

𝛾 β

M45 = L13 + L13 L14 I (𝛾 = 0) + L13 L14 (2K67 (β ))N β K69 I (𝛾 > 0) .

(8.12)

Remark 8.1.5. The parameter ε22 can be obtained with the use of formulas given in Re­ mark 5.1.14 for the parameter ε5 . It takes the following form, ε22 = min(ε19 , ε23 ), where the parameter ε23 , determined by relation (8.11), replaces the parameter ε4 penetrating Remark 5.1.14.

8.1.2 Convergence of option rewards for univariate Markov Gaussian log-price processes with bounded characteristics We impose the following convergence condition on pay-off functions: I8 : There exist sets Yn ∈ B1 , n = 0, . . . , N, such that the pay-off function g ε (n, e y ε ) → g0 (n, e y0 ) as ε → 0 for any y ε → y0 ∈ Yn as ε → 0, n = 0, . . . , N.

8.1 Univariate Markov Gaussian LPP

| 307

It is useful to note that, in this case, the log-price process Y ε,n has Gaussian one-step transition probabilities, i.e. for y ∈ R1 , A ∈ B1 , n = 1, 2, . . . , P ε,n (y, A) = P{y + μ ε,n (y) + σ ε,n(y) W ε,n ∈ A} .

(8.13)

We impose on parameters of the log-price processes Y ε,n the following conver­ gence condition: O1 : There exist sets Y n ∈ B1 , n = 0, . . . , N, such that (a) μ ε,n (y ε ) → μ 0,n (y0 ) and σ ε,n (y ε ) → σ 0,n (y0 ) as ε → 0 for any y ε → y0 ∈ Y n −1 as ε → 0, and n = 1, . . . , N;   (b) P{y0 + μ 0,n (y0 ) + σ 0,n (y0 )W0,n ∈ Yn ∩ Y n } = 1, for every y 0 ∈ Y n −1 ∩ Y n −1 , n = 1, . . . , N, where Yn , n = 0, . . . , N are sets introduced in the condition I8 . Let us consider two important examples, where the condition O1 (a) holds. The first model is, where the drift and volatility coefficients μ ε,n(y) = μ ε,n and σ ε,n(y) = σ ε,n, for y ∈ R1 , n = 1, . . . , N, i.e. these coefficients do not depend on y. In this case, the condition O1 (a) reduces to the relations μ ε,n → μ 0,n and σ ε,n → σ 0,n as ε → 0, for n = 1, . . . , N. Note that, in this case, the condition G5 holds. Another model is, where the parameters μ ε,n (y) = μ ε,n + μ  ε,n y and σ ε,n ( y ) =    σ ε,n + σ ε,n σ n (y), where |σ n (y) | ≤ |y| for y ∈ R1 , n = 1, . . . , N. In this case, the    condition O1 (a) reduces to the relations με,n → μ 0,n , μ  ε,n → μ 0,n and σ ε,n → σ 0,n ,   σ ε,n → σ 0,n as ε → 0, for n = 1, . . . , N. In this case, the condition G5 does not holds, but the weaker condition G6 formu­ lated in the next subsection, does hold. As far as the condition O1 (b) is concerned, it obviously holds, if, for every y0 ∈   Yn−1 ∩Y n −1 , n = 1, . . . , N, either σ 0,n ( y 0 ) > 0 and the sets Y n , Y n have zero Lebesgue measure, or σ 0,n (y0 ) = 0 and y0 + μ 0,n (y0 ) ∈ Yn ∩ Y n. The following theorem gives conditions of convergence for the log-reward func­ tions ϕ ε,n (y). Theorem 8.1.3. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the conditions G5 , I8 , and O1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place, for any y ε → y0 ∈ Yn ∩ Y n: ϕ ε,n (y ε ) → ϕ0,n (y0 ) as

ε → 0.

(8.14)

Proof. This theorem follows from Theorem 5.2.1. As was pointed out in the proof of Theorem 8.1.1, in this case, the conditions C6 [β¯ ] and B5 [𝛾¯] (their univariate variants for the model without index component) hold. Note also that we can always choose β ≥ 𝛾 if 𝛾 > 0 or β = 0 if 𝛾 = 0 that is required in Theorem 5.2.1. The condition I8 is a reduced version of the condition I1 (its univariate variant for the model without index component).

308 | 8 Convergence of rewards for Markov Gaussian LPP Also, the condition O1 (a) is equivalent to the condition J1 (a) (its univariate vari­ ant for the model without index component) since convergence of expectations and standard deviations for Gaussian random variables is equivalent to their convergence in distribution. The condition O1 (b) is a reformulation of the condition J1 (b) (its uni­ variate variant for the model without index component). Thus, all conditions of Theorem 5.2.1 hold. By applying this theorem we get rela­ tion (8.14). Let us now impose the following convergence condition for the initial distributions P ε,0(A) = P{Y ε,0 ∈ A}: K12 : (a) P ε,0(·) ⇒ P0,0 (·) as ε → 0;   (b) P0,0 (Y0 ∩ Y 0 ) = 1, where Y0 and Y0 are the sets introduced in the condi­ tions I8 and O1 . The following theorem gives conditions of convergence for the optimal expected re­ wards Φ ε . Theorem 8.1.4. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0. Also let the conditions G5 , I8 , O1 , and K12 hold. Then, the following relation takes place: (8.15) Φ ε → Φ0 as ε → 0 . Proof. This theorem follows from Theorem 5.2.3. Indeed, the condition K12 is a variant of the condition K1 (its one-dimensional variant for the model without index component). Other conditions of Theorem 5.2.3 also hold that was pointed out in the proofs of Theorems 8.1.1–8.1.3. Thus, Theorem 5.2.3 can be applied that yields relation (8.15).

8.1.3 Asymptotically uniform upper bounds for option rewards of univariate Markov Gaussian log-price processes with unbounded characteristics In this subsection, we give asymptotically uniform upper bounds for option rewards for the model, where the following condition holds: G6 : limε→0 max1≤n≤N supy∈R1 K71 < ∞.

| μ ε,n ( y )|+ σ 2ε,n ( y ) 1+ K 71 | y |

< K70 , for some 0 < K70 < ∞ and 0 ≤

It is worth to note that the condition G6 is weaker than the condition G5 and reduces to this condition in the case where the constant K71 = 0.

8.1 Univariate Markov Gaussian LPP | 309

In this case, according to the results presented in Subsection 4.4.2, one can use the second-type modulus of exponential moment compactness Ξ± β,A ( Y ε, · , N ), defined for β ≥ 0 1

Ξ± β,A ( Y ε, · ,

2

2

e±A N −n−1 (β)μ ε,n+1 (y) + 2 A N −n−1 (β)σ ε,n+1 (y) N ) = max sup , 0≤ n ≤ N −1 y ∈R1 e A(A N −n−1 (β))| y|

(8.16)

where A0 (β ) ≤ A1 (β ) ≤ A2 (β ) ≤ · · · is a nondecreasing sequence of functions generated by function A(β ) = K70 K71 (β + 21 β 2 ) , β ≥ 0 with the use of the following recurrence relation, for every β ≥ 0: ⎧ ⎨β , for n = 0 , (8.17) A n (β) = ⎩ A n−1 (β ) + A(A n−1 (β )), for n = 1, 2, . . . . We are interested in the following condition, assumed to hold for some β ≥ 0: E22 [β ]: Ξ± β,A ( Y ε, · , N ) < K 72 , for some 1 < K 72 < ∞. Lemma 8.1.2. Condition G6 implies that the condition E22 [β ] holds for any β ≥ 0. Proof. It is analogous to the proof of Lemma 4.4.2. The condition G6 implies that there exists ε24 ∈ (0, ε0 ] such that, for any y ∈ R1 and ε ∈ [0, ε24 ], max (|μ ε,n+1(y)| + σ 2ε,n+1(y)) ≤ K70 + K70 K71 |y| .

0≤ n ≤ N − 1

(8.18)

Relation (8.18) implies that, for any y ∈ R1 , n = 0, . . . , N − 1, ε ∈ [0, ε24 ] and β ≥ 0, 1

± A N −n−1 (β )μ ε,n+1(y) + 2 A2N −n−1 (β )σ 2ε,n+1(y) #  ≤ A N −n−1(β ) + 21 A2N −n−1 (β ) (K70 + K70 K71 |y|) #  1 = K70 A N −n−1 (β ) + 2 A2N −n−1 (β )  # + K70 K71 A N −n−1 (β ) + 21 A2N −n−1 (β ) |y| #  1 = K70 A N −n−1 (β ) + 2 A2N −n−1 (β ) + A(A N −n−1(β ))|y| .

(8.19)

Relations (8.16) and (8.19) imply that for any y ∈ R1 , n = 0, . . . , N − 1, ε ∈ [0, ε24 ] and β ≥ 0, 1

K 70 ( A N −n−1 ( β)+ 2 A N −n−1 ( β)) lim Ξ± β,A ( Y · , N ) ≤ lim max e ε →0

2

ε → 0 0≤ n ≤ N − 1


0) . (8.21) β Proof. This theorem follows from Theorem 6.1.1. Indeed, according to Lemma 8.1.2, the condition G6 implies that the condi­ tion E23 [β ] holds for any β ≥ 0. This is a variant of the condition E13 [β ] (its variant for the model without index component) for the univariate Markov Gaussian log-price process Y ε,n. According to Lemma 6.1.5 this condition implies that a reduced variant of the condition C9 [β ] (its variant for the model without index component) holds. Also, the condition B7 [𝛾] is a reduced variant of the condition B8 [𝛾] (its variant for the model without index component). Note that we can always choose β ≥ 𝛾 that is required in Theorem 6.1.1. Thus, Theorem 6.1.1 can be applied that yields inequalities (8.21). Remark 8.1.7. The explicit formulas for constants M46 , M47 (β ) can be obtained with the use of formulas given in Remarks 6.1.6 and 6.1.7. These formulas are analogous to formulas (8.12) and take the following form: M46 = L13 ,

𝛾

M47 (β ) = L13 L14 I (𝛾 = 0) + L13 L14 (2K72 (β ))N β I (𝛾 > 0) .

(8.22)

One can choose any β ≥ 𝛾 in formula (8.22). Remark 8.1.8. The parameter ε25 can be obtained with the use of formulas given in Remarks 6.1.6 and 6.1.7. This parameter is given by the formula ε25 = min(ε21 , ε24 ) analogous to those given in Remark 8.1.5 for the parameter ε22 . The following theorem gives asymptotically uniform upper bounds for optimal expected rewards Φ ε , for the model with unbounded characteristics. Theorem 8.1.6. Let the condition G6 holds and, also, the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 ≥ 0. Then, there exist 0 < ε26 ≤ ε0 and a constant 0 ≤ M48 < ∞ such that the following inequality holds for 0 ≤ ε ≤ ε26 : |Φ ε | ≤ M48 .

(8.23)

8.1 Univariate Markov Gaussian LPP | 311

Proof. This theorem follows from Theorem 6.1.3. All conditions of this theorems holds that was pointed out in the proof of Theo­ rems 8.1.2 and 8.1.5. Thus, Theorem 6.1.3 can be applied that yields inequality (8.23). Remark 8.1.9. The explicit formulas for the constant M48 is given by the following formula analogous to (8.12): 𝛾

𝛾 β

M48 = L13 + L13 L14 I (𝛾 = 0) + L13 L14 (2K72 (β ))N β K69 I (𝛾 > 0) .

(8.24)

Remark 8.1.10. Parameter ε26 can be obtained with the use of formulas given in Re­ marks 6.1.13 and 6.1.14. This parameter is given by the formula ε26 = min(ε25 , ε23 ) analogous to those given in Remark 8.1.5 for the parameter ε22 .

8.1.4 Convergence of option rewards for univariate Markov Gaussian log-price processes with unbounded characteristics The following theorem gives conditions of convergence for log-reward functions ϕ ε,n (y) for univariate Markov Gaussian log-price processes with unbounded char­ acteristics. Theorem 8.1.7. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the conditions G6 , I8 and O1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: (8.25) ϕ ε,n (y ε ) → ϕ0,n (y0 ) as ε → 0 . Proof. This theorem follows from Theorem 6.2.1. As was pointed out in the proof of Theorem 8.1.5, in this case, the conditions C9 [β ] and B8 [𝛾] (their variants for the model without index component) hold. The condition I8 is a variant of the condition I5 (its variant for the model without index component). The condition O1 (a) is equivalent to the condition J8 (a) (its variant for the model without index component). Also, the condition O1 (b) is, just, reformulation of the con­ dition J8 (b) (its variant for the model without index component). Note also that we can always choose β ≥ 𝛾 if 𝛾 > 0 or β = 0 if 𝛾 = 0 that is required in Theorem 6.2.1. Thus, Theorem 6.2.1 can be applied that yields relation (8.25). The following theorem gives conditions of convergence for optimal expected re­ wards Φ ε , for the model with unbounded characteristics.

312 | 8 Convergence of rewards for Markov Gaussian LPP Theorem 8.1.8. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0. Also let the conditions G6 , I8 , O1 , and K12 hold. Then, the following relation takes place: (8.26) Φ ε → Φ0 as ε → 0 . Proof. This theorem follows from Theorem 6.2.3. All conditions of this theorem hold that was pointed out in the proofs of Theorems 8.1.2, 8.1.4, 8.1.5, and 8.1.7. Thus, Theorem 6.2.3 can be applied that yields relation (8.26).

8.2 Multivariate modulated Markov Gaussian LPP In this subsection, we presents general results about convergence of rewards for Amer­ ican-type options for multivariate modulated Markov Gaussian log-price processes.

8.2.1 Asymptotically uniform upper bounds for option rewards of multivariate modulated Markov Gaussian log-price processes with bounded characteristics ε,n , X ε,n ), n = 0, 1, . . . Let us assume that a modulated log-price process  Z ε,n = (Y is, for every ε ∈ [0, ε0 ], a multivariate modulated Markov Gaussian log-price process given by the following stochastic transition dynamic relation: ⎧   ε,n−1 , X ε,n−1) + Σ ε,n (Y ε,n−1 , X ε,n−1)W  ε,n , ⎪ μ ε,n (Y ⎪ ⎪ Y ε,n = Y ε,n−1 +  ⎨ X ε,n = C ε,n (X ε,n−1, U ε,n ) , (8.27) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , ε,0 , X ε,0) = ((Y ε,0,1, . . . , Y ε,0,k)), X ε,0) is a random vector tak­ where (a)  Z ε,0 = (Y  ε,n , U ε,n ) = ((W ε,n,1, . . . , W ε,n,k), U ε,n ), ing values in the space Z = Rk × X, (b) (W n = 1, 2, . . . is a sequence of independent random vectors taking values in the space Rk × U, where U is a measurable space with a σ-algebra of measurable subsets BU , ε,0 , X ε,0) and the random sequence (W  ε,n , U ε,n ), n = 1, 2, . . . (c) the random vector (Y  are independent; (d) the random vector W ε,n = (W ε,n,1, . . . , W ε,n,k) has a standard multivariate normal distribution with EW ε,n,i = 0, EW ε,n,i W ε,n,j = I (i = j), i, j = 1, . . . , k, for every n = 1, 2, . . . , (e) the random variables U ε,1 , U ε,2 , . . . have a regular  ε,n = w conditional distribution G w (B) = P{U ε,n ∈ B/W  } (a probability measure in B ∈ BU for every w  ∈ Rk and a measurable function in w  ∈ Rk for every B ∈ BU ), μ ε,n( y , x) = (μ ε,n,1( y , x), . . . , μ ε,n,k( y , x)), n = 1, 2, . . . are for every n = 1, 2, . . . ; (e)  vector functions, which components are measurable functions acting from the space Z = Rk ×X to R1 , (f) Σ ε,n ( y , x) = σ ε,n,i,j( y , x) , n = 1, 2, . . . are k × k matrix functions

8.2 Multivariate modulated Markov Gaussian LPP

| 313

y , x), i, j = 1, . . . , k, which are measurable functions acting from with elements σ ε,n,i,j( the space Z = Rk × X to R1 , and (g) C ε,n (x, u ) is a measurable function acting from the space X × U to the space X. Z ε,n is a multivariate modulated Markov log-price process with an The process  index component. Therefore, a pay-off function g ε (n, ey , x) is also assumed to be a function of the argument  z = ( y , x) ∈ Z. In sequel, the log-reward functions ϕ ε,n( y , x) also are functions of the argument ( y , x) ∈ Z. In this subsection, we give upper bounds for option rewards for the model, where the following condition holds: G7 : limε→0 max1≤n≤N,i,j=1,...,k supz=(y ,x)∈Z(|μ ε,n,i( y , x)| + σ 2ε,n,i,j( y , x)) < K73 , for some 0 < K73 < ∞. z = ( y , x) ∈ Z, n = 0, 1, . . . , N − 1, i = 1, . . . , k, In this case, for every ε ∈ [0, ε0 ] and  there exists the moment-generating function Ez ,n e β(Y ε,n+1,i −Y ε,n,i ) , defined for β ∈ R1 , Ez ,n e β(Y ε,n+1,i −Y ε,n,i ) = Ee β(μ ε,n+1,i (y ,x)+ =e

1 βμ ε,n+1,i ( y ,x )+ 2

k

β

j =1

σ ε,n+1,i,j ( y ,x ) W ε,n+1,j )

k 2

j =1

σ 2ε,n+1,i,j ( y ,x )

.

(8.28)

The second-type modulus of exponential moment compactness Ξ± β ( Y ε,i, ·, N ) takes, according to formula (8.28), the following form, for β ≥ 0: Ξ± β ( Y ε,i, ·, N ) = max

1

sup

0≤ n ≤ N − 1  z=( y ,x )∈Z

e±βμ ε,n+1,i (y ,x)+ 2 β

2

k

j =1

σ 2ε,n+1,i,j ( y ,x )

.

(8.29)

We are interested in the following condition, assumed to hold for some vector pa­ rameter β¯ = (β 1 , . . . , β k ) with nonnegative components: E23 [β¯ ]: limε→0 Ξ± β i ( Y ε, · ,i , N ) < K 74,i , i = 1, . . . , k, for some 1 < K 74,i < ∞, i = 1, . . . , k. Lemma 8.2.1. The condition G7 implies that the condition E23 [β¯ ] holds for any vector parameter β¯ = (β 1 , . . . , β k ) with components β i ≥ 0, i = 1, . . . , k. Proof. It is analogous to the proof of Lemma 8.1.1. The condition G7 and relation (8.28) and (8.29) imply that for any β i ≥ 0, i = 1, . . . , k, lim Ξ± β i ( Y ε, · ,i , N ) = lim max ε →0

sup

ε → 0 0≤ n ≤ N − 1  z=( y ,x )∈Z


𝛾i > 0 or β i = 𝛾i = 0 that is required in Theorem 5.2.1.

316 | 8 Convergence of rewards for Markov Gaussian LPP The condition O2 (a) and (b) imply due to relation (8.33) that the following relation holds any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Z n −1 and n = 1, . . . , N: #

  ε,n , C ε,n(x ε , U ε,n ) yε +  μ ε,n ( y ε , x ε ) + Σ ε,n ( y ε , x ε )W #  d  0,n , C0,n (x0 , U0,n ) −→ y0 +  μ 0,n ( y0 , x0 ) + Σ0,n ( y0 , x0 )W

as

ε → 0.

(8.35)

Relation (8.35) is equivalent to the condition J3 (a). The conditions O2 (b) and (c) are reformulations, respectively, for the conditions J3 (b) and (c). Therefore, the condition J3 holds and, thus, by Lemma 5.2.1, the condition J1 holds for the log-price processes  Z ε,n . All conditions of Theorem 5.2.1 hold. By applying this theorem we get relation (8.34). Let us now impose the following convergence condition on the initial distributions Z ε,0 ∈ A}: P ε,0(A) = P{ K13 : (a) P ε,0(·) ⇒ P0,0 (·) as ε → 0;   (b) P0,0 (Z0 ∩ Z 0 ) = 1, where Z0 and Z0 are the sets introduced in the condi­ tions I1 and O2 . The following theorem gives conditions of convergence for optimal expected re­ wards Φ ε . Theorem 8.2.4. Let the conditions B5 [𝛾¯] and D15 [β¯ ] hold for some vector parameters β¯ = (β 1 , . . . , β k ) and 𝛾¯ = (𝛾1 , . . . , 𝛾k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0. Also let the conditions G7 , I1 , O2 , and K13 hold. Then, the following relation takes place: Φ ε → Φ0 as ε → 0 . (8.36) Proof. This theorem follows from Theorem 5.2.3. Indeed, the condition K13 is a variant of the condition K1 . Other conditions of Theorem 5.2.3 also hold that was pointed out in the proof of Theorem 8.2.3. Thus, Theorem 5.2.3 can be applied that yields relation (8.36).

8.2.3 Asymptotically uniform upper bounds for option rewards for modulated multivariate Markov Gaussian log-price processes with unbounded characteristics In this subsection, we give asymptotically uniform upper bounds for option rewards for modulated multivariate Markov Gaussian log-price processes in the case, where

8.2 Multivariate modulated Markov Gaussian LPP

| 317

the following condition holds: | μ ε,n,i ( y ,x )|+ σ 2ε,n,i,j ( y ,x ) k 1+ l=1 K 77,l | y l |

G8 : limε→0 max1≤n≤N, i,j=1,...,k supz=(y ,x)∈Z K76 < ∞ and 0 ≤ K77,1 , . . . , K77,k < ∞.

< K76 , for some 0
𝛾i > 0 or β i = 𝛾i = 0 that is required in Theorem 6.4.1. The condition I1 is required in both Theorems 8.2.7 and 6.4.1. As was shown in Theorem 8.2.3, the condition O2 implies that the condition J1 holds. Thus, all conditions of Theorem 6.4.1 hold. By applying this theorem we get rela­ tion (8.47).

8.3 Markov Gaussian LPP with estimated characteristics

|

321

The following theorem gives conditions of convergence for optimal expected re­ wards Φ ε . Theorem 8.2.8. Let the conditions B5 [𝛾¯] and D15 [β¯ ] hold for some vector parameters β¯ = (β 1 , . . . , β k ) and 𝛾¯ = (𝛾1 , . . . , 𝛾k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0. Also let the conditions G8 , I1 , O2 , and K13 hold. Then, the following relation takes place: (8.48) Φ ε → Φ0 as ε → 0 . Proof. This theorem follows from Theorem 6.4.3. All conditions of this theorem hold that was pointed out in the proofs of Theorems 8.2.3, 8.2.4, and 8.2.7. Thus, Theorem 6.4.3 can be applied that yields relation (8.48).

8.3 Markov Gaussian LPP with estimated characteristics In this section, we consider the model, where drift and volatility functional coefficients of a Markov Gaussian log-price process depend on unknown parameter, for which con­ sistent statistical estimates are given. In this case, we show that the log-reward func­ tions and the optimal expected rewards for the log-price processes with estimated co­ efficients are consistent estimates for the corresponding log-reward functions and the optimal expected rewards for the log-price processes with the true coefficients. We re­ strict our consideration by univariate models. However, analogous results can also be obtained for the multivariate modulated Markov Gaussian log-price processes.

8.3.1 Option rewards for univariate Markov Gaussian log-price processes with estimated bounded parameters Let Θ be a Polish metric space with a metric ρ Θ (θ , θ ). In this case the space Θ ×R1 is also a Polish metric space with the metric ρ ((θ , y ), (θ , y )) = (ρ 2Θ (θ , θ ) + |y − 1

y |2 ) 2 , (θ , y ), (θ , y ) ∈ Θ × R1 . Let also BΘ be the Borel σ-algebra of subsets for space Θ. Let us consider the model, where the univariate Markov Gaussian log-price pro­ cesses Y ε,n (θ), n = 0, 1, . . . is given, for every θ ∈ Θ and ε ∈ [0, ε0 ], by the following stochastic transition dynamic relation: Y ε,n(θ) = Y ε,n−1(θ) + μ ε,n(θ, Y ε,n−1(θ)) + σ ε,n(θ, Y ε,n−1(θ))W ε,n ,

n = 1, 2, . . . ,

(8.49)

322 | 8 Convergence of rewards for Markov Gaussian LPP where (a) Y ε,0 (θ) ≡ Y ε,0, θ ∈ Θ is a real-valued random variable, (b) W ε,n , n = 1, 2, . . . is a sequence of standard i.i.d. normal random variables with mean 0 and variance 1, (c) the random variable Y ε,0 and the sequence of random variables W ε,n , n = 1, 2, . . . are independent, (d) μ ε,n (θ, y) are real-valued Borel functions acting from the space Θ × R1 to R1 , for n = 1, 2, . . . , and (e) σ ε,n(θ, y) are real-valued Borel functions acting from the space Θ × R1 to R1 , for n = 1, 2, . . . . Let us assume that the following condition of boundedness for pay-off functions holds: | μ ( θ,y )|+ σ 2 ( θ,y ) G9 : limε→0 supθ ∈Θ max1≤n≤N supy∈R1 ε,n K  (θ ) ε,n < 1, where K  (θ) is a continu­ ous function acting from the space Θ to interval (0, ∞). The condition G9 implies that there exists 0 < ε ≤ ε0 such that the following relation holds for 0 ≤ ε ≤ ε : max sup (|μ ε,n (θ, y)| + σ 2ε,n(θ, y)) < K  (θ) ,

1≤ n ≤ N y ∈R1

θ ∈ Θ.

(8.50)

Relation (8.50) implies that the condition G5 holds for trend and volatility coeffi­ cients μ ε,n (θ, y) and σ ε,n (θ, y), for every θ ∈ Θ. Moreover, in this case the constant K  (θ), replacing the constant K66 , depends on the parameter θ, while the the parame­ ter ε , which guarantees holding of inequality (8.50), for 0 ≤ ε ≤ ε , does not depend on the parameter θ. Let us also assume that the condition B7 [𝛾] holds for the pay-off functions g ε (n, e y ). Theorem 8.1.1 implies that the log-reward functions ϕ ε,n (θ, y), y ∈ R1 , n = 0, 1, . . . , N are well defined for the log-price process Y ε,n(θ), for every θ ∈ Θ. More­ over, it follows from relation (8.50) that, in this case, parameters ε19 and ε22 penetrat­ ing Theorem 8.1.1 does not depend on parameter θ. The following theorem follows from Theorem 8.1.1 and the above remarks. Theorem 8.3.1. Let the condition G9 holds and, also, the condition B7 [𝛾] holds for some 𝛾 ≥ 0. Then, for every θ ∈ Θ, there exist 0 < ε32 ≤ ε0 and constants 0 ≤ M55 , M56 = M56 (θ) < ∞ such that the log-reward functions ϕ ε,n (θ, y) satisfy the following inequal­ ities for y ∈ R1 , 0 ≤ n ≤ N and 0 ≤ ε ≤ ε32 : |ϕ ε,n (θ, y)| ≤ M55 + M56 (θ)e𝛾| y| .

(8.51)

Remark 8.3.1. The constants M55 , M56 = M56 (θ) are given by the following formulas, which follows from formulas (8.9): 

M55 = L13 ,

M56 (θ) = L13 L14 I (𝛾 = 0) + L13 L14 1 + 2e

1 K  ( θ )( β+ 2 β2 )

 N β𝛾

I (𝛾 > 0) . (8.52)

One can choose any β ≥ 𝛾 in formulas (8.52).

8.3 Markov Gaussian LPP with estimated characteristics

| 323

Remark 8.3.2. The parameter ε32 is given by the following formula ε32 = min(ε , ε21 ), where the parameters ε21 and ε are determined by relations (8.8) and (8.50). Let us now assume that the condition D14 [β ] holds. Theorem 8.1.2 implies that the optimal expected rewards Φ ε (θ) is well defined for the log-price process Y ε,n (θ), for every θ ∈ Θ. Moreover, it follows from relation (8.50) that, in this case, the parameter ε22 pen­ etrating Theorem 8.1.2 does not depend on parameter θ. The following theorem follows from Theorem 8.1.2 and the above remarks. Theorem 8.3.2. Let the condition G9 holds and, also, the conditions B7 [𝛾] and D14 [β ] hold for some β ≥ 𝛾 ≥ 0. Then, for every θ ∈ Θ, there exist 0 < ε33 ≤ ε0 and a constant 0 ≤ M57 = M57 (θ) < ∞ such that the following inequality holds for 0 ≤ ε ≤ ε33 : |Φ ε (θ)| ≤ M57 (θ) .

(8.53)

Remark 8.3.3. The constant M57 (θ) is given by the following formula: 

1

𝛾

𝛾 β

M57 (θ) = L13 + L13 L14 I (𝛾 = 0) + L13 L14 (1 + 2e K (θ )(β+ 2 β ) )N β K69 I (𝛾 > 0) . 2

(8.54)

Remark 8.3.4. The parameter ε33 is given by the following formula ε33 = min(ε , ε21 , ε23 ), where the parameters ε21 , ε23 , and ε are determined by relations (8.8), (8.11), and (8.50).

8.3.2 Strong consistency of option rewards for univariate Markov Gaussian log-price processes with estimated bounded parameters Let us also assume the following convergence condition for drift and volatility coeffi­ cients of the log-price processes Y ε,n (θ) holds: O3 : There exist sets Θ0 ∈ BΘ and Y n ∈ B1 , n = 0, . . . , N such that (a) μ ε,n (θ ε , y ε ) → μ 0,n (θ0 , y0 ) and σ ε,n (θ ε , y ε ) → σ 0,n (θ0 , y0 ) as ε → 0, for any θ ε → θ0 ∈ Θ0 and y ε → y0 ∈ Y n −1 as ε → 0, n = 1, . . . , N; (b) P{y0 + μ 0,n (θ0 , y0 ) + σ 0,n (θ0 , y0 )W0,n ∈ Yn ∩ Y n } = 1, for every θ 0 ∈    Θ0 , y0 ∈ Yn−1 ∩ Yn−1 and n = 1, . . . , N, where Yn , n = 0, . . . , N are sets introduced in the condition I8 . Let  θˆ ε , ε ∈ [0, ε0 ] > be a family of random variables defined on some probability space  Ω , F  , P . We interpret random variables θˆ ε , for ε ∈ (0, ε0 ] as estimates of the random vari­ able θˆ0 and assume that the following consistency condition, coherent with conver­ gence condition O3 , for drift and volatility coefficients, holds:

324 | 8 Convergence of rewards for Markov Gaussian LPP a.s. P1 : (a) θˆ ε −→ θˆ0 as ε → 0; (b) P{θˆ0 ∈ Θ0 } = 1, where Θ0 is the set penetrating condition O3 .

In applications it can be shown that θˆ0 = const with probability 1. In this case, the condition P1 means that θˆ ε are strongly consistent estimates for the parameter ˆθ0 . Let us mention two important particular models. The first model, where the functions μ ε,n (y) = μ ε,n and σ ε,n (y) = σ ε,n, for y ∈ R1 , n = 1, . . . , N, where μ ε,n , n = 1, . . . , N and σ ε,n, n = 1, . . . , N are real-valued random variable defined on a probability space  Ω , F  , P . In this case, the random vectors θˆ ε = (μ ε,1 , σ ε,1 , . . . , μ ε,n , σ ε,n ), ε ∈ [0, ε0 ]. Note also that the trend and volatility coefficients can be defined as μ ε,n (y) = μ ε,N , σ ε,n (y) = σ ε,N , for y ∈ R1 , n > N. The a.s. a.s. condition P1 reduces to the relations μ ε,n −→ μ 0,n and σ ε,n −→ σ 0,n as ε → 0, for n = 1, . . . , N. The second model is where the parameters μ ε,n(y) = μ ε,n + μ  ε,n y and σ ε,n ( y ) =     σ ε,n + σ ε,n σ n (y), for y ∈ R1 , n = 1, . . . , N. In this model, μ ε,n , μ ε,n , n = 1, . . . , N and σ ε,n , σ  random variable defined on a probability space ε,n , n = 1, . . . , N are real-valued   Ω  , F  , P , while |σ n (y) | ≤ |y| are nonrandom real-valued measurable func­      tions. In this case, the random vectors θˆ ε = (μ ε,1 , μ  ε,1 , σ ε,1 , σ ε,1, . . . , μ ε,n , μ ε,n , σ ε,n ,  σ ε,n ), ε ∈ [0, ε0 ]. Note also that the trend and volatility coefficients can be defined as     μ ε,n (y) = μ ε,N , μ  ε,n ( y ) = μ ε,N , σ ε,n ( y ) = σ ε,N , σ ε,n ( y ) = σ ε,N , for y ∈ R1 , n > N. The a.s. a.s.     condition O1 (a) reduces to the relations με,n → μ 0,n , μ  ε,n −→ μ 0,n and σ ε,n −→ σ 0,n , a.s.  σ  ε,n −→ σ 0,n as ε → 0, for n = 1, . . . , N. The object of our interest are random variables ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and Φ ε (θˆ ε ). However, we would also like to analyze the stochastic processes Y ε,n (θˆ ε ), n = 0, 1, . . . . In this case, some additional assumptions about joint distributions of ran­ dom variables θˆ ε and Y ε,0, W ε,n , n = 1, 2, . . . should be made. It is clear that the superposition form ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and ˆ Φ ε (θ ε ) is based on some independence assumptions for the random variables θˆ ε and the sequence of random variables Y ε,0 , W ε,n , n = 1, 2, . . . . A natural variant to formal­ ize the above-mentioned independence assumption is to assume that the following condition holds: Q1 : (a) Random variables θˆ ε , ε ∈ [0, ε0 ] and Y ε,0 , W ε,n , n = 1, 2, . . . , ε ∈ (0, ε0 ] are defined on the same probability space  Ω, F , P; (b) families of random variables  θˆ ε , ε ∈ [0, ε0 ],  Y ε,0, ε ∈ [0, ε0 ],  W ε,n , ε ∈ [0, ε0 ], n = 1, 2, . . . are mutually independent . If the condition Q1 holds, then the log-price process Y ε,n (θˆ ε ), n = 0, 1, . . . is well defined, for every ε ∈ [0, ε0 ], by the following stochastic transition dynamic relation: Y ε,n (θˆ ε ) = Y ε,n−1(θˆ ε ) + μ ε,n(θˆ ε , Y ε,n−1(θˆ ε )) ˆ ε , Y ε,n−1(θˆ ε ))W ε,n , + σ ε,n(θ

n = 1, 2, . . . ,

(8.55)

8.3 Markov Gaussian LPP with estimated characteristics

| 325

In this case, the random variables ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and Φ ε (θˆ ε ) can be interpreted as estimates, respectively, for the log-reward functions and the op­ timal expected rewards for the log-price processes Y0,n (θ), n = 0, 1, . . . , with the drift and volatility coefficients μ0,n (θ, y) and σ 0,n (θ, y) depending on parameter θ, for which information about the true value θˆ0 is given in the form of consistent esti­ mates ˆθ ε . Let us describe a natural variant of model construction, which provides holding of the condition Q1 . Let us assume that the sequences of independent random variables Y ε,0 , W ε,n ,   n = 1, 2, . . . are defined on different probability spaces  Ω ε , F ε , P ε . In this case,   , W ε,n , n = 1, 2, . . . defined on some prob­ one can construct random sequences Y ε,0    ability space  Ω , F , P , for all ε ∈ [0, ε0 ], and such that the random sequence   Y ε,0 , W ε,n , n = 1, 2, . . . is equivalent in distributional sense to the random sequence Y ε,0 , W ε,n , n = 1, 2, . . . , for every ε ∈ [0, ε0 ]. Let us assume that a sequence of random variables ρ n , n = 0, 1, . . . , indepen­ dent and uniformly distributed on interval [0, 1], is defined on some probability space  Ω  , F  , P . Let G ε,0(u ) = P{Y ε,0 ≤ u } and G ε,n (u ) = P{W ε,n ≤ u }, n = 1, 2, . . . be dis­ 1 tribution functions of these random variables and G− ε,n ( v ) = inf ( u : G ε,n ( u ) > v ), n = 0, 1, . . . be the corresponding inverse functions. In this case, the random vari­ d

 1 = G− ables Y ε,0 ε,0 ( ρ 0 ) = Y ε,0 (have the same distribution), and the random vari­

d

 1 = G− ables W ε,n ε,n ( ρ n ) = W ε,n , n = 1, 2, . . . . By the definition, the random variables   Y ε,0 , W ε,n , n = 1, 2, . . . are defined on the same probability space  Ω , F  , P , for   all ε ∈ (0, ε0 ]. Also the families of random variables  Y ε,0 , ε ∈ (0, ε0 ] and  W ε,n , ε ∈ [0, ε0 ], n = 1, 2, . . . are mutually independent. It follows from the above remarks that the log-reward functions ϕ ε,n (θ, y) and the optimal expected rewards Φ ε (θ) are the same for the log-price process Y ε,n (θ),  (θ), n = 0, 1, . . . constructed with the n = 0, 1, . . . and the log-price process Y ε,n use of stochastic transition relation (8.49), in which the random variables Y ε,0 , W ε,n ,   n = 1, 2, . . . should be replaced by the corresponding random variables Y ε,0 , W ε,n , n = 1, 2, . . . . Now, let us define the probability space  Ω, F , P, where Ω = Ω × Ω , F = σ (F  × F  ) (that is the minimal σ-algebra of subsets of Ω containing all rectangles B × C, B ∈ F  , C ∈ F  ), and the probability measure P(A) is a product measure on F , which is uniquely defined by its values on rectangles P(B × C) = P (A) · P (B), B ∈ F  , C ∈ F  , via the measure continuation theorem. The random variable θˆ ε = θˆ ε (ω ), ω ∈ Ω can be considered as a function of ω = (ω , ω) ∈ Ω = Ω × Ω , which, in fact, are functions of ω ∈ Ω , for every ε ∈ [0, ε0 ]. Analogously, the random variables Y ε,0 = Y ε,0(ω ), ω ∈ Ω  and W ε,n = W ε,n (ω ), ω ∈ Ω can be considered as functions of ω = (ω , ω ) ∈ Ω = Ω × Ω , which, in fact, are functions of ω ∈ Ω , for every n = 1, 2, . . . and ε ∈ [0, ε0 ].

326 | 8 Convergence of rewards for Markov Gaussian LPP In this case, the condition Q1 holds. The conditions O3 and P1 are also consistent with the construction described above. In this construction let us consider ϕ ε,n (θˆ ε , y) and Φ ε (θˆ ε ) as random vari­ ables defined either on the probability space  Ω, F , P or on the probability space  Ω  , F  , P  . In fact, the quantities ϕ ε,n (θˆ ε (ω ), y) and Φ ε (θˆ ε (ω )) for given outcome ω ∈ Ω can be computed as the values of log-reward functions and optimal expected rewards for the log-price process Y ε,n (θˆ ε (ω )) given by the following stochastic transition re­ lation: Y ε,n (θˆ ε (ω )) = Y ε,n−1(θˆ ε (ω )) + μ ε,n (θˆ ε (ω ), Y ε,n−1(θˆ ε (ω ))) ˆ ε (ω ), Y ε,n−1(θˆ ε (ω )))W ε,n , + σ ε,n(θ

n = 1, 2, . . . ,

(8.56)

The following theorem takes place. Theorem 8.3.3. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the conditions Q1 , G9 , I8 , O3 , and P1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: a.s. ϕ ε,n (θˆ ε , y ε ) −→ ϕ0,n (θˆ0 , y0 )

as

ε → 0.

(8.57)

Proof. Let us choose an arbitrary n = 1, . . . , N and y ε → y0 ∈ Y n −1 as ε → 0.   Denote by A the set of those ω ∈ Ω , for which, ˆθ ε (ω ) → θˆ0 (ω )

as

ε → 0.

(8.58)

Also, denote by B the set of those ω ∈ Ω , for which θˆ0 (ω ) ∈ Θ0 .

(8.59)

Relations (8.58) – (8.59), and conditions O3 and P1 imply that for ω ∈ A ∩ B, μ ε,n (θˆ ε (ω ), y ε ) → μ 0,n (θˆ0 (ω ), y0 )

as

ε → 0,

(8.60)

σ ε,n (θˆ ε (ω ), y ε ) → σ 0,n (θˆ0 (ω ), y0 )

as

ε → 0,

(8.61)

and Y n −1

Since an arbitrary choice of n = 1, . . . , N and y ε → y0 ∈ as ε → 0, rela­ tions (8.60) and (8.61) imply that the condition O1 (a) holds for the log-price processes Y ε,n (θˆ ε (ω )) with trend and volatility coefficients μ ε,n (θˆ ε (ω ), y) and σ ε,n(θˆ ε (ω ), y), for every ω ∈ A ∩ B. Relations (8.59) and the condition P1 (b) imply that for ω ∈ B, P {y0 + μ 0,n (θˆ0 (ω ), y0 ) + σ 0,n (θˆ0 (ω ), y0 )W0,n ∈ Yn ∩ Y n} = 1.

(8.62)

8.3 Markov Gaussian LPP with estimated characteristics

|

327

Since an arbitrary choice of y0 ∈ Y n −1 as ε → 0 and n = 1, . . . , N, relation (8.62) implies that the condition O1 (b) holds for the log-price processes Y ε,n(θˆ ε (ω )), with drift and volatility coefficients μ ε,n(θˆ ε (ω ), y) and σ ε,n(θˆ ε (ω ), y), for every ω ∈ B. The above remarks imply that the condition O1 holds for the log-price pro­ cesses Y ε,n (θˆ ε (ω )), with the drift and volatility coefficients μ ε,n (θˆ ε (ω ), y) and σ ε,n(θˆ ε (ω ), y), for every ω ∈ C = A ∩ B. The condition G9 and relations (8.50) and (8.58) imply that the following relation holds, for any ω ∈ A: lim max sup (|μ ε,n (θˆ ε (ω ), y)| + σ 2ε,n(θˆ ε (ω ), y)) ε →0 1≤ n ≤ N y ∈R1

ˆ ε (ω )) = K  (θˆ0 (ω )) < ∞ . ≤ lim K  (θ ε →0

(8.63)

Relation (8.63) implies that the condition G5 holds for the log-price processes Y ε,n(θˆ ε (ω )), with the drift and volatility coefficients μ ε,n (θˆ ε (ω ), y) and σ ε,n (θˆ ε (ω ), y), for every ω ∈ A. We can now conclude that conditions in Theorem 8.1.3 hold for the log-price pro­ cesses Y ε,n (θˆ ε (ω )), for every ω ∈ C = A ∩ B. Therefore, for every n = 0, 1, . . . , N, the following relations holds for the log-re­ ward functions ϕ ε,n (θˆ ε (ω ), y), for any y ε → y0 ∈ Yn ∩ Y n: ϕ ε,n (θˆ ε (ω ), y ε ) → ϕ0,n (θˆ0 (ω ), y0 )

as

ε → 0,

ω ∈ C .

(8.64)

The condition P1 implies that P (A) = 1 and P (B) = 1. Thus, P (C) = 1. There­ fore, relation (8.64) implies relation (8.57) to hold. Let us assume that the following condition holds: K14 : (a) P ε,0(·) ⇒ P0,0 (·) as ε → 0;   (b) P0,0 (Y0 ∩ Y 0 ) = 1, where Y0 and Y0 are the sets introduced in the condi­ tions I8 and O3 . Theorem 8.3.4. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0, and, also, the conditions Q1 , G9 , I8 , O3 , P1 , and K14 hold. Then, the following relation takes place: a.s. Φ ε (θˆ ε ) −→ Φ0 (θˆ0 ) as ε → 0 . (8.65) Proof. All conditions of Theorem 8.1.4 hold for the log-price processes Y ε,n (θˆ ε (ω )) with the trend and volatility coefficients μ ε,n (θˆ ε (ω ), y) and σ ε,n(θˆ ε (ω ), y), for every ω ∈ A ∩ B. This was checked for the conditions G5 and O1 in the proof of Theorem 8.3.3. The conditions D14 [β ] and I8 are assumed in Theorem 8.1.4. The condition K14 is a variant of the condition K12 used in Theorem 8.1.4. Thus, conditions of Theorem 8.1.4 hold for the log-price processes Y ε,n (θˆ ε (ω )), for every ω ∈ C = A ∩ B.

328 | 8 Convergence of rewards for Markov Gaussian LPP By applying this theorem, we get the following relation, for ω ∈ C: Φ ε (θˆ ε (ω )) → Φ0 (θˆ0 (ω )) as

ε → 0.

(8.66)

Since P (C) = 1, relation (8.66) implies relation (8.65) to hold. Let us now assume that the object of our interest is just the asymptotic relations (8.57) and (8.65). The random variables ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and Φ ε (θˆ ε ) are super­ positions of the random variables θˆ ε and nonrandom functions ϕ ε,n (θ, y), (θ, y) ∈ Θ × R1 , n = 0, 1, . . . , N and Φ ε (θ), θ ∈ Θ. This makes it possible to analyze the ran­ dom variables ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and Φ ε (θˆ ε ) without any assump­ tions about joint distributions of the random variables θˆ ε and Y ε,0 , W ε,n , n = 1, 2, . . . . Recall that the condition P1 requires that  θˆ ε , ε ∈ [0, ε0 ] be a family of random variables defined on some probability space  Ω , F  , P . As far as sequences of ran­ dom variables Y ε,0 , W ε,n , n = 1, 2, . . . are concerned, they can be defined on different   probability spaces  Ω ε , F ε , P ε , for different values of ε ∈ [0, ε0 ]. In this case, the random variables ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and Φ ε (θˆ ε ) can be interpreted as estimates of the functions ϕ0,n (θ, y), y ∈ R1 , n = 0, 1, . . . , N and Φ0 (θ) depending on the parameter θ, whose true value θˆ0 is given in the form of consistent estimates θˆ ε . The following two theorems show that the condition Q1 can be omitted in Theo­ rem 8.3.3 and the asymptotic relation (8.57) still can be proved. Theorem 8.3.5. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the conditions G9 , I8 , O3 , and P1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: a.s. ϕ ε,n (θˆ ε , y ε ) −→ ϕ0,n (θˆ0 , y0 )

as

ε → 0.

(8.67)

Proof. Let us take an arbitrary nonrandom points θ ε ∈ Θ, ε ∈ [0, ε0 ] such that θ ε → θ0 ∈ Θ0

as

ε → 0.

(8.68)

Let us consider the log-price processes Y ε,n (θ ε ), with the trend and volatility co­ efficients μ ε,n(θ ε , y) and σ ε,n (θ ε , y). The conditions G9 and O3 imply that the following relation holds: lim max sup (|μ ε,n (θ ε , y)| + σ 2ε,n (θ ε , y)) ≤ lim K  (θ ε ) = K  (θ0 ) < ∞ . ε →0 1≤ n ≤ N y ∈R1

ε →0

(8.69)

Relation (8.69) implies that the condition G5 holds for the log-price processes Y ε,n(θ ε ). The condition O3 implies that the condition O1 holds for the log-price pro­ cesses Y ε,n(θ ε ).

8.3 Markov Gaussian LPP with estimated characteristics

| 329

Therefore, all conditions of Theorem 8.1.3 hold for the log-price processes Y ε,n (θ ε ). This theorem yields, in this case, that, for every n = 0, 1, . . . , N, the following asymp­ totic relation holds, for any y ε → y0 ∈ Yn ∩ Y n: ϕ ε,n (θ ε , y ε ) → ϕ0,n (θ0 , y0 )

as

ε → 0.

(8.70)

The following part of the proof reminds the corresponding part in the proof of Theorem 8.3.3. According to the condition P1 , the random variables θˆ ε , ε ∈ [0, ε0 ] are defined on the same probability space  Ω , F  , P . Let us denote by C the set of elementary events ω such that θˆ ε (ω ) → θˆ0 (ω ) as ε → 0 . (8.71) and (8.72) θˆ0 (ω ) ∈ Θ0 . The condition P1 implies that P (C) = 1. Relation (8.70) holds for any nonrandom points θ ε ∈ Θ, ε ∈ [0, ε0 ] satisfying assumption (8.68). Using this, we get, substituting points θˆ ε (ω ) in relation (8.70), that the following relation holds, for any ω ∈ C: ϕ ε,n (θˆ ε (ω ), y ε ) → ϕ0,n (θˆ0 (ω ), y0 )

as

ε → 0.

(8.73)



Since P (C) = 1, relation (8.73) implies relation (8.67) to hold. Theorem 8.3.6. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0, and, also, the conditions Q1 , G9 , I8 , O3 , P1 , and K14 hold. Then, the following relation takes place: a.s. (8.74) Φ ε (θˆ ε ) −→ Φ0 (θˆ0 ) as ε → 0 . Proof. Let us take an arbitrary nonrandom points θ ε ∈ Θ, ε ∈ [0, ε0 ] such that θ ε → θ0 ∈ Θ0 as ε → 0. As was shown in the proof of Theorem 8.3.5, the conditions G5 and O1 hold for the log-price processes Y ε,n (θ ε ) with trend and volatility coefficients μ ε,n (θ ε , y) and σ ε,n(θ ε , y). The conditions D14 [β ] and I8 are assumed in Theorem 8.1.4. The condition K14 is a variant of the condition K12 used in Theorem 8.1.4. Thus, all conditions of Theorem 8.1.4 hold for the log-price processes Y ε,n (θ ε ), for any nonrandom points θ ε ∈ Θ, ε ∈ [0, ε0 ] such that θ ε → θ0 ∈ Θ0 as ε → 0. By applying Theorem 8.1.4, we get the following relation, for any θ ε → θ0 ∈ Θ0 as ε → 0, (8.75) Φ ε (θ ε ) → Φ0 (θ0 ) as ε → 0 . By substituting points ˆθ ε (ω ), defined in relations (8.71) and (8.72), in relation (8.75), we get the following relation, which holds for any ω ∈ C: Φ ε (θˆ ε (ω )) → Φ0 (θˆ0 (ω ))

as

ε → 0.

Since P (C) = 1, relation (8.76) implies relation (8.74) to hold.

(8.76)

330 | 8 Convergence of rewards for Markov Gaussian LPP

8.3.3 Option rewards for univariate Markov Gaussian log-price processes with estimated unbounded parameters Analogous results can be obtained for the model where the condition G9 is replace by the weaker condition | μ ε,n ( θ,y )|+ σ 2 ( θ,y ) G10 : limε→0 supθ ∈Θ max1≤n≤N supy∈R1 K  (θ )(1+K 79ε,n| y|) < 1, where K  (θ) is a contin­ uous function acting from the space Θ to interval (0, ∞) and the constant 0 ≤ K79 < ∞. The condition G10 implies that there exists 0 < ε ≤ ε0 such that, for the following relation holds for 0 ≤ ε ≤ ε : max sup

1≤ n ≤ N y ∈R1

|μ ε,n (θ, y)| + σ 2ε,n (θ, y)

1 + K79 |y|

< K  (θ) ,

θ ∈ Θ.

(8.77)

Relation (8.77) implies that the condition G6 holds for trend and volatility coeffi­ cients μ ε,n (θ, y) and σ ε,n(θ, y), for every θ ∈ Θ. Moreover, in this case the constant K79 replaces the constant K71 , the constant K  (θ), replacing the constant K70 , depends on parameter θ, while the parameter ε , which guarantees holding of inequality (8.77), for 0 ≤ ε ≤ ε , does not depend on parameter θ. Let us also assume that the condition B7 [𝛾] holds for pay-off functions g ε (n, e y ). Theorem 8.1.5 implies that the log-reward functions ϕ ε,n (θ, y), y ∈ R1 , n = 0, 1, . . . , N and the optimal expected rewards Φ ε (θ) are well defined for the log-price process Y ε,n(θ), for every θ ∈ Θ. Moreover, it follows from relation (8.77) that, in this case, the parameters ε25 and ε26 penetrating, respectively, Theorems 8.1.5 and 8.1.6 do not depend on parameter θ. The following theorem follows from Theorem 8.1.5 and the above remarks. Theorem 8.3.7. Let the condition G10 holds and, also, the condition B7 [𝛾] holds for some 𝛾 ≥ 0. Then, for every θ ∈ Θ and β ≥ 𝛾, there exist 0 < ε34 ≤ ε0 and con­ stants 0 ≤ M58 , M59 = M59 (θ, β ) < ∞ such that the log-reward functions ϕ ε,n (θ, y) satisfy the following inequalities for y ∈ R1 , 0 ≤ n ≤ N and 0 ≤ ε ≤ ε34 : ) ( A N − n ( β )𝛾 |ϕ ε,n (y) | ≤ M58 + M59 (θ, β )I (𝛾 = 0) + M59 (θ, β ) exp |y| I (𝛾 > 0) . β (8.78) Remark 8.3.5. The explicit formulas for the constants M58 , M59 (θ, β ) are given by the following formulas, which follows from formulas (8.22): M58 = L13 ,

M59 (θ, β ) = L13 L14 I (𝛾 = 0) # N 𝛾 1  2 β I (𝛾 > 0) . + L13 L14 1 + 2e K (θ )((β+A N −1(β))+ 2 (β+A N −1 (β)) )

One can choose any β ≥ 𝛾 in formula (8.79).

(8.79)

8.3 Markov Gaussian LPP with estimated characteristics

| 331

Remark 8.3.6. The parameter ε34 is given by the following formula ε34 = min(ε , ε21 ), where the parameters ε21 and ε are determined by relations (8.8) and (8.77). Theorem 8.3.8. Let the condition G10 holds and, also, the conditions B7 [𝛾] and D14 [β ] hold for some β ≥ 𝛾 ≥ 0. Then, for every θ ∈ Θ, there exist 0 < ε35 ≤ ε0 and the constant 0 ≤ M60 = M60 (θ) < ∞ such that the following inequality holds for 0 ≤ ε ≤ ε35 : |Φ ε (θ)| ≤ M60 (θ) .

(8.80)

Remark 8.3.7. The constant M60 (θ) is given by the following formula: # N 𝛾 𝛾 1 2  β β M60 (θ) = L13 + L13 L14 I (𝛾 = 0) + L13 L14 1 + 2e K (θ )(β+ 2 β ) K69 I (𝛾 > 0) . (8.81)

Remark 8.3.8. The parameter ε35 is given by the following formula ε35 = min(ε , ε21 , ε23 ), where the parameters ε21 , ε23 , and ε are determined by relations (8.8), (8.11), and (8.77).

8.3.4 Strong consistency of option rewards for univariate Markov Gaussian log-price processes with estimated unbounded parameters As in the case of the model with estimated bounded parameters, two variants of con­ vergence results can be obtained. In the case, where the condition Q1 is assumed to hold, one can analyze of ran­ dom log-reward functions ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and random optimal expected rewards Φ ε (θˆ ε ) for the log-reward processes Y ε,n (θˆ ε ), n = 0, 1, . . . . The following theorem takes place. Theorem 8.3.9. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the conditions Q1 , G10 , I8 , O3 , and P1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: a.s.

ϕ ε,n (θˆ ε , y ε ) −→ ϕ0,n (θˆ0 , y0 )

as

ε → 0.

(8.82)

Proof. It is analogous to the proof of Theorem 8.3.3. The only difference is that the con­ dition G10 is used instead of the condition G9 . Thus let one check that all conditions of Theorem 8.1.7 hold, for the log-price process Y ε,n (θˆ ε (ω )), for every ω ∈ C defined in relations (8.71) and (8.72). It can be made in the same way, as holding of all conditions of Theorem 8.1.3 was checked in the proof of Theorem 8.3.3. So, Theorem 8.1.7 can be applied to the processes Y ε,n(θˆ ε (ω )), for ω ∈ C. Since, P (C) = 1, one get relation (8.82).

332 | 8 Convergence of rewards for Markov Gaussian LPP Theorem 8.3.10. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0, and, also, the conditions Q1 , G10 , I8 , O3 , P1 , and K14 hold. Then, the following relation takes place: a.s. (8.83) Φ ε (θˆ ε ) −→ Φ0 (θˆ0 ) as ε → 0 . Proof. It is analogous to the proof of Theorem 8.3.4. The only difference is that the condition G10 is used instead of the condition G10 . Thus let one check that all condi­ tions of Theorem 8.1.8 hold for log-price processes Y ε,n (θˆ ε (ω )), for any ω ∈ C such that P (C) = 1. It can be made in the same way, as holding of all conditions of Theo­ rem 8.1.4 was checked in the proof of Theorem 8.3.4. By applying Theorem 8.1.8 to the processes Y ε,n(θˆ ε (ω )), for ω ∈ C, and taking into account that P (C) = 1, one get relation (8.83). In the case, where the condition Q1 does not holds, the asymptotic relation (8.82) can also be proved by the direct analysis of superpositions ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N. Theorem 8.3.11. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the condi­ tions G10 , I8 , O3 , and P1 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n, a.s. ϕ ε,n (θˆ ε , y ε ) −→ ϕ0,n (θˆ0 , y0 )

as

ε → 0.

(8.84)

Proof. It is analogous to the proof of Theorem 8.3.5. The only difference is that the condition G10 is used instead of the condition G9 . Thus let one check that all condi­ tions of Theorem 8.1.7 hold, for log-price processes Y ε,n(θ ε ), for any nonrandom points θ ε → θ0 ∈ Θ0 , where Θ0 is the set penetrating condition O3 . It can be made in the same way, as holding of all conditions of Theorem 8.1.7 was checked in the proof of Theorem 8.3.5. By applying Theorem 8.1.7 to the log-price processes Y ε,n (θ ε ) one can get relation ϕ ε,n (θ ε , y ε ) → ϕ0,n (θ0 , y0 ) as ε → 0, for the corresponding nonrandom points θ ε and y ε . Then, using the condition P1 , one can get relation (8.67). Theorem 8.3.12. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0, and, also, the conditions G10 , I8 , O3 , P1 , and K14 hold. Then, the following relation takes place: a.s. (8.85) Φ ε (θˆ ε ) −→ Φ0 (θˆ0 ) as ε → 0 . Proof. It is analogous to the proof of Theorem 8.3.6. The only difference is that the condition G10 is used instead of the condition G9 . Thus let us check that all condi­ tions of Theorem 8.1.8 hold, for log-price processes Y ε,n (θ ε ), for any nonrandom points θ ε → θ0 ∈ Θ0 , where Θ0 is the set penetrating condition O3 . It can be made in the same way, as holding of all conditions of Theorem 8.1.8 was checked in the proof of Theorem 8.3.6. By applying Theorem 8.1.8 to the log-price processes Y ε,n (θ ε ) one can get relation ϕ ε,n (θ ε , y ε ) → ϕ0,n (θ0 , y0 ) as ε → 0, for the corresponding nonrandom points θ ε and y ε . Then, using the condition P1 , one can get relation (8.85).

8.3 Markov Gaussian LPP with estimated characteristics

|

333

8.3.5 Weak consistency of option rewards for univariate Markov Gaussian log-price processes with estimated parameters Let us now assume that random variables θˆ ε is defined on different probability spaces  Ω ε , Fε , Pε , for ε ∈ [0, ε0 ]. We interpret the random variables θˆ ε , for ε ∈ (0, ε0 ] as estimates for the random variable θˆ0 and assume that the following weak consistency condition, coherent with convergence condition O3 , for trend and volatility coefficients, holds: d P2 : (a) θˆ ε −→ ˆθ0 as ε → 0; (b) P{θˆ0 ∈ Θ0 } = 1, where Θ0 is the set penetrating condition O3 . In applications, it can be seen that ˆθ0 = const with probability 1. In this case, the P condition P2 means that θˆ ε −→ ˆθ0 as ε → 0, i.e. ˆθ ε are weakly consistent estimates for the parameter θˆ0 . The object of our interest are again the random variables ϕ ε,n (θˆ ε , y), y ∈ R1 , n = 0, 1, . . . , N and Φ ε (θˆ ε ). However, we would also like to analyze stochastic processes Y ε,n(θˆ ε ), n = 0, 1, . . . . In this case, some additional independence assumptions for random vari­ ables ˆθ ε and Y ε,0 , W ε,n , n = 1, 2, . . . should be made. A natural variant to formalize the above mentioned independence assumptions is to assume that the following condition holds: Q2 : For every ε ∈ [0, ε0 ]: (a) random variables ˆθ ε , Y ε,0 , W ε,n , n = 1, 2, . . . are defined on the same proba­ bility space  Ω ε , Fε , P ε ; (b) random variables ˆθ ε , Y ε,0, and W ε,n , n = 1, 2, . . . are mutually independent. If the condition Q2 holds, then the log-price process Y ε,n (θˆ ε ) is well defined, for every ε ∈ [0, ε0 ], by the following stochastic dynamic relation: Y ε,n(θˆ ε ) = Y ε,n−1(θˆ ε ) + μ ε,n (θˆ ε , Y ε,n−1(θˆ ε )) ˆ ε , Y ε,n−1(θˆ ε ))W ε,n , + σ ε,n(θ

n = 1, 2, . . . ,

(8.86)

Let us describe a natural variant of model construction, which provides holding of the condition Q2 . Let us assume that the sequence of independent random variables Y ε,0 , W ε,n , n =   1, 2, . . . is defined on a probability space  Ω ε , F ε , P ε , for ε ∈ [0, ε0 ]. Now, let us define, for every ε ∈ [0, ε0 ], the probability space  Ω ε , Fε , P ε , where   Ω ε = Ωε × Ω ε , F ε = σ (F ε × F ε ) (that is the minimal σ-algebra of subsets of Ω containing all rectangles B × C, B ∈ Fε , C ∈ Fε ), and the probability measure P ε (A) is a product measure on Fε , which is uniquely defined by its values on rectangles   P ε (B × C) = Pε (B)P ε ( C ), B ∈ F ε , C ∈ F ε , via the measure continuation theorem.

334 | 8 Convergence of rewards for Markov Gaussian LPP Let ε ∈ [0, ε0 ]. The random variable ˆθ ε = θˆ ε (ω ), ω ∈ Ωε can be considered as a function of ω = (ω , ω) ∈ Ω ε = Ωε × Ω ε , which, in fact, is a function of   ω ∈ Ω ε . Analogously, the random variables Y ε,0 = Y ε,0(ω ), ω ∈ Ω ε and W ε,n =     W ε,n (ω ), ω ∈ Ω ε can be considered as functions of ω = ( ω , ω ) ∈ Ω ε = Ω ε × Ω ε ,   which, in fact, are functions of ω ∈ Ω ε , for all n = 1, 2, . . . . In this case, the condition Q2 holds. The conditions O3 and P2 are also consistent with the construction described above. This construction enables one to consider ϕ ε,n (θˆ ε , y) and Φ ε (θˆ ε ) as random vari­ ables defined either on the probability space  Ω ε , Fε , P ε  or on the probability space  Ω ε , Fε , Pε . The following theorem takes place. Theorem 8.3.13. Let the condition B7 [𝛾] holds for some 𝛾 ≥ 0 and also the condi­ tions Q2 , G9 or G10 , I8 , O3 , and P2 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: d

ϕ ε,n (θˆ ε , y ε ) −→ ϕ0,n (θˆ0 , y0 )

as

ε → 0.

(8.87)

Theorem 8.3.14. Let the conditions B7 [𝛾] and D14 [β ] hold for some β > 𝛾 > 0 or β = 𝛾 = 0, and, also, the conditions Q2 , G9 or G10 , I8 , O3 , P2 , and K14 hold. Then, the following relation takes place: d (8.88) Φ ε (θˆ ε ) −→ Φ0 (θˆ0 ) as ε → 0 . Proof of Theorems 8.3.13–8.3.14. Since (8.87) is a relation of convergence in distribu­ tion, we can assume that the parameter ε → 0 runs only over some countable se­ quence of values. If we prove relation (8.87) under this additional assumption, then we will be able to prove relation (8.87) for any subsequence of values of the parameter ε → 0. This would prove relation (8.87) for any ε → 0. Let us assume that ρ, ρ 0 , ρ 1 , . . . is a sequence of independent uniformly dis­ tributed random variables defined of some probability space  Ω, F , P. According to the Skorokhod representation theorem, if the condition P2 holds, one can construct measurable functions f ε (v) acting from interval [0, 1] to the Polish d metric space Θ such that random variables θˆε = f ε (ρ ) = ˆθ ε , for every ε ∈ [0, ε0 ], and a.s. θˆε −→ θˆ0 as ε → 0.  1  −1 We again employ random variables Y ε,0 = G− ε,0 ( ρ 0 ) and W ε,n = G n ( ρ n ), n = d

d

  1, 2, . . . , defined in Subsection 8.3.2. By the definition, Y ε,0 = Y ε,0 and W ε,n = W ε,n , n = 1, 2, . . . , for every ε ∈ [0, ε0 ].   Obviously, random variables θˆε , Y ε,0 , W ε,n , n = 1, 2, . . . , ε ∈ [0, ε0 ] are defined on the same probability space  Ω, F , P and satisfy the condition Q1 stronger than the condition Q2 .

8.4 Skeleton reward approximations for Markov Gaussian LPP |

335

It follows from the above remarks that the log-reward functions ϕ ε,n (θ, y) and the optimal expected rewards Φ ε (θ) are the same for the log-price process Y ε,n (θ),  n = 0, 1, . . . and the log-price process Y ε,n (θ), n = 0, 1, . . . constructed with the use of stochastic transition relation (8.49), in which the random variables Y ε,0 , W ε,n ,   n = 1, 2, . . . should be replaced by the corresponding random variables Y ε,0 , W ε,n , n = 1, 2, . . . . d Also the random variables ϕ ε,n (θˆ ε , y) = ϕ ε,n (θˆε , y), for every y ∈ R1 , n = d 0, 1, . . . , N and ε ∈ [0, ε0 ] and the random variables Φ ε (θˆ ε ) = Φ ε (θˆε ). Obviously, the condition P1 holds for random variables θˆε , ε ∈ [0, ε0 ]. The above remarks imply that Theorem 8.3.3 or 8.3.9 (respectively, in the case, where the condition G9 or G10 is assumed to hold) can be applied to the log-price pro­  ˆε that yields the relation, cesses Y ε,n (θ) and random variables θ a.s.

ϕ ε,n (θˆε , y ε ) −→ ϕ0,n (θˆ0 , y0 )

as

ε → 0.

(8.89)

Since a.s. convergence is stronger than convergence in distribution and random d variables ϕ ε,n (θˆε , y ε ) = ϕ ε,n (θˆ ε , y ε ), relation (8.89) implies relation (8.87) to hold. The above remarks imply that Theorem 8.3.4 or 8.3.10 (respectively, in the case, where the condition G9 or G10 is assumed to hold) can be applied to the log-price pro­  ˆε that yields the relation, cesses Y ε,n (θ) and random variables θ a.s. Φ ε (θˆε ) −→ Φ0 (θˆ0 )

as

ε → 0.

(8.90)

d Since, random variables Φ ε (θˆε ) = Φ ε (θˆ ε ), relation (8.90) implies relation (8.88) to hold.

Remark 8.3.9. In the same way, as it was made for Theorems 8.3.5, 8.3.6, 8.3.11, and 8.3.12, one can prove that the condition Q2 can be omitted in these theorems. The cor­ responding proofs will only require to use the Skorokhod representation theorem for d a.s. constructing random variables θˆε = θˆ ε , for every ε ∈ [0, ε0 ], and such that θˆε −→ θˆ 0 as ε → 0. Then, the theorems mentioned above should be applied, in order, to yield a.s. a.s. relations ϕ ε,n (θˆε , y ε ) −→ ϕ0,n (θˆ0 , y0 ) as ε → 0 and Φ ε (θˆε ) −→ Φ0 (θˆ0 ) as ε → 0.

8.4 Skeleton reward approximations for Markov Gaussian LPP In this section, we present results about space-skeleton approximations for reward of American-type options for univariate and multivariate modulated Markov Gaussian log-price processes.

336 | 8 Convergence of rewards for Markov Gaussian LPP

8.4.1 Space-skeleton approximations for option rewards of univariate Markov Gaussian log-price processes Let the univariate Markov Gaussian log-price process Y0,n , n = 0, 1, . . . be defined by the following stochastic transition dynamic relation: Y 0,n = Y0,n−1 + μ 0,n (Y0,n−1) + σ 0,n (Y0,n−1)W0,n ,

n = 1, 2, . . . ,

(8.91)

where (a) Y 0,0 is a real-valued random variable, (b) W0,n , n = 1, 2, . . . is a sequence of standard i.i.d. normal random variables with mean 0 and variance 1, (c) the random variable Y 0,0 and sequence of random variables W0,n , n = 1, 2, . . . are independent, (d) μ 0,n (y), n = 1, 2, . . . are real-valued Borel functions defined on R1 , and (e) σ 0,n (y), n = 1, 2, . . . are real-valued Borel functions defined on R1 . Let us construct the corresponding space-skeleton approximating processes Y ε,n, n = 0, 1, . . . , for ε ∈ (0, ε0 ], according to the algorithm described in Subsection 7.2.1. Let us choose, for every ε ∈ (0, ε0 ], the parameters δ ε,n > 0, λ ε,n ∈ R1 , n = + 0, 1, . . . , and integers m− ε,n ≤ m ε,n , n = 0, 1, . . . . First, the corresponding intervals Iε,n,l, which play the role of skeleton sets Aε,n,l, + are constructed for l = m− ε,n , . . . , m ε,n , n = 0, 1, . . . , N, Iε,n,l = A ε,n,l ⎧ 1 ⎪ ⎪ (−∞, δ ε,n (m− ε,n + 2 )] + λ ε,n ⎪ ⎨ 1 = (δ ε,n (l − 2 ), δ ε,n (l + 21 )] + λ ε,n ⎪ ⎪ ⎪ ⎩(δ (m+ − 1 ), ∞) + λ ε,n

ε,n

2

ε,n

if l = m− ε,n , + if m− ε,n < l < m ε,n ,

(8.92)

if l = m+ ε,n .

Then, the skeleton points y ε,n,l ∈ Iε,n,l are chosen in the following way: y ε,n,l = lδ ε,n + λ ε,n ,

+ l = m− ε,n , . . . , m ε,n ,

n = 0, 1, . . . , N .

(8.93)

Finally, skeleton functions h ε,n (y), n = 0, 1, . . . , are defined by the following for­ mulas:  + (8.94) h ε,n (y) = y ε,n,l if y ∈ Iε,n,l , l = m− ε,n , . . . , m ε,n , The corresponding space-skeleton approximating Markov processes Y ε,n are de­ fined by the following stochastic transition dynamic relations: ⎧  ⎪ Y ε,n = h ε,n h ε,n−1(Y ε,n−1) + μ 0,n (h ε,n−1(Y ε,n−1)) ⎪ ⎪ ⎨ + σ 0,n (h ε,n−1(Y ε,n−1))W0,n , n = 1, 2, . . . , (8.95) ⎪ ⎪ ⎪ ⎩ Y = h (Y ) . ε,0

ε,0

0,0

8.4 Skeleton reward approximations for Markov Gaussian LPP

|

337

The approximating log-price process Y ε,n is, for every ε ∈ (0, ε0 ], the skeleton Markov chain with one-step transition probabilities,

P0,n (h ε,n−1(y), Iε,n,l) P ε,n (y, A) = y ε,n,l ∈ A

=



y ε,n,l ∈ A

 P h ε,n−1(y)  + μ 0,n (h ε,n−1(y)) + σ 0,n (h ε,n−1(y))W0,n ∈ Iε,n,l ,

and the initial distribution, P ε,0 (A) =



P0,0 (Iε,0,l)

(8.96)

(8.97)

y ε,0,l ∈ A

Note that the one-point transition probabilities P0,n (y, Iε,n,l) = P{y + μ 0,n (y) + σ 0,n (y) W0,n ∈ Iε,n,l} are interval Gaussian probabilities. Let us also assume that a pay-off function g ε (n, e y ) = g(n, e y ) does not depend on the parameter ε. Let us ϕ ε,n (y) be the corresponding log-reward functions for the log-price pro­ cesses Y ε,n . The following lemma is the direct corollary of Lemma 7.2.1. Lemma 8.4.1. Let, for every ε ∈ (0, ε0 ], the log-price process Y ε,n be a space-skele­ ton Markov chain with the transition probabilities and the initial distribution defined, respectively, in relations (8.96) and (8.97). Then the reward functions ϕ ε,n (y) and + ϕ ε,n+r(y ε,n+r,l), for points y ε,n+r,l, l = m− ε,n + r , . . . , m ε,n + r, r = 1, . . . N − n, given  − by formulas (8.93), are, for every y ∈ Iε,n,l, l = m ε,n , . . . , m+ ε,n , n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ + ⎪ ϕ ε,N (y ε,N,l) = g(N, e y ε,N,l ) , l = m− ⎪ ε,N , . . . , m ε,N , ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r(y ε,n+r,l) = max g(n + r, e y ε,n+r,l ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ m+ ε,n+r+1 ⎪

⎪ ⎪ ⎪ ⎪ ϕ ε,n+r+1(y ε,n+r+1,l)P0,n+r (y ε,n+r,l, Iε,n+r+1,l) , ⎪ ⎪ ⎪ ⎨ l =m− ε,n+r+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

+ l = m− r = N − n − 1, . . . , 1 , ε,n + r , . . . , m ε,n + r , & m+ ε,n+1

ϕ ε,n+1(y ε,n+1,l) ϕ ε,n (y) = max g(n, e y ),

l=m− ε,n+1

'

× P0,n (y ε,n,l , Iε,n+1,l) .

(8.98)

Let us now formulate conditions of convergence for the above log-reward functions. We are going to apply Theorem 7.2.1. We assume that the condition I6 holds for the pay-off function g(n, e y ). The condition J7 should be replaced in this case by the following condition:

338 | 8 Convergence of rewards for Markov Gaussian LPP O4 : There exist sets Y n ∈ B1 , n = 0, . . . , N such that: (a) μ 0,n (y ε ) → μ 0,n (y0 ) and σ 0,n (y ε ) → σ 0,n (y0 ) as ε → 0, for any y ε → y0 ∈ Y n −1 as ε → 0, and n = 1, . . . , N;  (b) P{y0 + μ 0,n (y0 ) + σ 0,n (y0 )W0,n ∈ Yn ∩ Y n } = 1, for every y 0 ∈ Y n −1 ∩   Yn−1 and n = 1, . . . , N, where Yn , n = 0, . . . , N are sets introduced in the condition I6 . The condition O4 (b) holds if, for every y0 ∈ Yn−1 ∩ Y n −1 , n = 1, . . . , N, either   σ 0,n (y0 ) > 0 and the sets Yn , Yn have zero Lebesgue measure, or σ 0,n (y0 ) = 0 and y0 + μ 0,n (y0 ) ∈ Yn ∩ Y n. The following theorem is a corollary of Theorems 7.2.1. Theorem 8.4.1. Let a Markov Gaussian log-price process Y0,n and the corresponding space-skeleton approximating Markov log-price processes Y ε,n are defined, respectively, by stochastic transition dynamic relations (8.91) and (8.95). Let also the conditions B4 [𝛾] holds for some 𝛾 ≥ 0, and, also, the conditions G1 , N1 , I6 , and O4 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n: ϕ ε,n (y ε ) → ϕ0,n (y0 )

as

ε → 0.

(8.99)

Proof. The proof can be obtained by application of Theorem 7.2.1. The fitting condition L2 holds since the corresponding transition probabilities are defined according to formula (8.96). The condition C11 [β ] holds for any β ≥ 0 that follows from Lemmas 4.1.8 and 4.4.1. We can always choose some β > 𝛾 if 𝛾 > 0 of β = 0 if 𝛾 = 0. The condition O4 (a) implies that the condition J9 (a) holds since convergence of parameters for normal random variables is equivalent to weak convergence for these random variables. The condition O4 (b) is just a variant of the condition J9 (b) . Thus, all conditions of Theorem 7.2.1 hold. By applying this theorem we get the convergence relation (8.99). Let us now formulate conditions of convergence for the corresponding optimal ex­ pected rewards. In this case the optimal expected reward Φ ε for the approximation log-price pro­ cesses Y ε,n can be found, for every ε ∈ (0, ε0 ], using the following formula: m+ ε,0

Φε =



ϕ ε,0 (y ε,0,l)p ε,0 ,

(8.100)

l=m− ε,0 + where p ε,l = P0,0 (Iε,0,l) = P{h ε,0(Y0,0 ) = y ε,0,l}, l = m− ε,0 , . . . , m ε,0 . We are going to use Theorem 7.2.2. The condition D10 [β ] used in Theorem 7.2.2 should be replaced by an analog of the condition D4 [β ], assumed to hold for some β ≥ 0: D16 [β ]: Ee β| Y0,0 | < K80 , for some 1 < K80 < ∞.

8.4 Skeleton reward approximations for Markov Gaussian LPP |

339

The condition K7 used in Theorem 7.2.2 should be replaced by the following condition imposed on the initial distribution P{Y0,0 ∈ A} = P0,0 (A):   K14 : P0,0 (Y0 ∩ Y 0 ) = 1, where Y0 and Y0 are the sets introduced, respectively, in the conditions I6 and O4 . Theorem 8.4.2. Let a Markov log-price process Y 0,n and the corresponding spaceskeleton approximating Markov log-price processes Y ε,n are defined, respectively, by stochastic transition dynamic relations (8.91) and (8.95). Let also the conditions B4 [𝛾] and D16 [β ] hold with parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions G1 , N1 , I6 , O4 , and K14 hold. Then, the following relation takes place, Φ ε → Φ0

as

ε → 0.

(8.101)

Proof. The fitting condition M2 holds since the corresponding initial distributions are defined according to formula (8.97). The condition K14 is just reformulation of the condition K8 . Other conditions of Theorem 7.2.2 also hold that were shown in the proof of Theo­ rem 8.4.1. By applying Theorem 7.2.2 we get convergence relations (8.125). Analogous results can be formulated for univariate Markov Gaussian log-price pro­ cesses with unbounded characteristics. In this case, Theorems 7.4.1 and 7.4.2 should be used instead Theorems 7.1.1 and 7.1.2. Theorem 8.4.3. Let a Markov Gaussian log-price process Y0,n and the corresponding space-skeleton approximating Markov log-price processes Y ε,n are defined, respectively, by stochastic transition dynamic relations (8.91) and (8.95). Let also the conditions B4 [𝛾] holds for some 𝛾 ≥ 0, and, also, the conditions G2 , N1 , I6 , and O4 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn ∩ Y n, ϕ ε,n (y ε ) → ϕ0,n (y0 ) as

ε → 0.

(8.102)

Proof. The proof repeats the proof of Theorem 8.4.1. The only change is that Theo­ rem 7.4.1 should be used instead of Theorem 7.2.1. In this case, the condition C15 [β ] holds for any β ≥ 0 according to Lemmas 4.3.6 and 4.4.2. We can always choose some β > 𝛾 if 𝛾 > 0 of β = 0 if 𝛾 = 0. Other conditions of Theorem 7.4.1 hold as it was shown in the proof of Theo­ rem 8.4.1. By applying Theorem 7.4.1 we get the convergence relation (8.102). The condition D12 [β ] used in Theorem 7.4.2 should be replaced by analog of the con­ dition D5 [β ] with the function A(β ) = K16 K17 (β + 21 β 2 ), for some β ≥ 0: D17 [β ]: E exp{A N (β )|Y0,0|} < K81 , for some 1 < K81 < ∞.

340 | 8 Convergence of rewards for Markov Gaussian LPP Theorem 8.4.4. Let a Markov log-price process Y 0,n and the corresponding spaceskeleton approximating Markov log-price processes Y ε,n are defined, respectively, by stochastic transition dynamic relations (8.91) and (8.95). Let also the conditions B4 [𝛾] and D17 [β ] hold with parameters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions G2 , N1 , I6 , O4 , and K14 hold. Then, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(8.103)

Proof. Theorem 8.4.4 is a corollary of Theorem 7.4.2. All conditions of the latter theo­ rem hold as was shown in the proofs of Theorems 8.4.1–8.4.3. By applying Theorem 7.4.2, we get the convergence relation (8.103).

8.4.2 Space-skeleton approximations for option rewards of multivariate modulated Markov Gaussian log-price processes 0,n , X0,n ), n = 0, 1, . . . is a multivari­ Let us assume that a log-price process  Z0,n = (Y ate modulated Markov Gaussian log-price process given by the following stochastic transition dynamic relation: ⎧  =Y ⎪ 0,n−1 +  0,n−1 , X0,n−1) + Σ0,n (Y 0,n−1 , X0,n−1)W  0,n , Y μ 0,n (Y ⎪ ⎪ ⎨ 0,n X0,n = C0,n (X0,n−1 , U0,n ) , (8.104) ⎪ ⎪ ⎪ ⎩ n = 1, 2, . . . , 0,0 , X0,0 ) = ((Y 0,0,1 , . . . , Y 0,0,k), X0,0 ) is a random vector tak­ where (a)  Z0,0 = (Y  0,n , U0,n ) = ((W0,n,1 , . . . , W0,n,k), U0,n ), ing values in the space Z = Rk × X, (b) (W n = 1, 2, . . . is a sequence of independent random vectors taking values in the space Rk × U, where U is a measurable space with a σ-algebra of measurable subsets BU , 0,0 , X0,0 ) and the random sequence (W  0,n , U0,n ), n = 1, 2, . . . (c) the random vector (Y  are independent, (d) the random vector W0,n = (W0,n,1 , . . . , W0,n,k) has a stan­ dard multivariate normal distribution with EW0,n,i = 0, EW0,n,i W0,n,j = I (i = j), i, j = 1, . . . , k, for every n = 1, 2, . . . , (e) the random variables U0,n have a regular  0,n = w  }, for every n = 1, 2, . . . , conditional distribution G w (B) = P{U0,n ∈ B/W (f)  μ 0,n ( y , x) = (μ 0,n,1( y , x), . . . , μ 0,n,k( y , x)), n = 1, 2, . . . are vector functions, which components are measurable functions acting from the space Z = Rk × X to R1 , y , x) = σ 0,n,i,j( y , x) , n = 1, 2, . . . are k × k matrix functions with elements (g) Σ0,n ( σ 0,n,i,j( y , x), i, j = 1, . . . , k, which are measurable functions acting from the space Z = Rk × X to R1 , and (h) C0,n (x, u ) is a measurable function acting from the space X × U to the space X. Let us construct the corresponding space-skeleton approximating processes  Z ε,n, n = 0, 1, . . . , for ε ∈ (0, ε0 ], according to the algorithm described in Subsection 7.2.3.

8.4 Skeleton reward approximations for Markov Gaussian LPP

|

341

+ Let δ ε,n,i > 0, λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . . and m− ε,n,j ≤ m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . be integer numbers. + First, the intervals Iε,n,i,l should be constructed for l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . ,

Iε,n,i,l ⎧ 1 − ⎪ ⎪ ⎪(−∞, δ ε,n,i(m ε,n,i + 2 )] + λ ε,n,i ⎨ = (δ ε,n,i(l − 21 ), δ ε,n,i(l + 21 )] + λ ε,n,i ⎪ ⎪ ⎪ 1 ⎩( δ (m+ − ), ∞) + λ ε,n,i

ε,n,i

ε,n,i

2

if l = m− ε,n,i , + if m− ε,n,i < l < m ε,n,i ,

(8.105)

if l = m+ ε,n,i .

+ Then cubes Iε,n,l1,...,l k = Iε,n,1,l1 × · · · × I ε,n,k,l k , l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . should be defined. By the definition, the skeleton points y ε,n,i,l i = l i δ ε,n,i + λ ε,n,i ∈ Iε,n,i,l i and, thus, + vector points (y ε,n,1,l1 , . . . , y ε,n,k,l k ) ∈ Iε,n,l1,...,l k , for l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . . + Second, nonempty sets Jε,n,l ⊂ BX , l = m− ε,n,0, . . . , m ε,n,0, n = 0, 1, . . . , such   that (a) Jε,n,l ∩ Jε,n,l = ∅, l ≠ l , n = 0, 1, . . . , N; (b) X = ∪m−ε,n,0 ≤l≤m+ε,n,0 Jε,n,l, n = 0, 1, . . . , should be constructed. Remind that one of our model assumption is the X which is a Polish metric space. The sets K ε,n , n = 0, 1, . . . , should be chosen and then the nonempty sets K ε,n,l ⊆ +   BX , l = m− ε,n,0 , . . . , m ε,n,0, n = 0, 1, . . . , such that (c) K ε,n,l  ∩ K ε,n,l  = ∅ , l ≠ l , n = 0, 1, . . . ; (d) ∪m−ε,n,0 ≤l≤m+ε,n,0 K ε,n,l = K ε,n, n = 0, 1, . . . . The sets Jε,n,l can be defined in the following way, for n = 0, 1, . . . , ⎧ + ⎨K ε,n,l if m− ε,n,0 ≤ l < m ε,n,0 , Jε,n,l = (8.106) ⎩K + ∪ K ε,n if l = m+ .

ε,n,m ε,n,0

ε,n,0

m− ε,n,0 ,

. . . , m+ Then, skeleton points x ε,n,l ∈ Jε,n,l, l = ε,n,0, n = 0, 1, . . . , N should be chosen. z ε,n,¯l ∈ Aε,n,¯l can be defined for Third, skeleton sets Aε,n,¯l and skeleton points,  + ¯l ∈ Lε,n = {¯l = (l0 , . . . , l k ), l j = m− , . . . , m , j = 0, . . . , k }, n = 0, 1, . . . , in the ε,n,j ε,n,j following way: A ε,n,¯l = Iε,n,l1,...,l k × Jε,n,l0 , (8.107) and  z ε,n,¯l = ( y ε,n,¯l , x ε,n,¯l) = ((y ε,n,1,l1 , . . . , y ε,n,k,l k ), x ε,n,l0) .

(8.108)

Fourth, skeleton functions, h ε,n,i(y), y ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . , should be defined by the following formulas:  + (8.109) h ε,n,i(y) = y ε,n,i,l if y ∈ Iε,n,i,l , m− ε,n,i ≤ l ≤ m ε,n,i , and skeleton functions h ε,n,0(x), x ∈ X, n = 0, 1, . . . , should be defined by the fol­ lowing formula:  + (8.110) h ε,n,0(x) = x ε,n,l if x ∈ Jε,n,l , m− ε,n,0 ≤ l ≤ m ε,n,0 .

342 | 8 Convergence of rewards for Markov Gaussian LPP ˆ ε,n ( Finally, vector skeleton functions h y ),  y = (y1 , . . . , y k ) ∈ Rk , n = 0, 1, . . . , N should be defined by the following formula: ˆ ε,n ( h y) = (h ε,n,1(y1 ), . . . , h ε,n,k(y k )) ,

(8.111)

z ),  z = ( y , x) ∈ Z, n = 0, 1, . . . , N can be and then vector skeleton functions h¯ ε,n ( defined by the following formula: ˆ ε,n ( h¯ ε,n ( z) = (h y ), h ε,n,0(x)) .

(8.112)

The corresponding space-skeleton approximating modulated Markov processes ε,n , X ε,n ) are defined by the following stochastic transition dynamic rela­  Z ε,n = (Y tions: # ⎧ ˆ ε,n h ˆ ε,n−1(Y ⎪ ε,n = h ε,n−1) ⎪Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆ ε,n−1(Y ε,n−1), h ε,n−1,0(X ε,n−1)) ⎪ μ 0,n (h +  ⎪ ⎪ ⎪  ⎪ ⎪ ⎨ ˆ ε,n−1(Y ε,n−1), h ε,n−1,0(X ε,n−1)) W  0,n , + Σ0,n (h (8.113)  ⎪ ⎪ ⎪ = h C ( h ( X ) , U ) , X ε,n ε,n,0 0,n ε,n − 1,0 ε,n − 1 0,n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n = 1, 2, . . . , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Y ˆ ε,0 (Y 0,0) , X ε,0 = h ε,0,0(X0,0 ) . ε,0 = h Z ε,n is, for every ε ∈ (0, ε0 ], a skeleton The approximating log-price process  Markov chain with one-step transition probabilities P ε,n ( z , A),  z = ( y , x) ∈ Z, A ∈ BZ , n = 1, 2, . . . given by the following formula:

P ε,n ( z , A) = P0,n (h¯ ε,n−1( z ), Aε,n,¯l)  z ε,n,¯l ∈ A

=



P

#

ˆ ε,n−1( ˆ ε,n−1( h y) +  μ 0,n (h y), h ε,n−1,0(x))

 z ε,n,¯l ∈ A

ˆ ε,n−1(  0,n , + Σ0,n (h y), h ε,n−1,0(x)) W   C0,n (h ε,n−1,0(x), U0,n ) ∈ Aε,n,¯l .

(8.114)

ε,0 ∈ A}, A ∈ BZ is concerned, it As far as the initial distribution P ε,0 (A) = P{Z takes, for every ε ∈ (0, ε0 ], the following form:

P ε,0(A) = P0,0 (Aε,0,¯l)  z ε,0,¯l ∈ A

=



0,0 , X0,0 ) ∈ A ε,0,¯l} . P{(Y

(8.115)

 z ε,0,¯l ∈ A

In this case, a pay-off function g ε (n, ey , x) = g(n, ey , x) does not depend on pa­ rameter ε. y , x) be the corresponding log-reward functions for the log-price pro­ Let ϕ ε,n ( cesses  Z ε,n. The following lemma is the direct corollary of Lemma 7.2.2.

8.4 Skeleton reward approximations for Markov Gaussian LPP |

343

Z ε,n is a space-skele­ Lemma 8.4.2. Let, for every ε ∈ (0, ε0 ], the log-price process  ton Markov chain with the transition probabilities and the initial distribution defined, respectively, in relations (8.114) and (8.115). Then the reward functions ϕ ε,n ( y , x) and y ε,n,¯l , x ε,n,¯l), for the points ( y ε,n,¯l, x ε,n,¯l), ¯l ∈ Lε,n+r, r = 1, . . . N − n, given by for­ ϕ n+r ( mulas (8.108), are, for every  z = ( y , x) ∈ Aε,n,¯l , ¯l ∈ Lε,n , n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ ⎪ ϕ ε,N ( y ε,N,¯l , x ε,N,¯l) = g(N, ey ε,N,l , x ε,N,¯l) , ¯l ∈ Lε,N , ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ( y ,x ) = max g(n + r, ey ε,n+r,¯l , x ε,n+r,¯l) , ⎪ ⎪ ε,n+r ε,n+r,¯l ε,n+r,¯l ⎪ ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ϕ ε,n+r+1( y ε,n+r+1,¯l , x ε,n+r+1,¯l) ⎪ ⎪ ⎪ ⎪ ¯l  ∈Lε,n+r+1 ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎨ × P0,n+r ( z ε,n+r,l, Aε,n+r+1,¯l) , (8.116) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯l ∈ Lε,n+r , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n ( y , x) = max g(n, ey , x) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ϕ ε,n+1( y ε,n+1,¯l , x ε,n+1,¯l)P0,n ( z ε,n,¯l , Aε,n+1,¯l) . ⎪ ⎪ ⎩ ¯ l ∈Lε,n+1

Let us now formulate conditions of convergence for the above log-reward functions. We are going to apply Theorem 7.2.3. We assume that the condition I7 holds for the pay-off function g(n, e y , x). The condition J8 should be replaced in this case by the following condition: O5 : There exist sets Z n ∈ BZ , n = 0, . . . , N and U n ∈ BU , n = 1, . . . , N such that (a)  μ 0,n ( yε , xε ) →  μ 0,n ( y0 , x0 ), Σ0,n ( y ε , x ε ) → Σ0,n ( y0 , x0 ), and C0,n (x ε , u ) → C0,n (x0 , u ) as ε → 0 for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Z n −1 as ε → 0, and u ∈ Un and n = 1, . . . , N; (b) P{# U0,n ∈ Un } = 1, n = 1, . . . , N;   0,n , C0,n (x0 , U0,n ) ∈ Zn ∩ Z y0 +  μ 0,n ( y0 , x0 ) + Σ0,n ( y0 , x0 )W (c) P{  n } = 1, for  every  z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Z and n = 1, . . . , N, where Z , n = 0, . . . , N n −1 n are sets introduced in the condition I7 . Using the definition of the model, we can write the following formula: P

#

   0,n , C0,n (x0 , U0,n ) ∈ Zn ∪ Z y0 +  μ 0,n ( y0 , x0 ) + Σ0,n ( y0 , x0 )W n

k w2i 1 ) √ e − i =1 2 d w , = g n (z0 , w k ( 2π ) Rk

(8.117)

344 | 8 Convergence of rewards for Markov Gaussian LPP where g n ( z0 , w ) =

#  I  y0 +  μ 0,n ( y0 , x0 ) + Σ0,n ( y0 , x0 ) w, U

   C0,n (x0 , u ) ∈ Zn ∪ Zn G w (du ) .

(8.118)

Let us also introduce sets Gn,z0 = { w : g n ( z0 , w  ) > 0} .

(8.119)

Then, condition O5 (c) can be rewritten in the form of the following relation, which z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Z should be required to hold for  n −1 and n = 1, . . . , N: L k (Gn,z0 ) = 0 .

(8.120)

This form of the condition O5 (c) reduces to the condition O4 (b) for the univariate model without index component considered in Subsection 8.4.1. The following theorem is a corollary of Theorems 7.3.1. Z0,n Theorem 8.4.5. Let the multivariate modulated Markov Gaussian log-price process  and the corresponding space-skeleton approximating Markov log-price processes  Z ε,n are defined, respectively, by stochastic transition dynamic relations (8.104) and (8.113). Let also the conditions B1 [𝛾¯] hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with non­ negative components, and, also, the conditions G3 , N2 , I7 , and O5 hold. Then, for ev­ ery n = 0, 1, . . . , N, the following relation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn ∩ Z , n ϕ ε,n ( y ε , x ε ) → ϕ0,n ( y0 , x0 )

as

ε → 0.

(8.121)

Proof. The fitting condition L4 holds since the corresponding transition probabilities are defined according to formula (8.114). The condition C12 [β¯ ] holds for any vector parameter β¯ = (β 1 , . . . , β k ) with non­ negative components that follows from Lemmas 4.1.8 and 4.5.1. We can always choose a vector β¯ such that, for every i = 1, . . . , k, either β i > 𝛾i if 𝛾i > 0 of β i = 0 if 𝛾i = 0. zε = The condition O5 (a) and (b) imply that the following relation holds any  yε , xε ) →  z0 = ( y0 , x0 ) ∈ Z as ε → 0, and n = 1, . . . , N: ( n #   0,n , C0,n (x ε , U0,n ) yε +  μ 0,n ( y ε , x ε ) + Σ0,n ( y ε , x ε )W #  a.s.  0,n , C0,n (x0 , U0,n ) −→ y0 +  μ 0,n ( y0 , x0 ) + Σ0,n ( y0 , x0 )W as ε → 0 . (8.122) zε = Relation (8.122) obviously implies that the following relation holds any  yε , xε ) →  z0 = ( y0 , x0 ) ∈ Z and n = 1, . . . , N: ( n #   0,n , C0,n (x ε , U0,n ) yε +  μ 0,n ( y ε , x ε ) + Σ0,n ( y ε , x ε )W #  d  0,n , C0,n (x0 , U0,n ) −→ y0 +  μ 0,n ( y0 , x0 ) + Σ0,n ( y0 , x0 )W as ε → 0 . (8.123)

8.4 Skeleton reward approximations for Markov Gaussian LPP |

345

Relation (8.123) is a variant of the condition J10 (a) . The condition O5 (b) is just a variant of the condition J10 (b) . Thus, all conditions of Theorem 7.3.1 hold. By applying this theorem we get the convergence relation (8.121). Let us now formulate conditions of convergence for the corresponding optimal ex­ pected rewards. In this case, the optimal expected reward functional Φ ε for the approximation log-price processes  Z ε,n can be found using the following formula:

ϕ ε,0 ( z ε,0,¯l)p ε,¯l , (8.124) Φε = ¯l∈Lε,0

Z0,0 ) =  z ε,0,¯l}, ¯l = (l0 , . . . , l k ) ∈ Lε,0. where p ε,¯l = P0,0 (Aε,0,¯l) = P{h¯ ε,0( We are going to apply Theorem 7.3.2. The condition D11 [β¯ ] should be replaced by the following condition, assumed to hold for some vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components: D18 [β¯ ]: Ee β i | Y0,0,i | < K82,i , i = 1, . . . , k, for some 1 < K82,i < ∞, i = 1, . . . , k. The condition K9 should be replaced by the following condition imposed on the initial distribution P{Z0,0 ∈ A} = P0,0 (A):   K15 : P0,0 (Z0 ∩ Z 0 ) = 1, where Z0 and Z0 are the sets introduced, respectively, in the conditions I7 and O5 . Theorem 8.4.6. Let the multivariate modulated Markov Gaussian log-price process  Z0,n and the corresponding space-skeleton approximating Markov log-price processes  Z ε,n are defined, respectively, by stochastic transition dynamic relations (8.104) and (8.113). Let also the conditions B1 [𝛾¯] and D18 [β¯ ] hold for some vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions G3 , N2 , I7 , O5 , and K15 hold. Then, the following relation takes place: Φ ε → Φ0 as ε → 0 . (8.125) Proof. The fitting condition M4 holds since the corresponding initial distributions are defined according to formula (8.115). The condition K15 is just reformulation of the condition K9 . Other conditions of Theorem 7.3.2 also hold that was shown in the proof of Theo­ rem 8.4.5. By applying Theorem 7.3.2 we get the convergence relations (8.125). Analogous results can be formulated for multidimensional modulated Markov Gaus­ sian log-price processes with unbounded characteristics. Z0,n Theorem 8.4.7. Let the multivariate modulated Markov Gaussian log-price process  and the corresponding space-skeleton approximating Markov log-price processes  Z ε,n

346 | 8 Convergence of rewards for Markov Gaussian LPP are defined, respectively, by stochastic transition dynamic relations (8.104) and (8.113). Let also the conditions B1 [𝛾¯] hold for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with non­ negative components, and, also, the conditions G4 , N2 , I7 , and O5 hold. Then, for ev­ z ε = ( yε , xε ) →  z0 = ery n = 0, 1, . . . , N, the following relation takes place for any  y0 , x0 ) ∈ Zn ∩ Z : ( n y ε , x ε ) → ϕ0,n ( y0 , x0 ) ϕ ε,n (

as

ε → 0.

(8.126)

Proof. The proof repeats the proof of Theorem 8.4.5. The only change is that Theo­ rem 7.5.3 should be used instead of Theorem 7.3.1. In this case, the condition C16 [β¯ ] holds for any vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components that follows from Lemmas 4.3.17 and 4.5.2. We can always choose a vector β¯ such that, for every i = 1, . . . , k, either β i > 𝛾i if 𝛾i > 0 of β i = 0 if 𝛾i = 0. Other conditions of Theorem 7.5.3 hold as it was shown in the proof of Theo­ rem 8.4.5. By applying Theorem 7.5.3 we get the convergence relation (8.126). Let us also give conditions of convergence for optimal expected rewards Φ ε .  (β¯ ) = (A1 (β¯ ), . . . , A k (β¯ )) given by the In this case, the corresponding functions A ¯ following formula, for β = (β 1 , . . . , β k ), β 1 , . . . , β k ≥ 0, A j (β¯ ) = K26 K27,j

k

1 (β l + 2 k 2 β 2l ) ,

j = 1, . . . , k ,

(8.127)

l =1

where K26 , K27,j , j = 1, . . . , k are constants penetrating condition G4 .  (β¯ ), should assumed to hold for The condition D6 [β¯ ], with the above functions A ¯ some vector parameter β = (β 1 , . . . , β k ) with nonnegative components and the corre­ sponding vectors β¯ i = (β i,1 , . . . , β i,k ) with components β i,j = I (i = j), i, j = 1, . . . , k. It takes, in this case, the following form:  D19 [β¯ ]: E exp{ kj=1 A N,j (β¯ i )|Y0,0,j|} < K83,i , i = 1, . . . , k, for some 1 < K83,i < ∞, i = 1, . . . , k. Theorem 8.4.8. Let the multivariate modulated Markov Gaussian log-price process  Z0,n and the corresponding space-skeleton approximating Markov log-price processes  Z ε,n are defined, respectively, by stochastic transition dynamic relations (8.104) and (8.113). Let also the conditions B1 [𝛾¯] and D19 [β¯ ] hold for some vector parameters 𝛾¯ = (𝛾1 , . . . , 𝛾k ) and β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k either β i > 𝛾i > 0 or β i = 𝛾i = 0, and also the conditions G4 , N2 , I7 , O5 , and K15 hold. Then, the following relation takes place: (8.128) Φ ε → Φ0 as ε → 0 . Proof. Theorem 8.4.8 is a corollary of Theorem 7.5.4. All conditions of the latter theo­ rem hold as was shown in the proofs of Theorems 8.4.5–8.4.7. By applying Theorem 7.5.4, we get the convergence relation (8.128).

8.5 LPP represented by Gaussian random walks

|

347

8.4.3 Space-skeleton approximations for option rewards of multivariate modulated autoregressive type log-price processes with Gaussian noise terms All models of multivariate modulated autoregressive type and autoregressive stochas­ tic volatility type log-price processes with Gaussian noise terms introduced in Sections 1.3 and 1.4 can be considered as particular cases of multivariate modulated Markov Gaussian log-price processes. Thus let one apply space-skeleton reward approximations presented in Subsec­ tions 8.4.1 and 8.4.2 to such autoregressive-type log-price processes. The correspond­ ing results will be given in the second volume of the present book.

8.5 LPP represented by Gaussian random walks In this section, we present results concerning space-skeleton approximations for re­ ward of American-type options for log-price processes represented by univariate and multivariate modulated Gaussian random walks .

8.5.1 Space-skeleton approximations for option rewards of log-price processes represented by univariate Gaussian random walks Let the univariate Markov Gaussian log-price process Y0,n , n = 0, 1, . . . be defined by the following stochastic transition dynamic relation: Y 0,n = Y0,n−1 + μ 0,n + σ 0,n W0,n ,

n = 1, 2, . . . ,

(8.129)

where (a) Y0,0 is a real-valued random variable, (b) W0,n , n = 1, 2, . . . is a sequence of standard i.i.d. normal random variables with mean 0 and variance 1, (c) the random variable Y0,0 and sequence of random variables W0,n , n = 1, 2, . . . are independent, and (d) μ0,n , σ 0,n , n = 1, 2, . . . are real numbers. In this case, an alternative space-skeleton approximation scheme with an additive skeleton structure can also be used. Let us construct the corresponding space-skeleton approximating processes Y ε,n , n = 0, 1, . . . , for ε ∈ (0, ε0 ], according to the algorithm described in Subsection 7.3.1. Let us take the space-skeleton parameters δ ε > 0, λ ε,n ∈ R1 , n = 0, 1, . . . , and + integers m− ε,n ≤ m ε,n , n = 0, 1, . . . .

348 | 8 Convergence of rewards for Markov Gaussian LPP + First, the intervals Iε,n,l, should be constructed for l = m− ε,n , . . . , m ε,n , n = 0, 1, . . . , N, ⎧ 1 − ⎪ if l = m− ⎪ ε,n , ⎪(−∞, δ ε (m ε,n + 2 )] + λ ε,n ⎨ 1 1 − (8.130) Iε,n,l = (δ ε (l − 2 ), δ ε (l + 2 )] + λ ε,n if m ε,n < l < m+ ε,n , ⎪ ⎪ ⎪ ⎩(δ (m+ − 1 ), ∞) + λ + if l = m .

ε

ε,n

ε,n

2

ε,n

and the skeleton points y ε,n,l ∈ Iε,n,l should be chosen in the following way: y ε,n,l = lδ ε + λ ε,n ,

+ l = m− ε,n , . . . , m ε,n ,

n = 0, 1, . . . , N .

Second, the skeleton functions h ε,n (y), n = 0, 1, . . . should be defined,  + h ε,n (y) = y ε,n,l if y ∈ Iε,n,l , m− ε,n ≤ l ≤ m ε,n .

(8.131)

(8.132)

The corresponding space-skeleton approximating Markov process Y ε,n is defined, for every ε ∈ (0, ε0 ], by the following stochastic transition dynamic relation: ⎧ ⎨ Y ε,n = Y ε,n−1 + h ε,n (μ 0,n + σ 0,n W0,n ), n = 1, 2, . . . , (8.133) ⎩ Y ε,0 = h ε,0(Y 0,0) . The approximating log-price process Y ε,n is a skeleton random walk with inde­ pendent random jumps W ε,n = h ε,n (μ 0,n + σ 0,n W0,n ), n = 1, 2, . . . , which have distri­ butions defined by the following formula, for A ∈ B1 , n = 1, 2, . . . : ˜ ε,n (A) = P{W ε,n ∈ A} = P{h ε,n (μ 0,n + σ 0,n W0,n ) ∈ A} P



˜ 0,n (Iε,n,l) = P P{μ 0,n + σ 0,n W0,n ∈ Iε,n,l} . = y ε,n,l ∈ A

(8.134)

y ε,n,l ∈ A

Note that one-point probabilities P{W ε,n = y ε,n,l} = P{μ 0,n + σ 0,n W0,n ∈ Iε,n,l} are interval Gaussian probabilities. As far as the initial random value Y ε,0 = h ε,0(Y0,0 ) is concerned, it has the distri­ bution defined by the following formula, for A ∈ B1 : P ε,0(A) = P{Y ε,0 ∈ A} = P{h ε,0 (Y0,0) ∈ A}



P0,0 (Iε,n,l) = P{Y0,0 ∈ Iε,n,l} . = y ε,0,l ∈ A

(8.135)

y ε,0,l ∈ A

In this case, a pay-off function g ε (n, e y ) = g(n, e y ) does not depend on parame­ ter ε. Let ϕ ε,n (y) be the corresponding log-reward functions for the log-price processes Y ε,n . Let us denote, y ε,n,n+r,l = δ ε l + λ ε,n,n+r ,

+ l = m− ε,n,n +r , . . . , m ε,n,n + r ,

(8.136)

8.5 LPP represented by Gaussian random walks |

349

where, for 0 ≤ n ≤ n + r < ∞, m± ε,n,n + r =

n

+r

m± ε,l ,

λ ε,n,n+r =

l = n +1

n

+r

λ ε,l .

(8.137)

l = n +1

The following lemma is a variant of Lemma 7.4.1. Lemma 8.5.1. Let, for every ε ∈ (0, ε0 ], the log-price process Y ε,n is a random walk with the distributions of jumps and the initial distribution defined, respectively, in rela­ tions (8.134) and (8.135). Then the reward functions ϕ ε,n (y) and ϕ ε,n+r (y + y ε,n,n+r,l), + for points y + y ε,n,n+r,l, l = m− ε,n,n + r, . . . , m ε,n,n + r, r = 1, . . . N − n, are, for every y ∈ R1 , n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ ϕ ε,N (y + y ε,n,N,l) = g(N, e y+y ε,n,N,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ l = m− ⎪ ε,n,N , . . . , m ε,n,N , ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ y + y ε,n,n+r,l ⎪ ⎪ ), ⎪ ⎪ ϕ ε,n+r(y + y ε,n,n+r,l) = max g(n + r, e ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ m+ ⎪ ε,n+r+1 ⎪

⎪ ⎪ ⎪ ϕ ε,n+r+1(y + y ε,n,n+r,l + y ε,n+r+1,l) ⎪ ⎪ ⎪ ⎪ ⎪ l =m− ε,n+r+1 ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

× P{μ 0,n+r+1 + σ 0,n+r+1 W0,n+r+1 ∈ I

ε,n + r +1,l 

' } ,

+ r = N − n − 1, . . . , 1 , l = m− ε,n,n + r , . . . , m ε,n,n +r , + & m ε,n+1

ϕ ε,n+1(y + y ε,n+1,l) ϕ ε,n (y) = max g(n, e y ),

l=m− ε,n+1

'

× P{μ 0,n+1 + σ 0,n+1 W0,n+1 ∈ Iε,n+1,l} .

(8.138)

Let us now formulate conditions of convergence for the above log-reward functions. We are going to apply Theorem 7.4.1. We assume that the condition I6 holds for the pay-off functions g(n, e y ). The condition J9 takes in this case the following form: O6 : P{y + μ 0,n + σ 0,n W0,n ∈ Yn } = 1, y ∈ Yn−1 , for n = 1, . . . , N, where Yn , n = 0, . . . , N are the sets introduced in the conditions I6 . The condition O6 obviously holds if, for every y ∈ Yn−1 , n = 1, . . . , N, either σ 0,n > 0  and the set Y n has zero Lebesgue measure or σ 0,n = 0 and y + μ 0,n ∈ Yn . The following theorem is a variant of Theorem 7.4.1 for log-price processes repre­ sented by Gaussian random walks.

350 | 8 Convergence of rewards for Markov Gaussian LPP Theorem 8.5.1. Let the log-price process Y 0,n is a Gaussian random walk defined by the stochastic transition dynamic relation (8.129) and the corresponding space-skeleton ap­ proximating processes Y ε,n is the random walk defined by stochastic transition dynamic relations (8.133). Let also the condition B4 [𝛾] holds for some 𝛾 ≥ 0, and also the condi­ tions N3 , I6 , and O6 hold. Then, for every n = 0, 1, . . . , N, the following relation takes place for any y ε → y0 ∈ Yn : ϕ ε,n (y ε ) → ϕ0,n (y0 )

as

ε → 0.

(8.139)

Proof. The condition L5 holds in this case that follows from relation (8.134). In this case, the condition G1 automatically holds. Thus, by Lemmas 4.1.8 and 4.5.1, the condition C13 [β ] holds for any β ≥ 0. We can always choose some β > 𝛾 if 𝛾 > 0 of β = 0 if 𝛾 = 0. As was mentioned above, the condition J9 takes in this case the form of the con­ dition O6 . Thus, all conditions of Theorem 7.4.1 hold. By applying this theorem we get the convergence relation (8.139). Let us now formulate conditions of convergence for the corresponding optimal ex­ pected rewards. In this case, the optimal expected reward functional Φ ε for the approximation log-price processes Y ε,n can be found, for every ε ∈ (0, ε0 ], using the following for­ mula: m+ ε,0

ϕ ε,0 (y ε,0,l)p ε,0 , (8.140) Φε = l=m− ε,0 + where p ε,l = P0,0 (Iε,0,l) = P{h ε,0(Y0,0 ) = y ε,0,l}, l = m− ε,0 , . . . , m ε,0 . We are going to apply Theorem 7.4.2. We assume that the condition D16 [β ] holds for some appropriately chosen param­ eter β ≥ 0. The condition K9 should be replaced by the following condition imposed on the initial distribution P{Y0,0 ∈ A} = P0,0 (A): K16 : P0,0 (Y0 ) = 1, where Y0 is the set introduced in the condition I6 .

Theorem 8.5.2. Let the log-price process Y 0,n is a Gaussian random walk defined by the stochastic transition dynamic relation (8.129) and the corresponding space-skeleton ap­ proximating log-price processes Y ε,n are random walks defined by stochastic transition dynamic relations (8.133). Let also the conditions B4 [𝛾] and D16 [β ] hold with parame­ ters 𝛾 and β such that either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions N3 , I6 , O6 , and K16 hold. Then, the following relation takes place: Φ ε → Φ0

as

ε → 0.

(8.141)

8.5 LPP represented by Gaussian random walks

|

351

Proof. The fitting condition M5 holds since the corresponding initial distributions are defined according to formula (8.135). The condition K16 is just reformulation of the condition K9 . Other conditions of Theorem 7.4.2 also hold that was shown in the proof of Theo­ rem 8.4.9. By applying Theorem 7.4.2 we get the convergence relations (8.141).

8.5.2 Space-skeleton approximations for option rewards for log-price processes represented by multivariate modulated Gaussian random walks 0,n , X0,n ), n = 0, 1, . . . is a multivari­ Let us assume that a log-price process  Z0,n = (Y ate modulated Markov Gaussian log-price process given by the following stochastic transition dynamic relation: ⎧ ⎨Y 0,n−1 +   0,n , 0,n = Y μ 0,n (X0,n−1) + Σ0,n (X0,n−1)W (8.142) ⎩ X0,n = C0,n (X0,n−1 , U0,n ), n = 1, 2, . . . , 0,0 , X0,0) = ((Y 0,0,1 , . . . , Y 0,0,k )), X0,0) is a random vector tak­ where (a)  Z0,0 = (Y  0,n , U0,n ) = ((W0,n,1 , . . . , W0,n,k ), U0,n ), n = ing values in space Z = Rk × X, (b) (W 1, 2, . . . is a sequence of independent random vectors taking values in space Rk × U, where U is a measurable space with a σ-algebra of measurable subsets BU , (c) the 0,0 , X0,0 ) and the random sequence (W  0,n , U0,n ), n = 1, 2, . . . are random vector (Y  independent, (d) the random vector W0,n = (W0,n,1 , . . . , W0,n,k ) has a standard mul­ tivariate normal distribution with EW0,n,i = 0, EW0,n,i W0,n,j = I (i = j), i, j = 1, . . . , k, for every n = 1, 2, . . . , (e) the random variables U0,n have a regular conditional dis­  0,n = w tribution G w (B) = P{U0,n ∈ B/W  } (a probability measure in B ∈ BU for every w  ∈ Rk and a measurable function in w  ∈ Rk for every B ∈ BU ), for every μ0,n (x) = (μ 0,n,1(x), . . . , μ 0,n,k(x)), n = 1, 2, . . . are vector func­ n = 1, 2, . . . , (f)  tions, which components are measurable functions acting from the space X to R1 , (g) Σ0,n (x) = σ 0,n,i,j(x) , n = 1, 2, . . . are k × k matrix functions with elements σ 0,n,i,j(x), i, j = 1, . . . , k, which are measurable functions acting from the space X to R1 ; (h) C0,n (x, u ) is a measurable function acting from the space X × U to the space X. In this case, a space-skeleton approximation scheme with the additive skeleton structure can also be used. Let us construct the corresponding space-skeleton approximating processes  Z ε,n , n = 0, 1, . . . , for ε ∈ (0, ε0 ], according to the algorithm described in Subsection 7.4.5. + Let δ ε,i > 0, λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 0, 1, . . . . and m− ε,n,j ≤ m ε,n,j, j = 0, . . . , k, n = 0, 1, . . . be integer numbers.

352 | 8 Convergence of rewards for Markov Gaussian LPP + First, the intervals Iε,n,i,l should be constructed for l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . , ⎧ 1 − ⎪ if l = m− ⎪ ε,n,i , ⎪(−∞, δ ε,i (m ε,n,i + 2 )] + λ ε,n,i ⎨ 1 1 − (8.143) Iε,n,i,l = (δ ε,i(l − 2 ), δ ε,i (l + 2 )] + λ ε,n,i if m ε,n,i < l < m+ ε,n,i , ⎪ ⎪ ⎪ + ⎩(δ (m+ − 1 ), ∞) + λ if l = m .

ε,i

ε,n,i

2

ε,n,i

ε,n,i

+ and cubes Iε,n,l1,...,l k , l i = m− ε,n,i , . . . , m ε,n,i , i = 1, . . . , k, n = 0, 1, . . . should be de­ fined, Iε,n,l1,...,l k = Iε,n,1,l1 × · · · × I ε,n,k,l k , (8.144) + Then, the skeleton points y ε,n,i,li ∈ Iε,n,i,li , l i = m− ε,n,i , . . . , m ε,n,i, i = 1, . . . , k, n = 0, 1, . . . should be defined,

y ε,n,i,li = l i δ ε,i + λ ε,n,i .

(8.145)

+ Second, nonempty sets Jε,n,l ⊂ BX , l = m− ε,n,0, . . . , m ε,n,0, n = 0, 1, . . . , such that (a) Jε,n,l ∩ Jε,n,l = ∅, l ≠ l , n = 0, 1, . . . , (b) X = ∪m−ε,n,0 ≤l≤m+ε,n,0 Jε,n,l, n = 0, 1, . . . , should be constructed. Recall that our model assumption is the X which is a Polish metric space. In this case, it is natural to assume that there exist some “large” sets K ε,n, n = 0, 1, . . . , + “small” nonempty sets K ε,n,l ⊆ BX , l = m− ε,n,0, . . . , m ε,n,0, n = 0, 1, . . . such that   − (c) K ε,n,l ∩ K ε,n,l = ∅, l ≠ l , n = 0, 1, . . . , (d) ∪m ε,n,0 ≤l≤m+ε,n,0 K ε,n,l = K ε,n , n = 0, 1, . . . . The sets Jε,n,l can be defined in the following way, for n = 0, 1, . . . : ⎧ + ⎨K ε,n,l if m− ε,n,0 ≤ l < m ε,n,0 , Jε,n,l = (8.146) ⎩K + ∪ Kε,n if l = m+ .

ε,n,m ε,n,0

ε,n,0

+ Then the skeleton points x ε,n,l0 , l0 = m− ε,n,0 , . . . , m ε,n,0, n = 0, 1, . . . should be chosen so that the following inclusion relation would hold, for every l0 = m− ε,n,0 , . . . , m+ , n = 0, 1, . . . : ε,n,0 x ε,n,l0 ∈ Jε,n,l0 . (8.147)

Let us introduce index sets, for n = 0, 1, . . . , + Lε,n = {¯l = (l0 , . . . , l k ), l j = m− ε,n.j , . . . , m ε,n,j , j = 0, . . . , k }

(8.148)

z ε,n,¯l ∈ Aε,n,¯l can be defined Third, the skeleton sets Aε,n,¯l and skeleton points,  for ¯l = (l0 , l1 , . . . , l k ) ∈ Lε,n , n = 1, 2, . . . , in the following way: A ε,n,¯l = Iε,n,l1,...,l k × Jε,n,l0 ,

(8.149)

 z ε,n,¯l = ( y ε,n,˜l, x ε,n,˜l) = ((y ε,n,1,l1 , . . . , y ε,n,k,l k ), x ε,n,l0) .

(8.150)

and

8.5 LPP represented by Gaussian random walks |

353

Fourth, the skeleton functions, h ε,n,i(y), y ∈ R1 , for ε ∈ (0, ε0 ] and i = 1, . . . , k, n = 0, 1, . . . should be defined,  + (8.151) h ε,n,i(y) =  y ε,n,l if y ∈ Iε,n,i,l , m− ε,n,i ≤ l ≤ m ε,n,i , ˆ ε,n ( y),  y = (y1 , . . . , y k ) ∈ Rk , for ε ∈ (0, ε0 ] and and the vector skeleton functions h n = 0, 1, . . . should be defined, ˆ ε,n ( h y) = (h ε,n,1(y1 ), . . . , h ε,n,k(y k )) .

(8.152)

Also, the skeleton functions h ε,n,0(x), x ∈ X, for ε ∈ (0, ε0 ] and n = 0, 1, . . . , should be defined  + (8.153) h ε,n,0(x) = x ε,n,l if x ∈ Jε,n,l , m− ε,n,0 ≤ l ≤ m ε,n,0 . The corresponding space-skeleton approximating modulated Markov processes Y ε,n are defined by the following stochastic transition dynamic relations: #  ⎧ ˆ ε,n   ε,n = Y ε,n−1 + h  0,n , ⎪ Y μ 0,n (h ε,n−1,0(X ε,n−1)) + Σ0,n (h ε,n−1,0(X ε,n−1))W ⎪ ⎪ ⎨ (8.154) X ε,n = h ε,n,0(C0,n (h ε,n−1,0(X ε,n−1), U0,n )) , n = 1, 2, . . . , ⎪ ⎪ ⎪ ⎩ ˆ ε,0(Y 0,0 ) , X ε,0 = h ε,0,0(X0,0) . Y ε,0 = h ε,n , X ε,n ) has one-step tran­ ε,n = (Y In this case, the Markov log-price process Z ˜ ε,n (x, A) = P{(W  ε,n , X ε,n ) ∈ A/X ε,n−1 = x} given for x ∈ X, sition probabilities P A ∈ BZ , n = 1, . . . by the following relation:

˜ 0,n (h ε,n−1,0(x), Aε,n,¯l) ˜ ε,n (x, A) = P P ( y ε,n,¯l ,x ε,n,¯l )∈ A

=



P

( #

 0,n ,  μ 0,n (h ε,n−1,0(x)) + Σ0,n (h ε,n−1,0(x))W

( y ε,n,¯l ,x ε,n,¯l )∈ A



)

C0,n (h ε,n−1,0(x), U0,n ) ∈ A ε,n,¯l =

&# I  μ 0,n (h ε,n−1,0(x)) + Σ0,n (h ε,n−1,0(x)) w,



( y ε,n,¯l ,x ε,n,¯l )∈ A R U k



'

k 1 − i =1 e C0,n (x, u ) ∈ A ε,n,¯l G w (du ) √ ( 2π )k

w2i 2

dw .

(8.155)

ε,0 , X ε,0) ∈ A}, A ∈ BZ is con­ As far as the initial distribution P ε,0(A) = P{(Y cerned, it takes the following form:

P ε,0(A) = P0,0 (Aε,0,¯l) ( y ε,0,¯l ,x ε,0,¯l )∈ A

=



( y ε,0,¯l ,x ε,0,¯l )∈ A

0,0 , X0,0 ) ∈ A ε,0,¯l} . P{(Y

(8.156)

354 | 8 Convergence of rewards for Markov Gaussian LPP In this case, the pay-off function g ε (n, ey , x) = g(n, ey , x) does not depend on the parameter ε. Let us ϕ ε,n ( y , x) be the corresponding log-reward functions for the log-price pro­ Z ε,n. cesses  + Let us denote, for ¯l = (l0 , . . . , l k ), l j = m− ε,n,n +r,j, . . . , m ε,n,n + r,j, j = 0, . . . , k,  y ε,n,n+r,¯l = (δ ε,1 l1 + λ ε,n,n+r,1, . . . , δ ε,k l k + λ ε,n,n+r,k) ,

(8.157)

x ε,n,n+r,¯l = x ε,n+r,¯l = x ε,n+r,l0 ,

(8.158)

and where, for j = 1, . . . , k, 0 ≤ n < n + r < ∞, m± ε,n,n + r,j =

n

+r

m± ε,l,j ,

λ ε,n,n+r,j =

l = n +1

n

+r

λ ε,l,j .

(8.159)

l = n +1

and, for 0 ≤ n < n + r < ∞, ± m± ε,n,n + r,0 = m ε,n + r,0 .

(8.160)

The following lemma is a variant of Lemma 7.4.2. Lemma 8.5.2. Let, for every ε ∈ (0, ε0 ], the space-skeleton approximating process  Z ε,n is represented modulated random walk with jump distributions probabilities and the initial distribution defined, respectively, in relations (8.155) and (8.156). Then the re­ y , x) and ϕ ε,n+r( y+ y ε,n,n+r,¯l, x ε,n+r,¯l) for points  z = ( y , x) = ward functions ϕ ε,n ( y + y ε,n,n+r,¯l, x ε,n+r,¯l), ¯l = (l0 , l1 , . . . , l k ), l j = m− ((y1 , . . . , y k ), x) and ( ε,n,n +r,j, + . . . , m ε,n,n+r,j, j = 0, . . . , k, r = 1, . . . N − n, are, for every  z ∈ Rk × Jε,n,l0 , l 0 = − + m ε,n,0, . . . , m ε,n,0, n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ ϕ ( y+ y ε,n,N,¯l, x ε,N,¯l) = g(N, ey+y ε,n,N,¯l , x ε,N,¯l) , ⎪ ⎪ ⎪ ε,N ⎪ ⎪ + ⎪ ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ j = 0, . . . , k , ⎪ ε,n,N,j, . . . , m ε,n,N,j , ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r( y+ y ε,n,n+r,¯l, x ε,n+r,¯l) ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪

⎪ ⎪ ⎪ ϕ ε,n+r+1( y+ y ε,n,n+r,¯l = max g(n + r, ey+y ε,n,n+r,¯l , x ε,n+r,¯l), ⎪ ⎪ ⎪ ⎪ ⎪ ¯l  ∈Lε,n+r+1 ⎪ ⎪ ⎪ ' ⎪ ⎪ ⎪ ⎪ ⎨ ˜ + y ε,n+r+1,¯l  , x ε,n+r+1,¯l  )P0,n+r+1(x ε,n+r,¯l, Aε,n+r+1,¯l) , ⎪ ⎪ ⎪ ⎪ + ¯l = (l0 , l1 , . . . , l k ) , l j = m− ⎪ ⎪ ε,n,n +r,j, . . . , m ε,n,n + r,j , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j = 0, . . . , k , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ( y , x ) = max g(n, ey , x), ⎪ ε,n ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎪ ⎪

⎪ ⎪ ⎪ ˜  ⎪ ϕ ε,n+1( y+ y ε,n+1,¯l  , x ε,n+1,¯l  )P0,n+1(x ε,n,l0 , Aε,n+1,¯l) . ⎪ ⎩ ¯l  ∈Lε,n+1

(8.161)

8.5 LPP represented by Gaussian random walks |

355

Let us now formulate conditions of convergence for the above log-reward functions. We are going to apply Theorem 7.4.5. The condition C14 [β¯ ] should be replaced in this case by the following reduced version of the condition G3 : G11 : max1≤n≤N, i,j=1,...,k supx∈X(|μ 0,n,i(x)| + σ 20,n,i,j(x)) < K84 , for some 0 < K84 < ∞. We assume that the condition I7 holds for pay-off functions g(n, ey , x). The condition J12 takes in this case the following form: O7 : There exist sets Xn ∈ BX , n = 0, . . . , N and Un ∈ BU , n = 1, . . . , N such that: μ 0,n (x ε ) →  μ 0,n (x0 ), Σ0,n (x ε ) → Σ0,n (x0 ) and C0,n (x ε , u ) → C0,n (x0 , u ) as (a)  ε → 0, for any x ε → x0 ∈ Xn−1 as ε → 0, u ∈ Un , n = 1, . . . , N; (b) P{U0,n ∈ Un } = 1, n = 1, . . . , N;  0,n , C0,n (x0 , U0,n )) ∈ Zn ∩ Z y0 + μ 0,n (x0 ) + Σ0,n (x0 )W (c) P{( n } = 1 for every    z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Zn−1 and n = 1, . . . , N, where Zn , n = 0, . . . , N are sets introduced in the condition I7 and Z n = Rk × X n , n = 0, . . . , N. 0,n is a multivariate modulated Gaussian ran­ Theorem 8.5.3. Let the log-price process Z dom walk defined by the stochastic transition dynamic relation (8.142) and the corre­ Z ε,n are de­ sponding space skeleton approximating Markov log-price process processes  fined by the stochastic transition dynamic relations (8.154). Let also the condition B1 [𝛾¯] holds for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components, and also the conditions G11 , N4 , I7 , and O7 hold. Then, for every n = 0, 1, . . . , N, the following re­ lation takes place for any  z ε = ( yε , xε ) →  z0 = ( y0 , x0 ) ∈ Zn−1 ∩ Z n −1 :

y ε , x ε ) → ϕ0,n ( y0 , x0 ) ϕ ε,n (

as

ε → 0.

(8.162)

Proof. The condition L6 holds in this case that follows from relation (8.155). The condition G11 implies, by Lemmas 4.2.1 and 4.5.1, that the condition C14 [β¯ ] holds for any vector parameter β¯ = (β 1 , . . . , β k ) with nonnegative components. We can always choose parameter β¯ = (β 1 , . . . , β k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0. As was mentioned above, the condition J12 takes in this case the form of the con­ dition O7 . Thus, all conditions of Theorem 7.4.5 hold. By applying this theorem we get the convergence relation (8.162). Let us now formulate conditions of convergence for the corresponding optimal ex­ pected rewards. We are going to apply Theorem 7.4.6. In this case, the optimal expected reward Φ ε for the approximation log-price pro­ cesses  Z ε,n can be found using the following formula:

ϕ ε,0 ( z ε,0,¯l)p ε,¯l , (8.163) Φε = ¯l∈Lε,0

356 | 8 Convergence of rewards for Markov Gaussian LPP ˆ ε,0 (Y 0,0) =  0,0) = x ε,0,¯l} = P0,0 (A ε,0,¯l), ¯l = (l0 , . . . , where p ε,¯l = P{h y ε,0,¯l , h ε,0(X l k ) ∈ Lε,0. We assume that the condition D18 [β¯ ], replacing in this case the condition D11 [β¯ ], holds for some appropriately chosen vector parameter β¯ = (β 1 , . . . , β k ) with nonneg­ ative components. The condition K1 should be replaced by the following condition imposed on the 0,0 } = P0,0 (A): initial distribution P{Z   K17 : P0,0 (Z0 ∩ Z0 ) = 1, where Z0 and Z 0 are the sets introduced, respectively, in the conditions I7 and O7 . Theorem 8.5.4. Let the conditions B1 [𝛾¯] and D18 [β¯ ] hold for some vector parameters β¯ = (β 1 , . . . , β k ) and 𝛾¯ = (𝛾1 , . . . , 𝛾k ) such that, for every i = 1, . . . , k, either β i > 𝛾i > 0 or β i = 𝛾i = 0. Also let the conditions G11 , N4 , I7 , O7 , and K17 hold. Then, the following relation takes place: (8.164) Φ ε → Φ0 as ε → 0 . Proof. The fitting condition M6 holds since the corresponding initial distributions are defined according to formula (8.156). The condition K17 is just reformulation of the condition K10 . Other conditions of Theorem 7.4.6 also hold that was shown in the proof of Theo­ rem 8.5.3. By applying Theorem 7.4.6, we get the convergence relations (8.164).

9 Tree-type approximations for Markov Gaussian LPP In this chapter, we investigate problems of fitting parameters for discrete time Markov log-price processes with binomial or trinomial transition probabilities to parameters of a discrete time Markov log-price processes with Gaussian transition probabilities. The corresponding fitting procedures are based on exact or asymptotic equalizing of moments for the corresponding processes (first two moments in the univariate case or vectors of expectations and covariance matrices, in the multivariate case). It should also be mentioned that this moment fitting should be realized under additional re­ combining conditions that should be imposed on parameters of the corresponding bi­ nomial or trinomial models, in order the backward recurrence tree reward algorithms would be effective for these models due to polynomial rate of growth for the number of tree nodes as functions of the number of steps in corresponding trees. In Section 9.1, we present well-known exact and asymptotic fitting formulas for univariate log-price processes represented by homogeneous and inhomogeneous in time binomial random walks. In Section 9.2, we present fitting formulas for multivariate log-price processes rep­ resented by homogeneous in time binomial random walks. In the case of multivariate models with dimension k ≥ 3, the special attention should also be given to the prob­ lem of construction of multivariate binomial distributions of jumps with given expec­ tations and covariances. In Section 9.3, we present fitting formulas for multivariate log-price processes rep­ resented by inhomogeneous in time trinomial random walks. It should be mentioned that in the case of multivariate inhomogeneous in time models the binomial random walks may possess not enough free parameters for exact or even asymptotic fitting. However, such fitting can be realized by using trinomial models. As in the case of bi­ nomial models, if dimension k ≥ 3, the special attention should also be given to the problem of construction of multivariate trinomial distributions of jumps with given expectations and covariances. In Section 9.4, we present fitting formulas for univariate log-price processes repre­ sented by homogeneous in time but inhomogeneous in space binomial random walks. In Section 9.5, we obtain fitting formulas for multivariate log-price processes rep­ resented by inhomogeneous in time and space trinomial random walks. In the case of Markov Gaussian-type log-price processes, the large number of bino­ mial, trinomial and multinomial tree approximations has been proposed in the works related to reward approximation models for both discrete and continuous time pro­ cesses. It should be noted that discrete time models and the corresponding approxi­ mations have their own important value. At the same time, they can be effectively used as approximations for the corresponding continuous time models.

358 | 9 Tree-type approximations for Markov Gaussian LPP We think that the fitting procedures for multivariate models, inhomogeneous in time or time and space, given in Lemmas 9.3.1 and 9.5.2, are the most interesting in this chapter, and to our knowledge, these lammas contain new results. Results about the convergence of the corresponding reward binomial and trino­ mial approximations are given in Chapter 10.

9.1 Univariate binomial tree approximations In this section, we present fitting formulas for univariate binomial tree approximation models inhomogeneous in time but homogeneous is space.

9.1.1 Exact fitting of parameters for a classical binomial tree model Let us consider the model of homogeneous in time Gaussian random walk, which can be considered as a discrete time analog of the Wienner process (Brownian motion). In this case, a log-price process Y0,n is given by the following stochastic transition dynamic relation: (9.1) Y0,n = Y0,n−1 + W0,n , n = 1, 2, . . . . where (a) Y0,0 is a real-valued random variable, (b) W0,n , n = 1, 2, . . . is a sequence of i.i.d. normal random variables with a mean EW0,1 = μ and a variance VarW0,1 = σ 2 , and (c) the random variable Y0,0 and the random sequence W0,n , n = 1, 2, . . . are independent. Also, usual assumptions are (d) μ ∈ R1 and (e) σ > 0. Note that the assumption (e) excludes the trivial case of degenerated log-price process Y0,n . Indeed, if σ = 0, then Y0,n = Y0,0 + μn, n = 0, 1, . . . . We shall approximate the log-price process Y0,n by log-price processes Y ε,n rep­ resented by homogeneous in time binomial random walk. In this case, the log-price process Y ε,n is given, for every ε ∈ (0, ε0 ], by the following dynamic transition rela­ tion: (9.2) Y ε,n = Y ε,n−1 + W ε,n , n = 1, 2, . . . . where (a) Y ε,0 is a real-valued random variable, (b) W ε,n = W ε,n,1 + · · · + W ε,n,r ε , n = 1, 2, . . . , where W ε,n,l , l, n = 1, 2, . . . is a family of i.i.d. binary random binomial variables, which take values δ ε,+ and δ ε,− with probabilities, respectively, p ε,+ and p ε,−, and (c) the random variable Y ε,0 and the family of random variables W ε,n,l , l, n = 1, 2, . . . are independent. Also, usual assumptions are (d) δ ε,− < δ ε,+, (e) 0 ≤ p ε,+ = 1 − p ε,− ≤ 1, and (f) r ε is a positive integer number.

9.1 Univariate binomial tree approximations

|

359

It is useful to note that W ε,n , n = 1, 2, . . . are binomial random variables taking values w ε,r ε ,l = lδ ε,+ + (r ε − l)δ ε,− , l = 0, . . . , r ε with the corresponding probabilities r −l p ε,l = C lr ε p lε,+ p ε,ε − , l = 0, . . . , r ε . We use the process Y ε,n for constructing approximation algorithms for rewards of American-type options for the process Y0,n . As a matter of fact, rewards for the logprice processes Y ε,n well approximate the rewards for log-price process Y0,n , if we fit in a natural way parameters of these processes. A usual approach is to choose the parameters δ ε,± , p ε,±, and r ε in such a way that the following exact moment-fitting equalities hold, for every ε ∈ (0, ε0 ]: { EW ε,1 = EW0,1 ,

VarW ε,1 = VarW0,1 .

(9.3)

Using the sum-representation for binomial random variables W ε,n used in the defining transition relation (9.2), one can easily transform equalities (9.3) to the equiv­ alent form of the following system of equations, for every ε ∈ (0, ε0 ]: ⎧ ⎪ ⎨ EW ε,1 = r ε (δ ε, + p ε, + + δ ε, − p ε, −) = μ , (9.4) μ ⎪ ⎩ VarW ε,1 = r ε (δ2ε, + p ε, + + δ2ε, − p ε, − − ( )2 ) = σ 2 . rε It is convenient to symmetrize the values of jumps by representing binary random variables W ε,n,l , l, n = 1, 2, . . . in the following form: ⎧ ⎨ δ ε with probability p ε, + ,  (9.5) W ε,n,l = λ ε + W ε,n,l = λε + ⎩ −δ ε with probability p ε, − , where

δ ε,+ − δ ε,− δ ε,+ + δ ε,− , λε = . (9.6) 2 2 It follows from relation (9.5) that the transition dynamic relation (9.2) can be rewritten in the following form: δε =

 Y ε,n = Y ε,n−1 + λε + W ε,n ,

n = 1, 2, . . . .

(9.7)

   where (a) Y ε,0 is a real-valued random variable, (b) W ε,n = W ε,n,1 + · · · + W ε,n,r , ε  n = 1, 2, . . . , where W ε,n,l , l, n = 1, 2, . . . is a family of i.i.d. binary random variables, which takes values δ ε and −δ ε with probabilities, respectively, p ε,+ and p ε,−, (c) the  , l, n = 1, 2, . . . are random variable Y ε,0 and the family of random variables W ε,n,l  independent, and (d) λ ε = r ε λ ε . Indeed, it follows from the following representation that takes place for the ran­ dom variable W ε,n , for every n = 1, 2, . . . :    + · · · + W ε,n,r = λε + W ε,n . W ε,n = W ε,n,1 + · · · + W ε,n,r ε = r ε λ ε + W ε,n,1 ε

The system (9.4) can be rewritten in following form: ⎧ μ ⎪ ⎪ ⎪ ⎨ δ ε (p ε, + − p ε, −) + λ ε = r ε , ⎪ μ σ2 ⎪ ⎪ . δ2ε − ( − λ ε )2 = ⎩ rε rε

(9.8)

(9.9)

360 | 9 Tree-type approximations for Markov Gaussian LPP The system (9.9) should be supplemented by the following consistency relations that should hold for admissible values of parameters: p ε,± ≥ 0 ,

p ε,+ + p ε,− = 1 ,

δε > 0 ,

λ ε ∈ R1 ,

r ε = 1, 2, . . . .

(9.10)

As was pointed out in Chapter 3, the recombining condition should hold for the process Y ε,n , in order the corresponding algorithms for computing of option rewards for the log-price processes Y ε,n would be effective. As was pointed in Subsection 3.3.1, the recombining condition holds for any ho­ mogeneous in time random walk with bounded jumps taking values on a grid of points λ ε r ε ± δ ε l, l = 0, 1, . . . . The binomial random walk Y ε,n belongs to this class, for any parameters δ ε > 0, λ ε ∈ R1 , and r ε = 1, 2, . . . . The system (9.9) has infinitely many solutions given by the following formulas: ⎧ ⎪ σ2 μ ⎪ ⎪ + ( − λ ε )2 , ⎪ δε = ⎪ ⎨ rε rε ⎛ ⎞ (9.11) μ ⎪ ⎪ 1⎝ rε − λε ⎠ ⎪ ⎪ ⎪ , ⎩ p ε, ± = 2 1 ± δ ε

where solutions are parameterized by the parameters λ ε ∈ R1 and r ε = 1, 2, . . . . Note that relations (9.10) hold for solutions given by formulas (9.11). Let now take ı = +, − , δ ≥ σ > 0 and choose, √ ı δ2 − σ2 μ + . (9.12) λε = √ rε rε Substituting λ ε defined by relation (9.12) in relation (9.11), we get the following ⎧ formulas: δ ⎪ ⎪ ⎪ δε = √ , ⎪ ⎨ rε & ' √ (9.13) ⎪ 1 ı δ2 − σ2 ⎪ ⎪ ⎪ 1∓ , ⎩ p ε, ± = 2 δ Thus, the following formulas give solutions of the system (9.9): & ' √ √ ı δ2 − σ2 μ δ 1 ı δ2 − σ2 + , δ = , p = 1 ∓ , λε = √ √ ε ε, ± rε rε rε 2 δ

(9.14)

parameterized by the parameters√ı = +, − , δ ≥ σ and r ε = 1, 2, . . . . μ δ2 −σ2 In this case, δ ε,± = ± √δr ε + ı √ + r ε . Thus, δ ε, − < δ ε, +. rε Note also that the consistency relations (9.10) hold for solutions given by formulas (9.14) for any ı = +, − , δ ≥ σ and r ε = 1, 2, . . . . Formulas (9.14) take the simplest form in the case, where the parameter δ = σ. In this case, we get the following formulas for the corresponding solutions of the system (9.9): μ σ 1 , δ ε = √ , p ε,± = , λε = (9.15) rε rε 2 parameterized by the parameter r ε = 1, 2, . . . .

9.1 Univariate binomial tree approximations

| 361

In this case, δ ε,± = ± √σr ε . Thus, δ ε,− < 0 < δ ε,+. There are also alternative variants of solutions with simplified parametrization for the system (9.9). For example, we can choose the parameter λ ε = 0. In this case we get the following formulas for solutions of the system (9.9): ⎛ ⎞ 2 ⎟ σ 1 μ 1⎜ μ 1 ⎟, 1+ , p ε,± = ⎜ 1± √ . (9.16) λε = 0 , δε = √ ⎠ 2 rε r ε σ2 2⎝ σ rε 1 μ 1 + rε σ2 where solutions are parameterized by the parameter r ε = 1, 2, . . . . . In this case, δ ε,± = ± √σr ε

1+

1 μ2 rε σ2 .

Thus, δ ε,− < 0 < δ ε,+.

Note also that the consistency relations (9.10) hold for solutions given by formulas (9.16) for any r ε = 1, 2, . . . . The above remarks can be summarized in the following lemma. Lemma 9.1.1. The log-price process Y ε,n represented, for every ε ∈ (0, ε0 ], by bino­ mial random walk, which is defined by the stochastic transition dynamic relation (9.2), realizes the exact moment fitting given by relation (9.3) for the log-price processes Y0,n represented by Gaussian random walk, which is defined by the stochastic transition dy­ namic relation (9.1). This holds if the corresponding parameters for the process Y ε,n are given by one of relations (9.11), (9.14), (9.15), or (9.16). It should be pointed out that the important recombining condition (which guarantees not more than the polynomial rate of growth for number of nodes in the correspond­ ing binomial tree representation for the above model) discussed in Chapter 3 holds automatically for any values of the parameters δ ε and λ ε as well as the parameters p ε,± and r ε , in particular, those given by relations (9.11), (9.14), (9.15), or (9.16). The corresponding tree T ε,(y,n) generated by the trajectories of the log-price pro­ cess Y ε,n+m, m = 0, 1, . . . with the value Y ε,n = y has the total number of points after r steps, (9.17) L ε,n,n+r ≤ 1 + 2r ε + 1 + · · · + 2r ε r + 1 = r ε r2 + (r ε + 1)r + 1 . The expression on the right-hand side in (9.17) is a polynomial (in this case quadratic) function of r.

9.1.2 Exact fitting of parameters for an univariate inhomogeneous in time binomial tree model Let us consider the model of inhomogeneous in time Gaussian random walk, which can be considered as a discrete time analog of a Gaussian process with independent increments. In this case, a log-price process Y0,n is given by the following stochastic transition dynamic relation: Y0,n = Y0,n−1 + W0,n ,

n = 1, 2, . . . .

(9.18)

362 | 9 Tree-type approximations for Markov Gaussian LPP where (a) Y0,0 is a real-valued random variable, (b) W0,n , n = 1, 2, . . . is a sequence of independent normal random variables with means EW0,n = μ n , n = 1, 2, . . . and vari­ ances VarW0,n = σ 2n , n = 1, 2, . . . , and (c) the random variable Y0,0 and the random sequence W0,n , n = 1, 2, . . . are independent. Also, usual assumptions are (d) μ n ∈ R1 , n = 1, 2, . . . and (e) σ n > 0, n = 1, 2, . . . . We approximate the log-price process Y0,n by log-price processes Y ε,n represented by inhomogeneous in time binomial random walks. In this case, the log-price process Y ε,n is given, for every ε ∈ (0, ε0 ], by the following stochastic transition dynamic relation: (9.19) Y ε,n = Y ε,n−1 + W ε,n , n = 1, 2, . . . . where (a) Y ε,0 is a real-valued random variable, (b) W ε,n = W ε,n,1 + · · · + W ε,n,r ε,n , n = 1, 2, . . . , where W ε,n,l , l, n = 1, 2, . . . is a family of independent binary random variables, which take, for every n = 1, 2, . . . , values δ ε,n,+ and δ ε,n,− with probabili­ ties, respectively, p ε,n,+ and p ε,n,−, and (c) the random variable Y ε,0 and the family of random variables W ε,n,l , l, n = 1, 2, . . . are independent. Also, usual assumptions are (d) δ ε,n,− < δ ε,n,+, n = 1, 2, . . . , (e) 0 ≤ p ε,n,+ = 1 − p ε,n,− ≤ 1, n = 1, 2, . . . , and (f) r ε,n , n = 1, 2, . . . are positive integer numbers. It is useful to note that W ε,n , n = 1, 2, . . . are binomial random variables taking, for every n = 1, 2, . . . , values w ε,r ε,n ,l = lδ ε,n,+ + (r ε,n − l)δ ε,n,−, l = 0, . . . , r ε,n with r

−l

ε,n the corresponding probabilities p ε,n,l = C lr ε,n p lε,n,+p ε,n, − , l = 0, . . . , r ε,n . The fitting should usually be made for a finite discrete time interval [1, . . . , N ]. We can try to choose parameters δ ε,n,±, p ε,n,±, r ε,n , n = 1, . . . , N in such a way that the following exact moment-fitting equalities hold, for every ε ∈ (0, ε0 ]:  EW ε,n = EW0,n , VarW ε,n = VarW0,n , n = 1, . . . , N . (9.20)

Using the sum-representation for binomial random variables W ε,n used in the defining transition relation (9.19), one can easily transform equalities (9.20) in the equivalent form of the following system of equations, for every ε ∈ (0, ε0 ]: ⎧ EW ε,n = r ε,n (δ ε,n,+p ε,n,+ + δ ε,n,−p ε,n,−) = μ n , ⎪ ⎪ ⎪ ⎪ ⎛ ⎪ & '2 ⎞ ⎪ ⎨ μ n ⎠ = σ2 , VarW ε,n = r ε,n ⎝δ2ε,n,+p ε,n,+ + δ2ε,n,−p ε,n,− − (9.21) n ⎪ r ε,n ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, . . . , N . As in the case of homogeneous in time model, it is convenient to symmetrize the values of jumps by representing binary random variables W ε,n,l in the following form, for l, n = 1, 2, . . . : ⎧ ⎨ +δ ε,n with probability p ε,n, + ,  = λ ε,n + (9.22) W ε,n,l = λ ε,n + W ε,n,l ⎩ −δ ε,n with probability p ε,n, − , where δ ε,n =

δ ε,n,+ − δ ε,n,− , 2

λ ε,n =

δ ε,n,+ + δ ε,n,− . 2

(9.23)

9.1 Univariate binomial tree approximations

| 363

It follows from relation (9.22) that the transition dynamic relation (9.19) can be rewritten in the following form:  , Y ε,n = Y ε,n−1 + λε,n + W ε,n

n = 1, 2, . . . .

(9.24)

   = W ε,n,1 + · · · + W ε,n,r , where (a) Y ε,0 is a real-valued random variable, (b) W ε,n ε,n  n = 1, 2, . . . , where W ε,n,l , l, n = 1, 2, . . . is a family of i.i.d. binary random variables, which take values δ ε,n and −δ ε,n with probabilities, respectively, p ε,n,+ and p ε,n,−,  (c) the random variable Y ε,0 and the family of random variables W ε,n,l , l, n = 1, 2, . . . are independent; (d) λε,n = r ε,n λ ε,n , n = 1, 2, . . . . The system (9.21) can be rewritten, for every ε ∈ (0, ε0 ], in the following form: ⎧ μn ⎪ ⎪ δ ε,n (p ε,n,+ − p ε,n,−) + λ ε,n = , ⎪ ⎪ r ε,n ⎨ '2 & (9.25) ⎪ μn σ 2n ⎪ 2 ⎪ ⎪ − λ ε,n = δ ε,n − ⎩ r ε,n r ε,n

The system (9.25) should be supplemented by the following consistency relations that should hold for admissible values of parameters, for every ε ∈ (0, ε0 ]: p ε,n,± ≥ 0 ,

p ε,n,+ + p ε,n,− = 1, δ ε,n > 0 ,

λ ε,n ∈ R1 , r ε,n = 1, 2, . . . ,

n = 1, . . . , N .

(9.26)

In the case of inhomogeneous in time model, the recombining condition does not hold automatically. As was shown in Subsection 3.3.2, the recombining condition holds for random walk Y ε,n if the parameter δ ε,n does not depend on n, i.e. the follow­ ing additional condition holds: δ ε,n = δ ε > 0 ,

n = 1, 2, . . . .

(9.27)

If relation (9.27) holds then the recombining condition holds for any values of the parameters δ ε and λ ε,n , p ε,n,±, r ε,n , n = 1, . . . . The system (9.25) takes in this case the following form: ⎧ μn ⎪ ⎪ δ ε (p ε,n,+ − p ε,n,−) + λ ε,n = , ⎪ ⎪ r ε,n ⎨ & '2 (9.28) ⎪ σ 2n μn ⎪ 2 ⎪ ⎪ − λ ε,n = , n = 1, . . . , N . δε − ⎩ r ε,n r ε,n The system of equations (9.28) can be transformed to the following equivalent form: ⎧ ⎪ σ 2n μn ⎪ ⎪ ⎪ δ = +( − λ ε,n )2 , ε ⎪ ⎪ r r ⎪ ε,n ε,n ⎪ ⎪ ⎛ ⎞ ⎨ μn 1⎝ (9.29) r ε,n − λ ε,n ⎠, ⎪ 1± p ε,n,± = ⎪ ⎪ ⎪ 2 δ ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, . . . , N .

364 | 9 Tree-type approximations for Markov Gaussian LPP Let us denote σ + = max σ n . 1≤ n ≤ N

Let us now take ı n = +, −, n = 1, . . . , N, δ ≥ σ + > 0 and choose,  ı n δ2 − σ 2n μn λ ε,n = + , n = 1, . . . , N . √ r ε,n r ε,n

(9.30)

(9.31)

Substituting λ ε,n , n = 1, . . . , N defined by relation (9.31) in relations (9.29), we get the following formulas: ⎧ δ ⎪ ⎪ ⎪ δε = √ , ⎪ ⎪ r ε,n ⎪ ⎪ ⎪  ⎪ ⎛ ⎞ ⎨ ı δ2 − σ 2n n 1⎝ (9.32) ⎠, ⎪ 1∓ p ε,n,± = ⎪ ⎪ ⎪ 2 δ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n = 1, . . . , N . This relation implies that we should choose, for every ε ∈ (0, ε0 ], the parameters r ε,n independent on n = 1, . . . , N, i.e. r ε,n = r ε ,

n = 1, . . . , N .

(9.33)

In this case, the following formulas give solutions of the system (9.25):   ⎛ ⎞ ı n δ2 − σ 2n ı n δ2 − σ 2n δ μn 1⎝ ⎠, + , p ε,n,± = 1∓ δ ε = √ , λ ε,n = √ rε rε rε 2 δ n = 1, . . . , N ,

(9.34)

parameterized by the parameters ı n = +, −, n = 1, . . . , N, δ ≥ σ + and r ε = 1, 2, . . . . Note that the consistency relations (9.26) hold for any solution given by formulas (9.34). The above remarks can be summarized in the following lemma. Lemma 9.1.2. The log-price process Y ε,n represented, for every ε ∈ (0, ε0 ], by the bino­ mial random walk, which is defined by the stochastic transition dynamic relation (9.19), realizes the exact moment fitting given by relation (9.20) for the log-price processes Y0,n represented by the Gaussian random walk, which is defined by the stochastic transition dynamic relation (9.18). This holds if the corresponding parameters of the process Y ε,n are given by relations (9.34). It should be pointed out that the recombining condition (which guarantees not more than polynomial rate of growth for the number of nodes in the corresponding binomial tree representation) holds automatically for any values of parameters of the log-price process Y ε,n given by relation (9.34).

9.1 Univariate binomial tree approximations

| 365

As in the homogeneous case, the corresponding tree T ε,(y,n) generated by the tra­ jectories of the log-price process Y ε,n+r, r = 0, 1, . . . with the value Y ε,n = y has the total number of points after r steps, L ε,n,n+r ≤ 1 + 2r ε + 1 + · · · + 2r ε r + 1 = r ε r2 + (r ε + 1)r + 1 .

(9.35)

The expression on the right-hand side in (9.35) is a polynomial (in this case quadratic) function of r.

9.1.3 Asymptotic fitting of parameters for a univariate inhomogeneous in time binomial tree model We use the log-price processes Y ε,n for approximation of the log-price process Y0,n asymptotically, as ε → 0, in the sense of weak convergence and convergence for the corresponding reward functions. In order to realize the above approximation, we can weaken the condition of ex­ act fitting replacing it by condition of asymptotic moment fitting for the log-price pro­ cesses Y ε,n and Y0,n . In this case, the exact fitting relations (9.20) are replaced by the following weaker asymptotic relations: EW ε,n → EW0,n ,

VarW ε,n → VarW0,n

as ε → 0 ,

n = 1, . . . , N .

(9.36)

Let us return back to the system of equations (9.29). Let us choose λ ε,n =

μn r ε,n ,

n = 1, . . . , N and represent δ ε > 0 in the form δ ε = √δr ε , where δ, r ε > 0. In this case, system (9.29) takes the following form: ⎧ ⎪ σ 2n δ ⎪ ⎪ ⎪ , √ = ⎪ ⎪ ⎪ rε r ε,n ⎨ 1 (9.37) ⎪ p ε,n,± = , ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ n = 1, . . . , N . The second equation in (9.37) yields the following expressions for the parameters r ε,n : r ε,n = r ε

σ 2n , δ2

n = 1, . . . , N .

(9.38)

The problem is that quantities r ε,n , n = 1, . . . , N given by relation (9.38) may not take positive integer values, except some special cases for values of the parameters σ 2n , n = 1, . . . , N. Let us show that the above asymptotic moment fitting can be achieved by comput­ ing real-valued parameters r∗ ε,n , n = 1, . . . , N using relation (9.38) and then taking as

366 | 9 Tree-type approximations for Markov Gaussian LPP r ε,n , n = 1, . . . , N the corresponding nearest positive integer numbers that are larger or equal than r∗ ε,n , n = 1, . . . , N. Summarizing the above remarks, let us choose parameters of the approximating process Y ε,n in the following way, for every ε ∈ (0, ε0 ]: δ δε = √ , rε

λ ε,n =

μn , r ε,n

p ε,n,± =

1 , 2

r ε,n = [r ε

σ 2n ]+1, δ2

n = 1, . . . , N , (9.39)

where one can take any positive real values for parameters δ and r ε and then pass the parameter r ε → ∞ as ε → 0. The exact fitting equalities take place, for every ε ∈ (0, ε0 ]: & ' δ 1 δ 1 μn = μ n , n = 1, . . . , N , −√ + (9.40) EW ε,n = r ε,n √ rε 2 r ε 2 r ε,n and the following asymptotic fitting relations take place if r ε → ∞ as ε → 0: ' & σ2 δ2 δ2 → σ 2n as ε → 0 , n = 1, . . . , N . VarW ε,n = r ε,n = [r ε 2n ] + 1 rε δ rε

(9.41)

The above remarks can be summarized in the following lemma. Lemma 9.1.3. The log-price process Y ε,n represented, for every ε ∈ (0, ε0 ], by the bino­ mial random walk, which is defined by the stochastic transition dynamic relation (9.19), realizes the asymptotic moment fitting given by relation (9.36) for the log-price processes Y 0,n represented by the Gaussian random walk, which is defined by the stochastic tran­ sition dynamic relation (9.18). This holds if the corresponding parameters of the process Y ε,n are given by relation (9.39) and the parameter r ε → 0 as ε → 0. It should be pointed out that the recombining condition (which guarantees not more than polynomial rate of growth for the number of nodes in the corresponding bino­ mial tree representation for the above model) holds automatically for any values of parameters of the log-price process Y ε,n given by relation (9.39). The corresponding tree T ε,(y,n) generated by the trajectories of the log-price pro­ cess Y ε,n+r, r = 0, 1, . . . with the value Y ε,n = y has the total number of points after r steps,  L ε,n,n+r ≤ 1 + 2r ε,n+1 + 1 + · · · + 2 r ε,n+1 + · · · + r ε,n+r + 1 & ' & ' σ 2n+r σ 2n+1 σ 2n+1 ≤ 1 + 2 rε 2 + 1 + 1 + · · · + 2 rε 2 + 1 + · · · + rε 2 + 1 + 1 δ δ δ & & ' ' σ2 σ2 = r ε 2 + 1 r2 + r ε 2 + 2 r + 1 , (9.42) δ δ where σ 2 = max1≤n≤N σ 2n . The expression on the right-hand side in (9.42) is a polynomial (in this case quadratic) function of r.

9.2 Multivariate binomial tree approximations

| 367

9.2 Multivariate binomial tree approximations In this section, we present fitting formulas for multivariate binomial tree approxima­ tion models homogeneous in time and space.

9.2.1 Exact fitting of parameters for a multivariate homogeneous in time binomial tree model Let us consider the model of multivariate homogeneous in time Gaussian random walk, which can be considered as a discrete time analog of the multivariate Wien­ 0,n is given by the following ner process (Brownian motion). In this case, a log-price Y stochastic transition dynamic relation: 0,n−1 + W  0,n , 0,n = Y Y

n = 1, 2, . . . .

(9.43)

0,0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking values in the space Rk , where (a) Y  (b) W0,n = (W0,n,1 , . . . , W0,n,k ), n = 1, 2, . . . is a sequence of i.i.d. k-dimensional nor­ mal random vectors with mean values EW0,n,i = μ i , i = 1, . . . , k, variances VarW0,n,i = σ 2i , i = 1, . . . , k, and covariances E(W0,n,i − μ i )(W0,n,i − μ i ) = ρ i,j σ i σ j , i, j = 1, . . . , k, 0,0 and the family of the random vectors W  0,n , n = 1, 2, . . . and (c) the random vector Y are independent. Usual assumptions are: (d) μ i ∈ R1 , i = 1, . . . ; (d) σ i > 0, i = 1, . . . , k; (e) |ρ i,j | ≤ 1 and ρ i,i = 1, i, j = 1, . . . , k; (f) Σ = σ i,j , where σ i,j = ρ i,j σ i σ j , i, j = 1, . . . , k, is a nonnegatively defined k × k matrix. We shall approximate the log-price process Y0,n by multivariate homogeneous in time binomial random walks. In this case, the log-price process Y ε,n is given, for every ε ∈ (0, ε0 ], by the following stochastic transition dynamic relation: ε,n−1 + W  ε,n ,  ε,n = Y Y

n = 1, 2, . . . .

(9.44)

 ε,0 = (Y ε,1 , . . . , Y ε,k ) is a random vector taking values in the space Rk , where (a) Y  ε,n,1 + · · · + W  ε,n,r ε , n = 1, 2, . . . , where  (b) W ε,n = (W ε,n,1, . . . , W ε,n,k) = W  W ε,n,l = (W ε,n,l,1, . . . , W ε,n,l,k), l, n = 1, 2, . . . is a family of i.i.d. binary random vectors taking values δ¯ ε,¯ı = (δ ε,1,ı1 , . . . , δ ε,k,ı k ) with the corresponding probabilities p ε,¯ı = p ε (ı1 , . . . ı k ) for ¯ı ∈ Ik = {¯ı =  ı1 , . . . , ı k  : ı1 , . . . ı k = ±}, (c) the random vector  ε,0 and the family of the random vectors W  ε,n,l , l, n = 1, 2, . . . are independent. Y Also, usual assumptions are (d) −∞ < δ ε,i, − < δ ε,i, + < ∞ , i = 1, . . . , k,  (e) p ε,¯ı ≥ 0, ¯ı ∈ Ik , ¯ı ,∈Ik p ε,¯ı = 1, and (f) r ε is a positive integer number. ε,n for approximation of the log-price process Y 0,n We use the log-price processes Y

as ε → 0. In this case, a usual approach is to fit the corresponding parameters, i.e. to try choosing the parameters δ¯ ε,¯ı , ¯ı , ∈ Ik , probabilities p ε,¯ı , ¯ı , ∈ Ik and r ε in such a way

368 | 9 Tree-type approximations for Markov Gaussian LPP that the following exact fitting equalities holds, for every ε ∈ (0, ε0 ]: ⎧ ⎪ EW ε,1,i = EW0,1,i , i = 1, . . . , k , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ VarW ε,1,i = VarW0,1,i , i = 1, . . . , k , ⎪ ⎪ E(W ε,1,i − EW ε,1,i)(W ε,1,i − EW ε,1,j) ⎪ ⎪ ⎪ ⎪ ⎩ = E(W0,1,i − EW0,1,i )(W0,1,i − EW0,1,j ) , 1 ≤ i < j ≤ k .

(9.45)

 ε,n used in the defin­ Using the sum-representation for binomial random vectors W ing dynamic relation (9.44), one can easily transform, for every ε ∈ (0, ε0 ], equalities (9.45) to the equivalent form of the following system of equations: ⎧  ⎪ EW ε,1,i = r ε δ ε,i, +p ε,i, + + δ ε,i, −p ε,i, − = μ i , i = 1, . . . , k ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ μi 2 ⎪ 2 2 ⎪ = σ 2i , i = 1, . . . , k , = r δ p + δ p − ( ) VarW ⎪ ε,1,i ε ε,i, + ε,i, + ε,i, − ε,i, − ⎪ ⎨ rε (9.46) ⎪ E(W ε,1,i − EW ε,1,i)(W ε,1,j − EW ε,1,j) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = r ε δ ε,i, +δ ε,j, +(p ε,i,j,+, + + p ε,i,j, −,− − p ε,i,j, +,− − p ε,i,j, −,+) − μ i μ j ⎪ ⎪ ⎪ ⎪ ⎩ = ρ i,j σ i σ j , 1 ≤ i < j ≤ k ,

where p ε,i,j,±,± = P{W ε,1,1,i = δ ε,i, ±, W ε,1,1,j = δ ε,j,±} ,

1≤i < j ≤ k,

(9.47)

and p ε,i, ± = P{W ε,1,1,i = δ ε,i, ±} ,

1≤ i ≤ k.

(9.48)

The system of equations (9.46) should be supplemented by the following proba­ bility inequalities: 0 ≤ p ε,i,j,+,+, p ε,i,j,+,−, p ε,i,j,−,+, p ε,i,j,−,− ≤ 1 ,

1≤i < j ≤ k,

(9.49)

and the following system of consistency relations for probabilities p ε,i,j,±,±, 1 ≤ i < j ≤ k and p ε,i, ±, 1 ≤ i ≤ k, ⎧ ⎪ ⎪ ⎪ p ε,i,j,±,+ + p ε,i,j,±,− = p ε,i, ± , 1 ≤ i < j ≤ k , ⎨ p ε,i,j,+,± + p ε,i,j,−,± = p ε,j,± , 1 ≤ i < j ≤ k , (9.50) ⎪ ⎪ ⎪ ⎩ +p = 1, 1≤ i ≤ k. p ε,i, +

ε,i −

ε,n = (Y ε,n,1, . . . , Y ε,n,k) possesses recombining The multivariate random walk Y property since every component of this random walk Y ε,n,i, which is a univariate ho­ mogeneous in time binomial random walk, possesses this property. Thus, the recom­ bining condition do not impose any additional assumptions on parameters of the pro­ ε,n . cess Y

9.2 Multivariate binomial tree approximations

|

369

As in the one-dimensional case, it is convenient to symmetrize the values of jumps  ε,n,l in the following form, for l, n = 1, 2, . . . : by representing binary random vectors W    ε,n,l  ε,n,l = λ¯ ε + W W = λ¯ ε + δ¯ ε,¯ı with probability p ε,¯ı , ¯ı ∈ Ik (9.51) where λ¯ ε = (λ ε,1 , . . . , λ ε,k ) , and δ ε,i =

δ¯ ε,¯ı = (ı1 δ ε,1 , . . . , ı k δ ε,k ),

δ ε,i, + − δ ε,i, − , 2

λ ε,i =

¯ı =  ı1 , . . . , ı k  ∈ Ik ,

δ ε,i, + + δ ε,i, − , 2

i = 1, . . . , k .

(9.52)

(9.53)

It follows from relation (9.52) that the transition dynamic relation (9.44) can be rewritten in the following form:  ε,n−1 + λ¯ε + W  ε,n  ε,n = Y Y ,

n = 1, 2, . . . .

(9.54)

 ε,0 = (Y ε,1 , . . . , Y ε,k ) is a random vector taking values in the space Rk , where (a) Y        ε,n,1  ε,n,r  ε,n  ε,n,l (b) W = (W ε,n,1 , . . . , W ε,n,k )=W + ··· + W , n = 1, 2, . . . , where W = ε

  (W ε,n,l,1 , . . . , W ε,n,l,k ), l, n = 1, 2, . . . is a family of i.i.d. binary random vectors taking  ¯ values δ ε,¯ı with the corresponding probabilities p ε,¯ı , for ¯ı ∈ Ik , (c) the random vec­   ε,0 and the family of the random vectors W  ε,n,l tor Y , l, n = 1, 2, . . . are independent; (d) λ¯ε = r ε λ¯ ε . According to the first equation in the system (9.46), for every i = 1, . . . , k,  + λ ε,i = δ ε,i (p ε,i, + − p ε,i, −) + λ ε,i = EW ε,1,1,i = EW ε,1,1,i

μj . rε

(9.55)

Also, p ε,i, + + p ε,i, − = 1, i = 1, . . . , k. Thus, for every i = 1, . . . , k,  VarW ε,1,1,i = VarW ε,1,1,i

= δ2ε,i (p ε,i, + + p ε,i, −) − (

μi μi − λ ε,i )2 = δ2ε,i − ( − λ ε,i )2 . rε rε

(9.56)

and, for every 1 ≤ i < j ≤ k,          μj μj μi μi   E W ε,1,1,i − W ε,1,1,j − − − λ ε,i W ε,1,1,j − − λ ε,j = E W ε,1,1,i rε rε rε rε    μj μi   = EWq ε,1,1,i W ε,1,1,j − − λ ε,i − λ ε,j , rε rε # = δ ε,i δ ε,j p ε,i,j, +,+ · p ε,i,j,+, + + p ε,i,j,−, − · p ε,i,j, −,−  − p ε,i,j, +,− · p ε,i,j,−, + − p ε,i,j,−, + · p ε,i,j, +,−    μj μi − − λ ε,i − λ ε,j . (9.57) rε rε

370 | 9 Tree-type approximations for Markov Gaussian LPP Therefore, the system (9.46) takes the following form: ⎧  μi ⎪ δ ε,i p ε,i, + − p ε,i, − + λ ε,i = , i = 1, . . . , k , ⎪ ⎪ ⎪ rε ⎪ ⎪ ⎪ 2  ⎪ ⎪ σ2 μi ⎪ ⎪ ⎪ δ2ε,i − − λ ε,i = i , i = 1, . . . , k , ⎨ rε rε    ⎪ μj μi ⎪ ⎪ ⎪ δ ( p + p − p − p ) − − λ − λ δ ε,i ε,j ε,i,j, + , + ε,i,j, − , − ε,i,j, + , − ε,i,j, − , + ε,i ε,j ⎪ ⎪ rε rε ⎪ ⎪ ⎪ ⎪ ρ i,j σ i σ j ⎪ ⎪ ⎩ (9.58) = , 1≤i 0, i = 1, . . . , k and choose quantities λ ε,i given by the following formulas:  ı i δ2i − σ 2i μi λ ε,i = + , i = 1, . . . , k . (9.64) √ rε rε Substituting λ ε,i , i = 1, . . . , k given by formulas (9.64) in relations (9.61), we get solutions of system of equations (9.58) in the following form:  ⎧ ⎪ ı δ2i − σ 2i ⎪ i δ μi i ⎪ ⎪ δ = , λ = + , i = 1, . . . , k . ⎪ √ √ ε,i ε,i ⎪ ⎪ rε rε rε ⎪ ⎪ ⎪   ⎛ ⎞ ⎪ ⎪ ⎪ 2 2 ⎪ ı δ2j − σ 2j ı δ − σ j ⎪ σ σ i 1 1 1 i i i j ⎪ ⎠ ⎪ p ε,i,j,+,+ = + ρ i,j + ⎝− − ⎪ ⎪ ⎪ 4 4 δi δj 4 δi δj ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ı δ2i − σ 2i ı j δ2j − σ 2j ⎪ i 1 ⎪ ⎪ ⎪ + , 1≤i < j ≤ k, ⎪ ⎪ 4 δi δj ⎪ ⎪ ⎪   ⎪ ⎛ ⎞ ⎪ ⎪ 2 2 ⎪ ı j δ2j − σ 2j ⎪ ı δ − σ σ σ i 1 1 1 ⎪ i i i j ⎪p ⎠ ⎪ − ρ i,j + ⎝− + ε,i,j,+ ,− = ⎪ ⎪ ⎪ 4 4 δi δj 4 δi δj ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ 2 2 2 2 ⎨ 1 ıi δi − σi ıj δj − σj (9.65) − , 1≤i < j ≤ k, ⎪ ⎪ 4 δi δj ⎪ ⎪ ⎪  ⎪ ⎛  ⎞ ⎪ ⎪ ⎪ ı j δ2j − σ 2j ı i δ2i − σ 2i ⎪ σ σ 1 1 1 i j ⎪ ⎪ ⎠ ⎪ p ε,i,j,−,+ = − ρ i,j + ⎝ − ⎪ ⎪ 4 4 δi δj 4 δi δj ⎪ ⎪ ⎪ ⎪  ⎪  ⎪ ⎪ ⎪ ⎪ ı i δ2i − σ 2i ı j δ2j − σ 2j ⎪ 1 ⎪ ⎪ − , 1≤i < j ≤ k, ⎪ ⎪ ⎪ 4 δi δj ⎪ ⎪ ⎪   ⎛ ⎞ ⎪ ⎪ 2 2 ⎪ ⎪ ı δ2j − σ 2j ı δ − σ j σ σ i ⎪ 1 1 1 i i i j ⎪ ⎝ ⎠ ⎪ + + p ε,i,j,−,− = + ρ i,j ⎪ ⎪ ⎪ 4 4 δ δ 4 δ δ i j i j ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ 2 2 ⎪ ı δ − σ ı δ2j − σ 2j ⎪ i j i i 1 ⎪ ⎪ ⎩ + , 1≤i < j ≤ k, 4 δi δj parameterized by the parameters ı i = +, − , δ i ≥ σ i , i = 1, . . . , k and r ε = 1, 2, . . . . It is easy to check that solutions given by the formulas (9.65) satisfy the consis­ tency relations (9.50).

9.2 Multivariate binomial tree approximations

| 373

For example, the following formulas can be obtained for quantities p ε,i, ± by ap­ plying the corresponding summation formulas given in consistency relations (9.50) to quantities p ε,i,j,ı i,ı j , ı i , ı j = +, − given by formulas (9.65)  ı i δ2i − σ 2i 1 p ε,i, ± = (1 ∓ ) , i = 1, . . . , k . (9.66) 2 δi The question about holding the probability inequalities (9.49) for quantities p ε,i,j,±,± is more complicated. These quantities, given by formulas (9.65), can take negative values for large values of parameters δ i , i = 1, . . . , k Let us introduce the parameter ρ = max |ρ i,j| ∈ [0, 1] . 1≤ i < j ≤ k

(9.67)

It follows from formulas (9.65) that inequalities (9.49) for quantities p ε,i,j,ıi,ı j , ı i , ı j = +, − hold if the following inequalities hold, for every 1 ≤ i < j ≤ k:     δ2j − σ 2j δ2i − σ 2i δ2j − σ 2j δ2i − σ 2i σi σj 1 (1 − ρ − − + ) ≥ 0. (9.68) 4 δi δj δi δj δi δj For the moment, let us use parameters κi =

δi ≥ 1, σi

i = 1, . . . , k .

(9.69)

Inequalities (9.68) can be rewritten in the following equivalent form:   (κ i − κ 2i − 1)(κ j − κ 2j − 1) ≥ ρ . (9.70) √ A simple calculus calculations show that functions f (κ ) = κ − κ 2 − 1, κ ≥ 1 takes value f (1) = 1, strictly decrease, and f (κ ) → 0 as κ → ∞. √ That is why, f (κ ) ≥ ρ if and only if 1 ≤ κ ≤ κ (ρ ), where κ (ρ ) is the root of equation,   (9.71) κ(ρ) − κ2 (ρ) − 1 = ρ . This equation has the root κ(ρ) =

1+ρ √ ≥ 1. 2 ρ

(9.72)

Inequality (9.70) and the above remarks imply that inequalities (9.49) hold if the following condition holds for the parameters δ i , i = 1, . . . , k: σ i ≤ δ i ≤ κ(ρ)σ i ,

i = 1, . . . , k .

(9.73)

At the same time, the above remarks imply that inequalities (9.49) do not hold if there exist a pair of indices 1 ≤ i < j ≤ k such that ρ i,j = −ρ and δ i > κ (ρ )σ i ,

374 | 9 Tree-type approximations for Markov Gaussian LPP δ j > κ (ρ )σ j . In this case, at least one of the quantities p ε,i,j,ı i,ı j , ı i , ı j = +, −, given by formulas (9.65), takes a negative value. It is useful to note that condition (9.73) let one choose parameters δ i = σ i , i = 1, . . . , k. In this case, formulas (9.65) reduce to simpler formulas (9.62). Let us introduce the condition: R1 : ρ < 1. If the condition R1 holds, then κ (ρ ) > 1, and, thus, all intervals [σ i , κ (ρ )σ i ], i = 1, . . . , k, for admissible values of the parameters δ i , i = 1, . . . , k have nonzero lengths. If the condition R1 does not hold, i.e. ρ = 1, then κ (ρ ) = 1. In this case the only values δ i = σ i , i = 1, . . . , k are admissible. The fitting procedure described above, yields the one-dimensional and two-di­  ε,1,1 = (W ε,1,1,1, mensional distributions for components of the random vectors W . . . , W ε,1,1,k), respectively, p ε,i, ±, i = 1, . . . , k and p ε,i,j,±,±, 1 ≤ i < j ≤ k. These distributions are consistent in the sense of relations (9.47), (9.48), and (9.50). The analysis of the fitting problem presented above is complete in the case of twodimensional binomial random walks. However, in the case of k-dimensional binomial random walk, with k ≥ 3, the ad­ ditional question, about existence of a multivariate distribution of the random vector  ε,1,1 (with the above one- and two-dimensional distributions p ε,i, ±, 1 ≤ i ≤ k and W p ε,i,j,±,±, 1 ≤ i < j ≤ k) arises. This question has an affirmative answer. Moreover, the algorithm for construct­ ing of the corresponding k-dimensional distributions is given below in Lemma 9.2.2, which should be applied to the above one- and two-dimensional distributions, for ev­ ery ε ∈ (0, ε0 ]. The above remarks can be summarized in the following lemma.  ε,n represented, for ε ∈ (0, ε0 ], by the multivari­ Lemma 9.2.1. The log-price process Y ate binomial random walk, which is defined by the stochastic transition dynamic relation (9.44), realizes the exact moment fitting given by relation (9.45) for the log-price process Y0,n represented by the Gaussian random walk defined by the stochastic transition dy­  ε,n namely, δ ε,i , λ ε,i , p ε,i, ±, 1 ≤ i ≤ namic relation (9.43). Parameters of the process Y k, p ε,i,j,±,±, 1 ≤ i < i ≤ k, r ε , can be chosen, for every ε ∈ (0, ε0 ], according to rela­ tions (9.62) and (9.63) and an arbitrary choice of r ε = 1, 2, . . . , or according to relations (9.65) and (9.66), the choice of the parameters δ i , i = 1, . . . , k satisfying condition (9.73) and an arbitrary choice of r ε = 1, 2, . . . . The corresponding high-order distributions for  ε,1,1 can be constructed with the use of algorithm de­ components of the random vectors W scribed in Lemma 9.2.2, which should be applied to the above one- and two-dimensional distributions p ε,i, ±, i = 1, . . . , k and p ε,i,j,±,±, 1 ≤ i < j ≤ k, for every ε ∈ (0, ε0 ].

It is worth noting that the fitting procedure based on expectations and covariances  ε,1,1. involves only one- and two-dimensional distributions for the random vectors W

9.2 Multivariate binomial tree approximations

| 375

 ε,1,1, with the corresponding In fact, we need only know that the random vectors W fitted one- and two-dimensional distributions, do exist. The same is valid for the corresponding convergence results for option rewards presented in the next chapter. The only expectations and covariances and, thus, only  ε,1,1 are involved one- and two-dimensional distributions for the random vectors W in the corresponding convergence conditions, which provide convergence of the re­ ε,n to the reward functions for ward functions for the binomial log-price processes Y  the Gaussian log-price process Y0,n .  ε,1,1 (for the However, the high-dimensional distributions for the random vectors W case, k ≥ 3) are explicitly used in the backward recurrence relations for computing of  ε,n. the reward functions for the approximating binomial log-price processes Y It should be pointed out that the recombining condition (which guarantees not more than polynomial rate of growth for number of nodes in the corresponding bino­ mial tree representation for the above model) discussed in Chapter 3 holds automati­ ε,n given Lemma 9.2.1. cally for any values of parameters for the log-price process Y The corresponding tree T ε,(y,n) generated by the trajectories of the log-price pro­ ε,n =  cess Y ε,n+l, l = 0, 1, . . . with the value Y y has the total number of points after r steps,

L ε,n,n+r ≤ 1 + (2r ε + 1)k + · · · + (2r ε r + 1)k r +1

≤

(2r ε x + 1)k dx = 0

(2r ε (r + 1) + 1)k+1 − 1 . 2r ε (k + 1)

(9.74)

The expression on the right-hand side in (9.74) is a polynomial function of r of order k + 1.

9.2.2 Multivariate binary distributions with given one- and two-dimensional distributions  = (W1 , . . . , W k ) be a k-dimensional binary random vector, which Let k ≥ 3 and W ¯ takes values δ¯ı = (δ1,ı1 , . . . , δ k,ı k ), where δ i,ı i , ı i = +, −, i = 1, . . . , k are real numbers and ¯ı =  ı1 , . . . , ı k  ∈ Ik = {¯ı : ı1 , . . . , ı k = +, −}. Also, let 1 ≤ i1 < · · · < i l ≤ k and p i1 ,...,i l ,ı i1 ,...,ı il = P{W i1 = δ i1 ,ı i1 , . . . , W i l = δ i l ,ı il }, ı i1 , . . . , ı i l = +, − be the corresponding joint l-dimensional distributions for  components of the random vector W. The distributions p i1 ,...,i l ,ı i1 ,...,ı il should satisfy the following consistency relations: ⎧ ⎪ p i1 ,...,i l ,ı i1 ,...,ı il ≥ 0 , ⎪ ⎪

⎪ ⎪ ⎨ p i1 ,...,i l ,ı i1 ,...,ı il = p i1 ,...,i j−1 ,i j+1 ,...,i l ,ı i1 ,...,ı ij−1 ,ı ij+1 ,...,ı il (9.75) ⎪ ⎪ ı i j =+, − ⎪ ⎪ ⎪ ⎩ ı i 1 , . . . , ı i l = +, − , 1 ≤ i1 < · · · < i l ≤ k , 1 ≤ j ≤ l ≤ k ,

376 | 9 Tree-type approximations for Markov Gaussian LPP where the quantity p = p i1 ,...,i l ,ı i1 ,...,ı il should be counted as 1 for the case l = 0. By the Kolmogorov theorem about consistent finite-dimensional distributions,  with any consistent family of joint distributions of there exists a random vector W components satisfying relations (9.75).  = (W1 , . . . , W k ) can be constructed in the following way. The random vector W One can order indices + and − (using the ordering symbol ) as +  − and then vectors ¯ı =  ı1 , . . . , ı k  ∈ Ik (using the ordering symbol ) in a natural way, by count­   ing ¯ı  ¯ı , if there exists 1 ≤ j ≤ k such that ıi = ı i , for i < j, but ı j = +, ı j = −. − + Then, one can define disjoint subintervals I¯ı = [q¯ı , q¯ı ), ¯ı ∈ Ik of interval [0, 1) − such that: (a) q¯+ ı ∈ Ik , (b) interval I¯ı is located to the right of ı − q¯ı = p 1,...,k,ı 1 ,...,ı k , ¯    interval I¯ı , if ¯ı  ¯ı .  = (W1 , . . . , W k ) can be defined as Finally, the random vector W

 = δ¯¯ı I(ρ ∈ I¯ı ) , W (9.76) ¯ı∈Ik

where ρ is the random variable uniformly distributed in the interval [0, 1). The fitting algorithm presented in Subsection 9.2.1 does require to answer a slightly different question. Let us assume that some one-dimensional and two-dimensional distributions, re­ spectively p i,ı i , ı i = +, −, 1 ≤ i ≤ k and p i,j,ı i,ı j , ı i , ı j = +, −, 1 ≤ i < j ≤ k are given and satisfy the consistency relations, which are the corresponding relations in (9.75) related to one- and two-dimensional distributions, ⎧ ⎪ p i,j,ı i,ı j ≥ 0 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p i,j,ı i , + + p i,j,ı i, − = p i,ı i , p i,j, +,ıj + p i,j, −,ı j = p j,ı j , (9.77) ⎪ ⎪ ı i , ı j = +, − , 1 ≤ i < j ≤ k , ⎪ ⎪ ⎪ ⎪ ⎩ p i,+ + p i,− = 1 , 1 ≤ i ≤ k .  = (W1 , . . . , W k ), which The following question arises. Does a random vector W has the prescribed above consistent one- and two-dimensional distributions of com­ ponents always exists, for any for k ≥ 3? Taking into account the above remarks, one can reduce this question to the fol­ lowing equivalent question. Do multivariate distributions p i1 ,...,i l ,ı i1 ,...,ı il , ı i1 , . . . , ı i l = +, − , 1 ≤ i1 < · · · < i l ≤ k, 1 ≤ l ≤ k, which have prescribed one- and two-dimen­ sional distributions consistent in the sense of relations (9.77), always exist, for any for k ≥ 3? In the following, we describe the algorithm for constructing high-order (three-, four-, etc.) distributions with the above one- and two-dimensional distributions con­ sistent in the sense of relations (9.75). Let us describe how the corresponding three-dimensional distributions can be constructed. It is enough to show how to choose, for any prescribed indices 1 ≤ i < j < r ≤ k, values for eight probabilities p i,j,r,ıi,ı j ,ı r , ı i , ı j , ı r = +, − such that the following consis­

9.2 Multivariate binomial tree approximations

tency relation hold: ⎧ ⎪ p i,j,r,ıi,ı j ,ı r ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p i,j,r,+,ı j,ı r + p i,j,r,−,ı j,ı r ⎪ ⎪ p i,j,r,ı i,+,ı r + p i,j,r,ıi,−,ı r ⎪ ⎪ ⎪ ⎪ ⎩ p i,j,r,ı i,ı j ,+ + p ε,i,j,r,ı i,ı j ,−

≥ 0,

| 377

ıi , ıj , ır = ± ,

= p j,r,ı j,ı r ,

ıj , ır = ± ,

= p i,r,ı i,ı r ,

ıi , ır = ± ,

= p i,j,ı i,ı j ,

ıi , ıj = ± .

(9.78)

Let us order triplets of indices  ı i , ı j , ı r  (using the ordering symbol ) in the way described above, i.e. +, +, +  +, +, −  +, −, +  · · ·  −, −, − .

(9.79)

One should begin by choosing a value for probability p i,j,r,+,+,+. It follows from relations (9.78) that p ε,i,j,r,+,+,+ ≤ p i,j,+,+, p i,r,+,+, p j,r,+,+. Thus, one can choose for this probability any values satisfying inequality, 0 ≤ p i,j,r,+,+,+ ≤ p i,j,+,+ ∧ p i,r,+,+ ∧ p j,r,+,+ .

(9.80)

Then, one should define probabilities p i,j,r,ı i,ı j ,ı r , sequentially, in three groups. First, the group of probabilities p i,j,r,+,+,−, p i,j,r,+,−,+, p ε,i,j,r,−,+,+, which index triplets differ just by one index from triplet +, +, +, should be defined. Second, the group of probabilities p i,j,r,+,−,−, p i,j,r,−,+,−, p i,j,r,−,−,+, which index triplets differ by two indices from triplet +, +, +, should be defined. Third, the group includeing one probability p i,j,r,−,−,−, which index triplet differs by three indices from triplet +, +, +, should be defined. The order established by relation (9.79) is used within the above groups. Relation (9.78) imply the following formulas: ⎧ p = p i,j, +, + − p i,j,r,+, +,+ , ⎪ ⎪ ⎪ i,j,r,+, +,− ⎪ ⎪ ⎪ ⎪ p i,j,r,+, −,+ = p i,r, + , + − p i,j,r,+ ,+,+ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p i,j,r,−, +,+ = p j,r, +, + − p i,j,r,+,+,+ , ⎪ ⎨ p i,j,r,+,−,− = p i,j,+,− − p i,j,r,+,−,+ = p i,r,+,− − p i,j,r,+,+,− , (9.81) ⎪ ⎪ ⎪ ⎪ ⎪ p = p − p = p − p , i,j,r,−, +,− i,j, − , + i,j,r,−, +,+ j,r, + , − i,j,r,+,+ ,− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p = p − p = p − p ⎪ i,r, − , + i,j,r,− ,+,+ j,r, − , + i,j,r,+,− ,+ , ⎪ ⎪ i,j,r,−, −,+ ⎪ ⎩ p i,j,r,−,−,− = p i,j,−,− − p i,j,r,−,−,+ = p i,r,−,− − p i,j,r,−,+,− = p j,r,−,− − p i,j,r,+,−,− . One can check that alternative formulas for the above probabilities follows from consistency of the corresponding one- and two-dimensional probabilities. For example, the equality p i,j,+,− − p i,j,r,+,−,+ = p i,r,+,− − p i,j,r,+,+,− is equivalent, due to first two equalities in (9.81), to the equality p i,j,+,− − (p i,r,+,+ − p i,j,r,+,+,+) = p i,r,+,− − (p i,j,+,+ − p i,j,r,+,+,+) that is equivalent to the equality p i,j,+,− + p i,j,+,+ = p i,r,+,− + p i,r,+,+ = p i,+. Also, one can easily check that probabilities p ε,i,j,r,+,+,+ . . . , p ε,i,j,r,−,−,− defined by relations (9.80) and (9.81) satisfy all consistency inequalities and equations in (9.78).

378 | 9 Tree-type approximations for Markov Gaussian LPP It follows from relations (9.80) and (9.81) that there is one semifree parameter p i,j,r,+,+,+ ∈ [0, p i,j,+,+ ∧ p i,r,+,+ ∧ p j,r,+,+] in these relations. The algorithm described above should be repeated for every 1 ≤ i < j < r ≤ k. In total, there are C3k semifree parameters p i,j,r,+,+,+, 1 ≤ i < j < r ≤ k for the family of three-dimensional distributions. As soon as the three-dimensional distributions consistent with given one- and two-dimensional distributions are constructed, one can construct, in the similar way, the corresponding four-dimensional distributions, consistent with the corresponding one-, two- and three-dimensional distributions. Note only that, again, there are C4k semifree parameters p i,j,r,s,+,+,+,+ ∈ [0, p i,j,r,+,++ ∧ p i,j,s,+,+,+ ∧ p i,r,s,+,+,+ ∧ p j,r,s,+,+,+], for the family of four-dimensional distributions, for every chosen indices 1 ≤ i < j < r < s ≤ k. In total, there are C3k + C4k semifree parameters p i,j,r,+,+,+, 1 ≤ i < j < r ≤ k and p i,j,r,s,+,+,+,+, 1 ≤ i < j < r < s ≤ k for the family of three- and four-dimensional distributions. The procedure can be repeated for five-dimensional distributions, etc. It will be completed when the corresponding k-dimensional distributions will be constructed. As soon as k-dimensional distributions p1,...,k,ı1,...,ı k , ı1 , . . . , ı k = +, − are con­  = (W1 , . . . , W k ) with the above distri­ structed, one can construct a random vector W butions, as this was described above. Let us summarize the above remarks in the following lemma.  = (W1 , . . . , W k ), with any Lemma 9.2.2. A k-dimensional binary random vector W prescribed consistent one- and two-dimensional distributions, exists. The correspond­ ing high-order distributions for this vector can be constructed according to algorithm described in the above relations (9.78)–(9.81) and related remarks.

Now, one can construct, for every ε ∈ (0, ε0 ], the random walk Y ε,n defined by the transition dynamic relation (9.44). First, one should choose jump values δ¯ ε,¯ı = (δ ε,ı1 , . . . , δ ε,ı k ), ¯ı =  ı i1 , . . . , ı i k  ∈ Ik ∈ Ik and one- and two-dimensional distributions, respectively, p ε,i.ı i , ı i = +, −, 1 ≤  ε,1,1, according to i ≤ k and p ε,i,j,ıi.ı j , ı i , ı j = +, −, 1 ≤ i < j ≤ k for random vector W relations pointed in Lemma 9.2.1. Second, one should construct the corresponding consistent k-dimensional distri­ butions p ε,1,...,k,ı1,...,ı k , ¯ı ∈ Ik , with the above one- and two dimensional distributions, using the algorithm described in Lemma 9.2.2, and then to define the corresponding intervals I¯ı , = Iε,¯ı , ¯ı ∈ Ik , which, in this case, also depend on parameter ε. Third, one can define the corresponding sequence of i.i.d. random variables  ε,n,l , n, l = 1, 2, . . . used in the transition dynamic relation (9.44) as W

 ε,n,l = δ¯ ε,¯ı I(ρ n,l ∈ Iε,¯ı ) , n, l = 1, 2, . . . , W (9.82) ¯ı∈Ik

where ρ n,l , n, l = 1, 2, . . . is a family of i.i.d. random variables uniformly distributed in the interval [0, 1).

9.3 Multivariate trinomial tree approximations

|

379

9.3 Multivariate trinomial tree approximations In this section, we present fitting formulas for multivariate trinomial tree approxima­ tion models inhomogeneous in time but homogeneous is space.

9.3.1 Exact fitting of parameters for a multivariate inhomogeneous in time trinomial tree model Let us consider the model of multivariate inhomogeneous in time Gaussian random walk, which can be considered as a discrete time analog of a multivariate Gaussian 0,n is given by the fol­ process with independent increments. In this case, a log-price Y lowing stochastic transition dynamic relation: 0,n−1 + W  0,n , 0,n = Y Y

n = 1, 2, . . . .

(9.83)

0,0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking values in the space Rk , where (a) Y  (b) W0,n = (W0,n,1 , . . . , W0,n,k), n = 1, 2, . . . is a sequence of independent k-di­ mensional normal random vectors with mean values EW0,n,i = μ n,i , i = 1, . . . , k, n = 1, 2, . . . , variances VarW0,n,i = σ 2n,i, i = 1, . . . , k, and covariances E(W0,n,i − μ n,i)(W0,n,i − μ n,i ) = ρ n,i,jσ n,i σ n,j , i, j = 1, . . . , k, n = 1, 2, . . . , and (c) the random vec­ 0,0 and the family of the random vectors W  0,n,l , l, n = 1, 2, . . . are independent. tor Y Usual assumptions are: (d) μ n,i ∈ R1 , i = 1, . . . , n = 1, 2, . . . , (d) σ n,i > 0, i = 1, . . . , k, n = 1, 2, . . . , (e) |ρ n,i,j| ≤ 1, ρ n,i,i = 1, i, j = 1, . . . , k, n = 1, 2, . . . , (f) Σ n = σ n,i,j , n = 1, 2, . . . , where σ n,i,j = ρ n,i,jσ n,i σ n,j , i, j = 1, . . . , k, are nonnegatively defined k × k matrices. We shall approximate the log-price process Y0,n by multivariate inhomogeneous in time trinomial random walks. In this case, the log-price process Y ε,n is given, for every ε ∈ (0, ε0 ], by the fol­ lowing stochastic transition dynamic relation: ε,n−1 + W  ε,n ,  ε,n = Y Y

n = 1, 2, . . . .

(9.84)

 ε,0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking values in the space Rk , where (a) Y  ε,n,1 + · · · + W  ε,n,r ε,n , n = 1, 2, . . . , where W  ε,n = (W ε,n,1 , . . . , W ε,n,k) = W  ε,n,l = (b) W (W ε,n,l,1, . . . , W ε,n,l,k), l, n = 1, 2, . . . is a family of independent random vectors tak­ ing values δ¯ ε,n,¯ȷ = (δ ε,n,1,ȷ1 , . . . , δ ε,n,k,ȷ k ) with the corresponding probabilities p ε,n,¯ȷ = p ε,n (ȷ1 , . . . ȷ k ) for ¯ȷ ∈ Jk = {¯ȷ =  ȷ1 , . . . , ȷ k  : ȷ j , . . . ȷ k = +, ◦, −}, (c) the random vec­ ε,0 and the family of the random vectors W  ε,n,l, l, n = 1, 2, . . . are independent. tor Y Usual assumptions are: (d) −∞ < δ ε,n,i, − < δ ε,n,i,◦ < δ ε,n,i,+ < ∞ , i = 1, . . . , k,  n = 1, 2, . . . ; (e) p ε,n,¯ȷ ≥ 0, ¯ȷ ∈ Jk , ¯ȷ∈Jk p ε,n,¯ȷ = 1, n = 1, 2, . . . ; and (f) r ε,n , n = 1, 2, . . . are positive integer numbers.

380 | 9 Tree-type approximations for Markov Gaussian LPP  ε,n for approximation of the log-price process Y 0,n We use the log-price processes Y as ε → 0. In this case, a usual approach is to fit the corresponding parameters, i.e. to try to choose parameters δ¯ ε,n,¯ȷ, ¯ȷ ∈ Jk , n = 1, . . . , N, probabilities p ε,n,¯ȷ , ¯ȷ ∈ Jk , n = 1, . . . , N, and r ε,n , n = 1, . . . , N in such a way that the following exact fitting equalities holds, for every ε ∈ (0, ε0 ]: ⎧ ⎪ EW ε,n,i = EW0,n,i , i = 1, . . . , k, n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ VarW ε,n,i = VarW0,1,i , i = 1, . . . , k, n = 1, . . . , N , (9.85) ⎪ ⎪ E(W ε,n,i − EW ε,n,i)(W ε,n,i − EW ε,n,j) = E(W0,n,i − EW0,n,i) ⎪ ⎪ ⎪ ⎪ ⎩ × (W0,n,i − EW0,n,j), 1 ≤ i < j ≤ k, n = 1, . . . , N .

As pointed out in Subsection 3.3.2, an inhomogeneous in time univariate ran­ dom walk with bounded jumps possess the recombining property if nth jump of this random walk take values on a grid of points of the following form δl + δ n,◦, l = 0, ±1, ±2, . . . for every n = 1, 2, . . . , where δ > 0, δ n,◦ ∈ R1 , n = 1, 2, . . . . The multivariate random walk possesses the recombining property if and only if every its component, which is an inhomogeneous in time univariate random walk, possesses this property. Thus, the above condition should hold for every component of the multivariate random walk, with parameters δ and δ n,◦, n = 1, 2, . . . specific for every component. As follows from the above remarks, in the case, of multivariate inhomogeneous in time trinomial random walk Y ε,n considered in this section, the above recombin­ ing condition reduces to the following assumption that the parameters δ ε,n,i,+, δ ε,n,i,◦, δ ε,n,i,−, n = 1, 2, . . . have, for every ε ∈ (0, ε0 ], the following form: δ ε,n,i,± = ±δ ε,i + λ ε,n,i ,

δ ε,n,i,◦ = λ ε,n,i ,

i = 1, . . . , k ,

n = 1, 2, . . . ,

(9.86)

where δ ε,i > 0, λ ε,n,i ∈ R1 , i = 1, . . . , k, n = 1, 2, . . . . As in the one-dimensional case, it is convenient to symmetrize the values of jumps  ε,n,l in the following form, for l, n = 1, 2, . . . : by representing binary random vectors W    ε,n,l  ε,n,l = λ¯ ε,n + W W = λ¯ ε,n + δ¯ ε,¯ȷ with probability p ε,n,¯ȷ , ¯ȷ ∈ Jk , (9.87) where λ¯ ε,n = (λ ε,n,1, . . . , λ ε,n,k) ,

n = 1, 2, . . . ,

(9.88)

¯ȷ =  ȷ1 , . . . , ȷ k  ∈ Jk ,

(9.89)

and δ¯ ε,¯ȷ = (δε,1,ȷ1 , . . . , δε,k,ȷ k ) , where δε,i, + = δ ε,i ,

δε,i, ◦ = 0 ,

δε,i, − = −δ ε,i ,

i = 1, . . . , k .

(9.90)

9.3 Multivariate trinomial tree approximations

|

381

It follows from relation (9.87) that the transition dynamic relation (9.83) can be rewritten in the following form:   ε,n−1 + λ¯ ε,n + W  ε,n ε,n = Y Y ,

n = 1, 2, . . . .

(9.91)

 ε,0 = (Y 0,1 , . . . , Y 0,k ) is a random vector taking values in the space Rk , where (a) Y        ε,n,1  ε,n,r   ε,n,l (b) W ε,n = (W ε,n,1 , . . . , W ε,n,k )=W +···+ W , n = 1, 2, . . . , where W = ε,n   (W ε,n,l,1, . . . , W ε,n,l,k), l, n = 1, 2, . . . is a family of i.i.d. trinary random vectors taking values δ¯ ε,n,¯ȷ with the corresponding probabilities p ε,n,¯ȷ, for ¯ȷ ∈ Jk , n = 1, . . . , and   ε,0 and the family of the random vectors W  ε,n,l (c) the random vector Y , l, n = 1, 2, . . .    are independent; (d) λ ε,n = r ε,n λ ε,n , n = 1, 2, . . . . Calculations analogous to those made in relations (9.55)–(9.57) yield the following formulas:  EW ε,n,1,i = EW ε,n,1,i + λ ε,n,i

= δ ε,i (p ε,n,i, + − p ε,n,i, −) + λ ε,n,i =

μ n,i , r ε,n

i = 1, . . . , k ,

n = 1, . . . ,

 = δ2ε,i (1 − p ε,n,i,◦) VarW ε,n,1,i = VarW ε,n,1,i & '2 μ n,i − λ ε,n,i , i = 1, . . . , k , n = 1, . . . , − r ε,n '  & μ n,j μ n,i W ε,n,1,j − E W ε,n,1,i − rε r ε,n & & & '' & '' μ n,j μ n,i   = E W ε,n,1,i − − λ ε,n,i W ε,n,1,j − − λ ε,n,j r ε,n r ε,n

(9.92)

= δ ε,i δ ε,j(p ε,n,i,j,+, + + p ε,n,i,j, −,− − p ε,n,i,j,+,− − p ε,n,i,j,−,+) '& ' & μ n,j μ n,i − λ ε,n,i − λ ε,n,j , 1 ≤ i < j ≤ k , n = 1, . . . , − r ε,n r ε,n

where   = δε,i,ȷ i , W ε,n,1,j = δε,n,j,ȷj }, p ε,n,i,j,ȷi,ȷ j = P{W ε,n,1,i

ȷ i , ȷ j = +, ◦ , −, 1 ≤ i < j ≤ k ,

n = 1, . . . ,

(9.93)

and  = δε,i,ȷ}, p ε,n,i,ȷ = P{W ε,n,1,i

ȷ = +, ◦ , − ,

i = 1, . . . , k ,

n = 1, . . . .

(9.94)

382 | 9 Tree-type approximations for Markov Gaussian LPP Relations (9.92) let us rewrite the fitting relations (9.85) in the following equivalent form: ⎧ μ n,i ⎪ − λ ε,n,i , ⎪ δ ε,i(p ε,n,i, + − p ε,n,i, −) = ⎪ ⎪ r ε,n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = 1, . . . , k, n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ '2 & ⎪ ⎪ ⎪ σ 2n,i μ n,i ⎪ 2 ⎪ ⎪ + − λ ε,n,i , δ (1 − p ε,n,i,◦) = ⎪ ⎪ r ε,n r ε,n ⎪ ε,i ⎨ (9.95) i = 1, . . . , k , n = 1, . . . , N , ⎪ ⎪ ⎪ #  ⎪ ⎪ ⎪ ⎪ δ ε,i δ ε,j p ε,n,i,j,+,+ + p ε,n,i,j,−,− − p ε,n,i,j,+,− − p ε,n,i,j, −,+ ⎪ ⎪ ⎪ ⎪ '& ' & ⎪ ⎪ ⎪ ρ n,i,jσ n,i σ n,j μ n,j μ n,i ⎪ ⎪ ⎪ = + − λ − λ ε,n,i ε,n,j , ⎪ ⎪ r ε,n r ε,n r ε,n ⎪ ⎪ ⎪ ⎪ ⎩ 1 ≤ i < j ≤ k , n = 1, . . . , N . The system of equations (9.95) should be supplemented by the following proba­ bility inequalities: 0 ≤ p ε,n,i,j,ȷ i,ȷ j ≤ 1,

ȷ i , ȷ j = +, ◦ , −,

1≤ i < j ≤ k,

n = 1, . . . , N ,

(9.96)

and the following system of consistency relations for probabilities p ε,i,j,ȷi,ȷ j and p ε,i,ȷ i , ȷ i , ȷ j = +, ◦, −, 1 ≤ i < j ≤ k, n = 1, . . . , N: ⎧ ⎪ ⎪ ⎪ p ε,n,i,j,ȷ i, + + p ε,n,i,j,ȷ i, ◦ + p ε,n,i,j,ȷ i, − = p ε,n,i,ȷ i, , ⎪ ⎪ ⎪ ⎪ ⎪ ȷ i = +, ◦ , −, 1 ≤ i < j ≤ k , n = 1, . . . , N , ⎪ ⎨ p ε,n,i,j, +,ȷj + p ε,n,i,j,◦,ȷ j + p ε,n,i,j,−,ȷj = p ε,n,j,ȷj , (9.97) ⎪ ⎪ ⎪ ⎪ ⎪ ȷ j = +, ◦, −, 1 ≤ i < j ≤ k , n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎩p 1 ≤ i ≤ k , n = 1, . . . , N . ε,n,i,+ + p ε,n,i,◦ + p ε,n,i, − = 1 , In order to simplify the fitting problem, let us choose the parameters r ε,n indepen­ dent on n, i.e. assume that they satisfy the following relation: r ε,n = r ε ,

n = 1, 2, . . . , N ,

(9.98)

and choose the parameters λ ε,n,i such that they compensate the drift of the log-price process Y ε,n , i.e. these parameters take the following form: μ n,i λ ε,n,i = , i = 1, . . . , k , n = 1, 2, . . . , N . (9.99) rε In this case, the system of equations (9.95) is essentially simplified. It takes the following form: ⎧ ⎪ p ε,n,i, + − p ε,n,i, − = 0 , i = 1, . . . , k , n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ 2n,i ⎪ 2 ⎪ ⎪ , i = 1, . . . , k , n = 1, . . . , N , ⎨ δ ε,i (1 − p ε,n,i, ◦) = rε (9.100) ⎪ ⎪ ⎪ δ ( p + p − p − p ) δ ε,i ε,j ε,n,i,j, + , + ε,n,i,j, − , − ε,n,i,j, + , − ε,n,i,j, − , + ⎪ ⎪ ⎪ ⎪ ⎪ ρ n,i,jσ n,i σ n,j ⎪ ⎪ ⎩ , 1 ≤ i < j ≤ k , n = 1, . . . , N . = rε

9.3 Multivariate trinomial tree approximations

| 383

Let us introduce the parameters σ i,+ = max σ n,i > 0 , 1≤ n ≤ N

i = 1, . . . , k .

(9.101)

For the second equation in (9.100) let us choose the parameters δ ε,i in the follow­ ing form: δi δ ε,i = √ , where δ i ≥ σ i,+ , i = 1, . . . , k , (9.102) rε and then get the following explicit formulas for quantities p ε,n,i, ◦, which, in this case, do not depend on the parameter ε ∈ (0, ε0 ]: p ε,n,i, ◦ = p n,i, ◦ = 1 −

σ 2n,i δ2i

,

i = 1, . . . , k ,

n = 1, . . . , N .

(9.103)

Note that the quantities p ε,n,i, ◦ ∈ [0, 1), i = 1, . . . , k, n = 1, . . . , N, i.e. they take values consistent with relations (9.96) and (9.97). Formula (9.103) gives us the hint to try to find values of quantities p ε,n,i,ȷ = p n,i,ȷ, ȷ = +, ◦ , −, i = 1, . . . , k and p ε,n,i,j,ȷi,ȷ j = p n,i,j,ȷ i,ȷ j , ȷ i , ȷ j = +, ◦ , − , 1 ≤ i < j ≤ k in the form independent on the parameter ε ∈ (0, ε0 ]. The first equation in (9.100) yields the following formulas for quantities p ε,n,i,ȷ: p ε,n,i, ± = p n,i, ± =

σ 2n,i 1 − p n,i, ◦ = , 2 2δ2i

i = 1, . . . , k ,

n = 1, . . . , N .

(9.104)

Relations (9.97) and (9.104) imply the following relation, for every 1 ≤ i < j ≤ k, n = 1, . . . , N: p n,i, + + p n,i, − = p n,i,j,+,+ + p n,i,j,+,◦ + p n,i,j,+,− + p n,i,j,−,+ + p n,i,j,−,◦ + p n,i,j,−,− = p n,i,j,+,+ + p n,i,j,+,− + p n,i,j,−,+ + p n,i,j,−, − + p n,j, ◦ − p n,i,j,◦, ◦ = 1 − p n,i, ◦

(9.105)

which can be rewritten in the following form: p n,i,j,+,+ + p n,i,j,+,− + p n,i,j,−,+ + p n,i,j,−,− = 1 − p n,i, ◦ − p n,j,◦ + p n,i,j,◦,◦

(9.106)

By substituting quantities δ ε,i given by relation (9.102) in the last equation in (9.100), we get the following equation, for every 1 ≤ i < j ≤ k, n = 1, . . . , N:   p n,i,j,+,+ + p n,i,j,−,− − p n,i,j,+,− − p n,i,j,−,+ = ρ n,i,j 1 − p n,i, ◦ 1 − p n,j,◦ . (9.107) Let us denote, for the moment, P n,i,j = p n,i,j,+,+ + p n,i,j,−,− and Q n,i,j = p n,i,j,+,− + p n,i,j,−,+. For relations (9.106) and (9.107), let us write the following system of linear equations for quantities P n,i,j , Q n,i,j, for every 1 ≤ i < j ≤ k, n = 1, . . . , N: ⎧ ⎨ P n,i,j + Q n,i,j = 1 − p n,i, ◦ − p n,j, ◦ + p n,i,j, ◦,◦ ,   (9.108) ⎩P n,i,j − Q n,i,j = ρ n,i,j 1 − p n,i, ◦ 1 − p n,j, ◦ .

384 | 9 Tree-type approximations for Markov Gaussian LPP By solving this system of equation, we get the following formulas, for every 1 ≤ i < j ≤ k, n = 1, . . . , N: ⎧    1# ⎪ ⎪ ρ n,i,j 1 − p n,i, ◦ 1 − p n,j,◦ + 1 − p n,i, ◦ − p n,j,◦ + p n,i,j,◦,◦ , ⎨ P n,i,j = 2 (9.109)  #   ⎪ ⎪ ⎩ Q n,i,j = 1 − ρ n,i,j 1 − p n,i, ◦ 1 − p n,j, ◦ + 1 − p n,i, ◦ − p n,j, ◦ + p n,i,j,◦,◦ . 2 Relations (9.96) and (9.97) imply that quantities P n,i,j and Q n,i,j should satisfy in­ equalities (a) P n,i,j ≥ 0, (b) Q n,i,j ≥ 0 and (c) P n,i,j + Q n,i,j ≤ 1, for every 1 ≤ i < j ≤ k, n = 1, . . . , N. In order to improve chances for holding inequalities (a) – (c), let us choose for probabilities p n,i,j,◦,◦ the maximal admissible value, which is obviously given by the following formula: p n,i,j,◦,◦ = p n,i, ◦ ∧ p n,j,◦ ,

1≤ i 0 representing it in the form δ ε = √δr ε , where δ, r ε are the positive real numbers. Substituting δ ε = √δr ε in relation (9.142) and taking into account that p ε,+(y) + p ε,−(y) = 1 we can transform relation (9.142) to the following equivalent form: ⎧ δ ⎪ ⎪ δε = √ , ⎪ ⎪ ⎪ rε ⎪ ⎪ ⎪ & ⎪ √ ' ⎨ 1 μ(y) r ε 1+ , p ε,+(y) = (9.143) ⎪ 2 r ε (y) δ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ δ2 2 ⎪ ⎩ r ε (y) − σ 2 (y)r ε (y) − μ 2 (y) = 0 , y ∈ R1 . rε The third equality in (9.143) is a quadratic equation with respect to the unknown r ε (y). Solving this equation and taking into account that r ε (y) > 0, we get the fol­ lowing formula for function r ε (y) : .   4δ 2 μ 2 ( y ) r ε σ2 (y) + σ4 (y) + r ε r ε (y) = , y ∈ R1 . (9.144) 2δ2 The problem is that quantity r ε (y) may not take positive integer values for all y ∈ R1 , except some special cases for functions μ (y) and σ 2 (y). We use the log-price processes Y ε,n for approximation of the log-price process Y0,n in the sense of weak convergence and convergence of the corresponding reward func­ tions. We shall show in Chapter 10 that in order to realize the above approximation, one should assume that the parameter r ε → ∞ as ε → 0 and to use asymptotic moment fitting for log-price processes Y ε,n and Y0,n , where the exact fitting relations (9.141) are replaced by the following weaker asymptotic relations: ⎧ ⎨ EW ε (y) → EW0,1 (y) as ε → 0 , y ∈ R1 , (9.145) ⎩ VarW ε,1 (y) → VarW0,1 (y) as ε → 0 , y ∈ R1 . Let us show that the above asymptotic moment fitting can be achieved, first, by computing real-valued r∗ ε ( y ) using relation (9.144), second, by computing probability

396 | 9 Tree-type approximations for Markov Gaussian LPP p ε,+(y) as function of r∗ ε ( y ) given by the second equality in relation (9.143), third, by taking as r ε (y) the nearest positive integer number that is larger or equal than r∗ ε (y) , and, fourth, by passing the parameter r ε → ∞ as ε → 0. Thus, according to the above remarks, we define the parameters of the approxi­ mating log-price process Y ε,n according to the following formulas: ⎧ δ ⎪ ⎪ δε = √ ⎪ ⎪ ⎪ rε ⎪ ⎪ & ⎪ √ ' ⎪ ⎪ ⎪ 1 μ(y) r ε ⎪ ⎪ 1+ , ⎪ p ε, +(y) = ⎨ 2 r∗ ε,y δ (9.146) .   ⎪ ⎪ 4δ 2 μ 2 ( y ) ⎪ 2 4 ⎪ r ε σ (y) + σ (y) + r ε ⎪ ⎪ ⎪ ⎪ r∗ ( y) = ⎪ , ⎪ ε ⎪ ⎪ 2δ2 ⎪ ⎪ ⎩ r ε ( y ) = [ r∗ y ∈ R1 . ε ( y )] + 1 , where δ, r ε > 0, and finally passing the parameter r ε → ∞ as ε → 0. √ μ (y) r ε and, It should also be noted that, according to relation (9.146), r∗ ε (y) ≥ δ thus, the second equality in (9.146) yields that 0 ≤ p ε,+(y) ≤ 1, for all y ∈ R1 . Let us show that the asymptotic moment-fitting relation (9.145) takes place in the case, where parameters of the approximating log-price process Y ε,n are chosen accord­ ing to relation (9.146). The second equality in relation (9.146) can be rewritten in the following equivalent form, for every ε ∈ (0, ε0 ] and y ∈ R1 : & ' δ δ ∗ (9.147) r ε,y √ p ε,+(y) − √ p ε,−(y) = μ (y) . rε rε Taking into account relation (9.147), inequality 0 ≤ r ε (y) − r∗ ε ( y ) ≤ 1, relation → ∞ as ε → 0, which follows from the third equality in (9.146), and the as­ sumption r ε → ∞ as ε → 0, we get the following relation, for every y ∈ R1 : & ' δ δ EW ε (y) = r ε (y) √ p ε,+(y) − √ p ε,− (y) rε rε & & & √ ' √ '' δ 1 μ(y) r ε δ 1 μ(y) r ε − = μ ( y ) + ( r ε ( y ) − r∗ ( y )) 1 + 1 − √ √ ε rε 2 r∗ rε 2 r∗ ε (y)δ ε (y) δ r∗ ε (y)

= μ(y) + μ(y)

r ε ( y ) − r∗ ε (y) → μ(y) ∗ r ε (y)

as

ε → 0.

(9.148)

Thus, the first asymptotic moment-fitting relation in (9.145) holds. The function r∗ ε ( y ), given by the third equality in relation (9.146), is a root of the quadratic equation. It is given by the third equality in relation (9.143), or, equivalently, in the form of the second equality in relation (9.142). Thus, r∗ ε ( y ) satisfies the following relation, for every ε ∈ (0, ε0 ] and y ∈ R1 : ⎛ & '2 ⎞ μ ( y ) ⎝δ2 − ⎠ = σ2 (y) . r∗ (9.149) ε (y) ε r∗ ε (y)

9.4 Inhomogeneous in space binomial approximations

|

397

Taking into account relation (9.149), inequality 0 ≤ r ε (y) − r∗ ε ( y ) ≤ 1, relation → ∞ as ε → 0, which follows from the third equality in (9.146), and the assumption that r ε → ∞ as ε → 0, we get the following relation, for every y ∈ R1 : ⎛ & '2 ⎞ δ2 μ(y) ⎠ ⎝ − VarW ε (y) = r ε (y) rε r ε (y) r∗ ε (y), r ε (y)

 δ2 = σ 2 ( y ) + r ε ( y ) − r∗ ε (y) rε 2 μ (y) + ( r ε ( y ) − r∗ → σ2 (y) ε ( y )) r ε ( y ) r∗ ε (y)

as

ε → 0.

(9.150)

Relation (9.150) is equivalent to the second asymptotic moment-fitting relation in (9.145). The above remarks can be summarized in the following lemma. Lemma 9.4.1. The log-price process Y ε,n represented, for every ε ∈ (0, ε0 ], by the homogeneous binomial Markov chain with transition probabilities defined by relation (9.140), realizes the asymptotic moment fitting given by relation (9.145), for the log-price processes Y0,n represented by the Gaussian Markov chain with transition probabilities defined by relation (9.139). This holds, if the corresponding parameters of the process Y ε,n are given by relation (9.146), and under the assumption that the parameter r ε → ∞ as ε → 0.

9.4.2 The recombining condition The question about recombining property of the binomial Markov chain Y ε,n does re­ quire a special analysis. Let us first consider the case, where the following condition holds: G12 : supy∈R1 (|μ (y)| + σ 2 (y)) < K85 , for some 0 < K85 < ∞. The condition G12 and relation (9.146) implies the following relation: .   4δ 2 μ 2 ( y ) 2 r ε σ (y) + σ4 (y) + r ε sup r ε (y) ≤ sup +1 2δ2 y ∈R1 y ∈R1 #  δ | μ ( y )| r ε σ 2 ( y) + √r ε ≤ sup +1 δ2 y ∈R1 ≤

K85 + K85 δ + δ2 r ε = K86 (δ)r ε . δ2

(9.151)

As follows from inequality (9.151), the corresponding tree T ε,(y,n) generated by the trajectories of the log-price process Y ε,n+m, m = 0, 1, . . . , with the initial value Y ε,n =

398 | 9 Tree-type approximations for Markov Gaussian LPP y, has the total number of points after r steps, L ε,n,n+r ≤ 1 + 2K86 (δ)r ε + 1 + · · · + 2K86 (δ)r ε r + 1 = K86 (δ)r ε r2 + (K86 (δ)r ε + 1)r + 1 ,

(9.152)

The expression on the right-hand side in 9.152) is a polynomial (in this case quadratic) function of r. Let us now assume that the condition G12 does not hold. In this case, the number of nodes, which appear in the corresponding tree T ε,(y,n) generated by the trajectories ε,n =  y , may possess of the log-price process Y ε,n+r , r = 0, 1, . . . , with the initial value Y an exponential rate of growth as a function of the number of steps. For example, let us assume that the coefficients μ (y) and σ 2 (y) have not less than linear rate of growth as functions of y, i.e. that following condition holds: | μ ( y )|+ σ 2( y ) G13 : inf y∈R1 1+K 88 | y| ≥ K87 , for some 0 < K87 , K88 < ∞. Let us assume for simplicity that (y, n) = (0, 0). Let us denote by y ε,r the maximal value that can take the random variable |Y ε,r |. Formula (9.146) and the condition G13 √ √ r K r K K y r K K imply that y ε,1 ≥ r εδK288 √δr ε = εδ 88 . Also, y ε,2 ≥ y ε,1 + ε 87δ288 ε,1 √δr ε = y ε,1(1 + ε δ87 88 ). Since, the parameter r ε → ∞ as ε → 0, we can always assume that r ε ≥ 1. In this case, the above inequalities imply that y ε,2 ≥ y ε,1(1 + L), where L = K 87δK 88 > 0. The repe­ tition of the above calculations yields inequalities y ε,r ≥ y ε,1(1 + L)r−1 , r = 1, 2, . . . . These inequalities imply that the number of nodes appearing in the tree T ε,(0,0), as the √ 2y r L ) r −1 result of rth step, is larger than or equal to ε,rδ ε = 2r ε (1+ , and, thus, this tree has δ2 the total number of nodes after r steps, 2r ε 2r ε (1 + L)r−1 2r ε ((1 + L)r − 1) + · · · = , (9.153) δ2 δ2 δ2 L Therefore, the tree has an exponential rate of growth for the number of nodes as a function of r. L ε,n,n+r ≥

9.5 Inhomogeneous in time and space trinomial approximations In this section, we present results about fitting of parameters for inhomogeneous in time and space trinomial tree approximation models.

9.5.1 Fitting of parameters for univariate inhomogeneous in time and space trinomial tree models Let us assume that the log-price process Y0,n is an inhomogeneous in time Gaussian Markov chain, with the phase space R1 , an initial distribution P0 (A), and transition

9.5 Inhomogeneous in time and space trinomial approximations

| 399

probabilities, given for y ∈ R1 , A ∈ B1 , n = 1, 2, . . . by the following relation: P0,n (y, A) = P{Y0,n ∈ A/Y0,n−1 = y} = P{y + W0,n (y) ∈ A} ,

(9.154)

where (a) W0,n (y) is, for every y ∈ R1 , n = 1, 2, . . . , a normal random variable with mean value EW0,n (y) = μ n (y) and variance VarW0,n (y) = σ 2n (y). Usual assumptions are: (b) μ n (y) is, for every n = 1, 2, . . . , a measurable function acting from R1 to R1 , (c) σ 2n (y) is, for every n = 1, 2, . . . ,a measurable function acting from R1 to (0, ∞). We shall approximate the log-price process Y0,n by the log-price process Y ε,n , which is, for every ε ∈ (0, ε0 ], an inhomogeneous trinomial Markov chain, with the phase space R1 , an initial distribution P ε (A), and transition probabilities, given for y ∈ R1 , A ∈ B1 , n = 1, 2, . . . by the following relation: P ε,n (y, A) = P{Y ε,n ∈ A/Y ε,n−1 = y} = P{y + W ε,n (y) ∈ A} ,

(9.155)

where (a) W ε,n (y) is, for every y ∈ R1 , n = 1, 2, . . . , a trinomial random variable, which can be represented in the form W ε,n (y) = W ε,n,1(y) + · · · + W ε,n,r ε,n(y) (y), where W ε,n,l(y), l = 1, 2, . . . are the i.i.d. random variables taking values δ ε,ȷ with probabilities p ε,n,ȷ(y), for ȷ = +, ◦, −}. Usual assumptions are: (b) δ ε,− = −δ ε < δ ε,◦ = 0 < δ ε,+ = +δ ε , (c) p ε,n,ȷ(y) , ȷ = +, ◦, − , n = 1, 2, . . . are the measurable functions acting from R1 to [0, 1] such  that ȷ=+,◦,− p ε,n,ȷ(y) = 1, y ∈ R1 , n = 1, 2, . . . ; (d) r ε,n (y), n = 1, 2, . . . are the measurable functions defined on the space R1 and taking positive integer values. Note that, in this case, the use of asymmetric jumps δ ε,n,ȷ = δ ε,ȷ + λ ε,n , n = 1, 2, . . . has no advantage, since any compensators λ ε,n ∈ R1 inhomogeneous in time cannot compensate an inhomogeneous in time and space drift μ n (y). Trinomial models have larger number of the parameters p ε,n,ȷ(y) than binomial models. This makes it possible to simplify the choice of the parameters r ε,n (y). We simplify the model by choosing the parameters r ε,n (y) of the simplest form r ε,n (y) = r ε ,

y ∈ R1 ,

n = 1, 2, . . . .

(9.156)

In this case, the Markov random walk Y ε,n satisfies the recombining condition. Indeed, the random variable W ε,n (y) can, for every y ∈ R1 , n = 1, 2, . . . , take with positive probabilities only values lδ ε , −r ε ≤ l ≤ r ε and, therefore, the random variable Y ε,n can, under condition that Y ε,0 = y, take with positive probabilities only values y + lδ ε , −nr ε ≤ l ≤ nr ε . The number of such values is 2nr ε + 1. We use the log-price processes Y ε,n for approximation of the log-price process Y0,n asymptotically, as ε → 0. In this case, the usual approach is to fit the corresponding parameters, i.e. to try choosing the parameters δ ε , probabilities p ε,n,ȷ(y), and the pa­ rameters r ε in such a way that the following exact fitting equalities holds, for every ε ∈ (0, ε0 ]: ⎧ ⎨ EW ε,n (y) = EW0,n (y) , (9.157) ⎩ VarW ε,n (y) = VarW0,n (y) , y ∈ R1 , n = 1, . . . , N .

400 | 9 Tree-type approximations for Markov Gaussian LPP Calculations analogous to those made in relations (9.55)–(9.57) yield the following formulas, for every y ∈ R1 , n = 1, . . . : EW ε,n,1(y) = δ ε (p ε,n,+(y) − p ε,n,−(y)) =

μ n (y) , rε

(9.158)

and 

VarW ε,n,1(y) = δ2ε (1 − p ε,n,◦(y)) −

μ n (y) rε

2 =

σ 2n (y) . rε

(9.159)

By analogy with the homogeneous in space case, we shall choose the parameter δ ε in the following form: δ (9.160) δε = √ . rε For relations (9.158) – (9.160), let us rewrite the fitting relations (9.157) in the fol­ lowing equivalent form: ⎧ μ n (y) ⎪ ⎪ ⎪ ⎨ p ε,n, +(y) − p ε,n, −(y) = δ√r , ε (9.161) 2 ⎪ σ ( y ) μ 2n (y) ⎪ n ⎪ ⎩ + 2 , y ∈ R1 , n = 1, . . . , N . 1 − p ε,n,◦(y) = δ2 δ rε The system of equations (9.161) should be supplemented by the following consis­ tency relations: ⎧ ⎨ p ε,n,ȷ(y) ≥ 0 , ȷ = +, ◦, − , (9.162) ⎩ p ε,n, +(y) + p ε,n, ◦(y) + p ε,n, −(y) = 1 , y ∈ R1 , n = 1, . . . , N . The system of equations (9.161) supplemented by the consistency equalities from relations (9.162) has the following solution: & ' ⎧ ⎪ 1 σ 2n (y) μ 2n (y) μ n (y) ⎪ ⎪ + 2 ± √ , ⎪ ⎨ p ε,n, ±(y) = 2 δ2 δ rε δ rε (9.163) ⎪ ⎪ σ 2n (y) μ 2n (y) ⎪ ⎪ ⎩ p ε,n, ◦(y) = 1 − − 2 , y ∈ R1 , n = 1, . . . , N . δ2 δ rε Let us assume that the following conditions hold: G14 : min1≤n≤N inf y∈R1 σ 2n (y) ≥ σ 2 , for some σ 2 > 0. and G15 : max1≤n≤N supy∈R1 (|μ n (y)| + σ 2n (y)) < K89 , for some 0 < K89 < ∞. In this case, the consistency inequalities given in relation (9.162) hold for probabili­ ties p ε,n,ȷ(y), ȷ = +, ◦, −, for every y ∈ R1 , n = 1, . . . , N, if to choose parameter δ satisfying the following inequality:  2 + K89 , (9.164) δ ≥ K89

9.5 Inhomogeneous in time and space trinomial approximations

| 401

and then to choose the parameter r ε satisfying the following inequality: 2 2 K89 δ . (9.165) 4 σ Indeed, the condition G15 and inequality (9.164) imply that, for every y ∈ R1 , n = 1, . . . , N,

rε ≥

y) = 1 − p ε,n,◦(

2 K89 σ 2n ( y) μ 2n (y) K89 − ≥ 1 − − ≥ 0, δ2 δ2 r ε δ2 δ2

(9.166)

Also, the conditions G14 and G15 , and inequality (9.165) imply that, for every y ∈ R1 , n = 1, . . . , N, & ' & ' 1 σ 2n (y) μ 2n (y) μ n (y) 1 σ2 K89 + 2 ± √ − √ (9.167) ≥ ≥ 0. p ε,n,±(y) = 2 δ2 δ rε δ rε 2 δ2 δ rε The above remarks can be summarized in the following lemma. Lemma 9.5.1. The log-price process Y ε,n represented, for every ε ∈ (0, ε0 ], by the in­ homogeneous trinomial Markov chain with transition probabilities defined by relation (9.155), realizes the exact moment fitting given by relation (9.157) for the log-price process Y0,n represented by the Gaussian–Markov chain with transition probabilities defined by relation (9.154). This holds, if the corresponding parameters of the process Y ε,n are given by relations (9.160) and (9.163), the conditions G14 and G15 hold, and parameters δ and r ε satisfy conditions (9.164) and (9.165). The corresponding tree T ε,(y,n) generated by the trajectories of the log-price process Y ε,n+r, r = 0, 1, . . . , with the initial value Y ε,n = y, has the total number of points after r steps, L ε,n,n+r ≤ 1 + 2r ε + 1 + · · · + 2r ε r + 1 = r ε r2 + (r ε + 1)r + 1 .

(9.168)

The expression on the right-hand side in (9.168) is a polynomial (in this case quadratic) function of r.

9.5.2 Fitting of parameters for multivariate inhomogeneous in time and space trinomial tree models 0,n is an inhomogeneous in time Gaussian Let us assume that a log-price process Y Markov chain with the phase space Rk , an initial distribution P0 (A), and transition y ∈ Rk , A ∈ Bk , n = 1, 2, . . . by the following relation: probabilities, given for  0,n ∈ A/Y 0,n−1 =   0,n ( y , A ) = P{ Y y } = P{ y+W y) ∈ A} , P0,n (

(9.169)

 0,n ( y) = (W0,n,1( y), . . . , W0,n,k ( y)) is, for every n = 1, 2, . . . ,  y ∈ Rk , where (a) W y) = μ n,i ( y), i = 1, . . . , k, vari­ a normal random variable with means EW0,n,i ( ances VarW0,n,i( y) = σ 2n,i( y ), i = 1, . . . , k, and covariances E(W0,n,i( y ) − μ n,i ( y)) y ) − μ n,j( y))) = ρ n,i,j( y)σ n,i ( y)σ n,j( y ), i, j = 1, . . . , k. ·(W0,n,j(

402 | 9 Tree-type approximations for Markov Gaussian LPP Usual assumptions are: (b) μ n,i( y ), i = 1, . . . , k, n = 1, 2, . . . are the measurable functions acting from Rk to R1 , (c) σ 2n,i ( y), i = 1, . . . , k, n = 1, 2, . . . are the mea­ surable function acting from R1 to (0, ∞); (d) ρ n,i,j( y), i, j = 1, . . . , k, n = 1, 2, . . . y) = 1,  y ∈ Rk , are the measurable functions acting from Rk to [−1, 1] and ρ n,i,i( i = 1, . . . , k, n = 1, 2, . . . ; (e) Σ( y) = σ n,i,j( y) ,  y ∈ Rk , n = 1, 2, . . . , where σ n,i,j( y) = ρ n,i,j( y)σ n,i ( y)σ n,j( y), i, j = 1, . . . , k, are nonnegatively defined k × k ma­ trices. 0,n by the log-price process Y ε,n which We shall approximate the log-price process Y is, for every ε ∈ (0, ε0 ], an inhomogeneous trinomial Markov chain with the phase y ∈ Rk , space Rk , an initial distribution P ε (A), and transition probabilities, given for  A ∈ Bk , n = 1, 2, . . . by the following relation: ε,n ∈ A/Y ε,n−1 =   ε,n ( y , A ) = P{ Y y} = P{ y+W y) ∈ A} , P ε,n (

(9.170)

 ε,n ( where (a) W y ) = (W ε,n,1( y), . . . , W ε,n,k( y)) is, for every  y ∈ Rk , n = 1, 2, . . . , a  ε,n,1(  ε,n ( trinomial random vector, which can be represented in the form W y) = W y) +  ε,n,r ε,n(y) (  ε,n,l( ··· + W y), where W y) = (W ε,n,l,1( y), . . . , W ε,n,l,k( y)), l = 1, 2, . . . are the i.i.d. random vectors taking, for every  y ∈ Rk and n = 1, 2, . . ., values δ¯ ε,¯ȷ = (δ ε,1,ȷ1 , . . . , δ ε,k,ȷ k ) with probabilities p ε,n,¯ȷ( y ) = p ε,n,ȷ1,...ȷ k ( y ), for ¯ȷ ∈ Jk = {¯ȷ =  ȷ1 , . . . , ȷ k  : ȷ j , . . . ȷ k = +, ◦, −}; Usual assumptions are: (b) δ ε,i, − = −δ ε,i < δ ε,i, ◦ = 0 < δ ε,i, + = +δ ε,i , i = 1, . . . , k; (c) p ε,n,¯ȷ( y), ¯ȷ ∈ Jk , n = 1, 2, . . . are measurable function acting from Rk to  [0, 1] such that ¯ȷ∈Jk p ε,n,¯ȷ( y) = 1,  y ∈ Rk , n = 1, 2, . . . ; (d) r ε,n ( y ), n = 1, 2, . . . are measurable functions defined on the space Rk and taking positive integer values. Note that, in this case, the use of asymmetric jumps δ ε,n,i,ȷ = δ ε,i,ȷ + λ ε,n,i, i = 1, . . . , k, n = 1, 2, . . . has no advantage, since any compensators λ ε,n,i ∈ R1 inho­ mogeneous in time cannot compensate an inhomogeneous in time and space drifts μ n,i ( y ). y) of the We also simplify the model by choosing positive integer parameters r ε,n ( simplest form, r ε,n ( y) = r ε ,  y ∈ Rk , n = 1, 2, . . . . (9.171)

As a matter of fact, trinomial models have larger number of parameters p ε,n,¯ȷ( y) than binomial models. This makes it possible to simplify the choice of the parameters r ε,n ( y ). ε,n satisfies the recombining condition. In this case, the Markov random walk Y  Indeed, the random vector W ε,n ( y) can, for every  y ∈ Rk , n = 1, 2, . . . , take with positive probabilities only values (l1 δ ε,1 , . . . , l k δ ε,k ), −r ε ≤ l1 , . . . , l k ≤ r ε and, there­ ε,n can, under condition that Y ε,0 =  y, take with positive fore, the random vector Y probabilities only values  y + (l1 δ ε,1 , . . . , l k δ ε,k ), −nr ε ≤ l1 , . . . , l k ≤ nr ε . The number of such values is (2nr ε + 1)k .  ε,n for approximation the log-price process Y 0,n We use the log-price processes Y asymptotically, as ε → 0. In this case, a usual approach is to fit the corresponding pa­

9.5 Inhomogeneous in time and space trinomial approximations

|

403

y) and a parameters rameters, i.e. to try to choose parameters δ ε,i , probabilities p ε,n,¯ȷ( r ε in such a way that the following exact fitting equalities hold, for every ε ∈ (0, ε0 ]: ⎧ EW ε,n,i( y) = EW0,n,i ( y) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y) = VarW0,n,i( y) , VarW ε,n,i( ⎪ ⎪ ⎨ i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎪ E(W ε,n,i( y) − EW ε,n,i( y))(W ε,n,i( y) − EW ε,n,j( y)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = E(W0,n,i( y ) − EW0,n,i( y))(W0,n,i( y) − EW0,n,j ( y)) , ⎪ ⎪ ⎪ ⎪ ⎩ y ∈ Rk , n = 1, . . . , N . 1≤i < j ≤ k, 

(9.172)

Calculations analogous to those made in relations (9.55)–(9.57) yield the following formulas: μ n,i ( y) y) = δ ε,i (p ε,n,i,+( y) − p ε,n,i, −( y)) = , EW ε,n,1,i( rε y ∈ Rk , i = 1, . . . , k, 

n = 1, . . . ,

y) = δ2ε,i (1 − p ε,n,i, ◦( y)) − ( VarW ε,n,1,i(

σ 2n,i( y) μ n,i( y) 2 ) = , rε rε

y ∈ Rk , n = 1, . . . , i = 1, . . . , k,  '& ' μ n,j( y) μ n,i ( y) E W ε,n,1,i( y) − W ε,n,1,j( y) − (9.173) rε rε #  y )+ p ε,n,i,j,−,−( y)− p ε,n,i,j,+,−( y)− p ε,n,i,j,−,+( y) = δ ε,i δ ε,j p ε,n,i,j,+, +( &

μ n,i ( y)μ n,j( y) r2ε ρ n,i,j( y)σ n,i ( y)σ n,j( y) = , rε −

y ∈ Rk , 1 ≤ i < j ≤ k, 

n = 1, . . . ,

where p ε,n,i,j,ȷi,ȷ j ( y) = P{W ε,n,1,i( y ) = δ ε,i,ȷ i , W ε,n,1,j( y) = δ ε,j,ȷ j }, ȷ i , ȷ j = +, ◦ , − ,

1 ≤ i < j ≤ k,  y ∈ Rk ,

n = 1, . . . ,

(9.174)

and p ε,n,i,ȷi = P{W ε,n,1,i = δ ε,i,ȷ i } ,

ȷ i = +, ◦ , − ,

i = 1, . . . , k ,

n = 1, . . . .

(9.175)

By analogy with the homogeneous in space case, we shall search the parameters δ ε,i in the following form: δi δ ε,i = √ , rε

i = 1, . . . , k .

(9.176)

404 | 9 Tree-type approximations for Markov Gaussian LPP For relations (9.173) and (9.176), let us rewrite the fitting relations (9.172) in the following equivalent form: ⎧ μ n,i ( y) ⎪ ⎪ ⎪ p ε,n,i,+( y ) − p ε,n,i,−( y) = √ , ⎪ ⎪ ⎪ δ r i ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ = 1, ...,k,  y ∈ Rk , n = 1, . . . , N , i ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ σ n,i ( y) μ n,i ( y) ⎪ ⎪ ⎪ y) = 1 − − , p ε,n,i, ◦( ⎪ 2 2 ⎪ ⎨ δi δi rε (9.177) ⎪ y ∈ Rk , n = 1, . . . , N , i = 1, . . . , k ,  ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p ε,n,i,j, +,+( y) + p ε,n,i,j,−,−( y) − p ε,n,i,j,+,−( y ) − p ε,n,i,j, −,+( y) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρ ( y ) σ ( y ) σ ( y ) μ ( y ) μ ( y ) ⎪ n,i,j n,i n,j n,i n,j ⎪ ⎪ + , = ⎪ ⎪ ⎪ δ δ δ δ r i j i j ε ⎪ ⎪ ⎪ ⎪ ⎩ y ∈ Rk , n = 1, . . . , N . 1≤i < j ≤ k,  The system of equations (9.95) should be supplemented by the following proba­ bility inequalities: 0 ≤ p ε,n,i,j,ȷ i,ȷ j ( y) ≤ 1 , 1≤ i < j ≤ k,

ȷ i , ȷ j = +, ◦ , − ,

 y ∈ Rk ,

n = 1, . . . , N ,

and the following system of consistency relations: ⎧ ⎪ p ε,n,i,j,ȷ i,+( y) + p ε,n,i,j,ȷi,◦ ( y) + p ε,n,i,j,ȷi,− ( y) = p ε,n,i,ȷ i ( y) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ p ε,n,i,j,+,ȷj ( y ) + p ε,n,i,j, ◦,ȷj ( y) + p ε,n,i,j,−,ȷ j ( y) = p ε,n,j,ȷj ( y) , ⎪ ⎪ ⎨ ȷ i , ȷ j = +, ◦ , − , 1 ≤ i < j ≤ k ,  y ∈ Rk , n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎪ p ε,n,i, +( y) + p ε,n,i,◦( y) + p ε,n,i−( y) = 1 , ⎪ ⎪ ⎪ ⎪ ⎩ 1≤ i ≤k,  y ∈ Rk , n = 1, . . . , N .

(9.178)

(9.179)

The first two series of equalities in (9.177) and the last series of equalities in (9.179) yield the following system of equations: ⎧ μ n,i( y) ⎪ ⎪ p ε,n,i,+( y) − p ε,n,i, −( y) = √ , ⎪ ⎪ ⎪ δi rε ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ σ 2n,i ( y) μ 2n,i ( y) y) = 1 − − , p ε,n,i, ◦( (9.180) 2 2 δi δi rε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y) + p ε,n,i−( y) = 1 − p ε,n,i, ◦( y) ⎪ p ε,n,i, +( ⎪ ⎪ ⎪ ⎩ i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N , which yields the following formulas: ⎧ & 2 ' ⎪ y ) μ 2n,i( y ) μ n,i ( 1 σ n,i( y) ⎪ ⎪ ⎪ p ε,n,i, ±( y) = + ± √ , ⎪ ⎪ 2 δi rε δ2i δ2i r ε ⎪ ⎪ ⎨ σ 2n,i ( y) μ 2n,i( y) ⎪ ( y ) = 1 − − , p ⎪ ε,n,i, ◦ 2 2 ⎪ ⎪ δi δi rε ⎪ ⎪ ⎪ ⎪ ⎩ y ∈ Rk , n = 1, . . . , N , i = 1, . . . , k , 

(9.181)

9.5 Inhomogeneous in time and space trinomial approximations

|

405

Let us assume that the following conditions hold: y) ≥ σ 2 , for some σ 2 > 0, G16 : min1≤i≤k,1≤n≤N infy∈Rk σ 2n,i ( and G17 : max1≤i≤k,1≤n≤N supy∈Rk (|μ n,i ( y)| + σ 2n,i( y )) < K90 , for some 0 < K90 < ∞. In this case, the following consistency relations hold: ⎧ ⎪ p ε,n,i, +( y), p ε,n,i,◦( y), p ε,n,i, −( y) ≥ 0 , ⎪ ⎪ ⎨ p ε,n,i, +( y) + p ε,n,i, ◦( y) + p ε,n,i, −( y) = 1 , ⎪ ⎪ ⎪ ⎩ 1≤ i ≤k,  y ∈ R , n = 1, . . . , N ,

(9.182)

k

if we choose the parameters δ i , i = 1, . . . , k satisfying the following inequalities:  2 + K90 , i = 1, . . . , k . (9.183) δ i ≥ K90 and then choose the parameter r ε satisfying the following inequality: 2 2 K90 δ , σ4

(9.184)

δ = max δ i .

(9.185)

rε ≥ where

1≤ i ≤ k

Indeed, the condition G17 and inequality (9.183) imply that, for every i = 1, . . . , k,  y ∈ Rk , n = 1, . . . , N, p ε,n,i, ◦( y) = 1 −

2 σ 2n,i( y ) μ 2n,i( y) K90 K90 − ≥ 1 − − ≥ 0, δ2i δ2i r ε δ2i δ2i

(9.186)

Also, the conditions G16 , G17 , and inequality (9.184) imply that, for every i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N, & 2 ' y) μ 2n,i ( y) μ n,i ( 1 σ n,i( y) p ε,n,i, ±( y) = + ± √ 2 δi rε δ2i δ2i r ε & ' 2 1 σ K90 ≥ 0. ≥ − √ (9.187) 2 δ2i δi rε Relations (9.179) imply the following relation, for every 1 ≤ i < j ≤ k , Rk , n = 1, . . . , N:

 y ∈

y) + p ε,n,i,−( y ) = p ε,n,i,j,+,+( y ) + p ε,n,i,j, +,◦( y) + p ε,n,i,j,+,−( y) p ε,n,i, +( + p ε,n,i,j, −,+( y) + p ε,n,i,j,−,◦( y) + p ε,n,i,j,−,−( y)

(9.188)

= p ε,n,i,j,+, +( y ) + p ε,n,i,j, +,−( y) + p ε,n,i,j,−,+( y) + p ε,n,i,j, −,−( y) + p ε,n,j,◦( y) − p ε,n,i,j,◦,◦( y ) = 1 − p ε,n,i, ◦( y) ,

406 | 9 Tree-type approximations for Markov Gaussian LPP which can be rewritten in the following form: p ε,n,i,j, +,+( y) + p ε,n,i,j,+,−( y) + p ε,n,i,j,−,+( y ) + p ε,n,i,j, −,−( y) = 1 − p ε,n,i, ◦( y) − p ε,n,j,◦( y) + p ε,n,i,j,◦,◦( y) =

σ 2n,i( y) δ2i

+

μ 2n,i ( y) δ2i r ε

+

σ 2n,j ( y) δ2j

+

μ 2n,j( y) δ2j r ε

− 1 + p ε,n,i,j,◦, ◦( y)

(9.189)

y ∈ Rk , n = 1, . . . , N, Let us denote, for 1 ≤ i < j ≤ k ,  P ε,n,i,j( y) = p ε,n,i,j,+,+( y) + p ε,n,i,j,−,−( y) , Q ε,n,i,j( y) = p ε,n,i,j,+,−( y) + p ε,n,i,j,−,+( y) ,

(9.190)

The last series of equations in (9.177) and relations (9.189) constitute the following system of equations: ⎧ ⎪ σ 2n,i( y ) μ 2n,i( y) ⎪ ⎪ ⎪ P ε,n,i,j( y ) + Q ( y ) = + ε,n,i,j ⎪ 2 2 ⎪ ⎪ δi δi rε ⎪ ⎪ ⎪ ⎪ 2 ⎪ σ ( y ) μ 2n,j ( y) ⎪ n,j ⎪ ⎨ + + − 1 + p ε,n,i,j,◦,◦( y) , 2 2 δj δj rε (9.191) ⎪ ⎪ ⎪ ⎪ ρ n,i,j( y)σ n,i ( y)σ n,j( y) μ n,i ( y)μ n,j ( y) ⎪ ⎪P ⎪ y ) − Q ε,n,i,j( y) = + , ⎪ ε,n,i,j( ⎪ ⎪ δ δ δ δ r i j i j ε ⎪ ⎪ ⎪ ⎪ ⎩ y ∈ Rk , n = 1, . . . , N 1≤i 𝛾 if 𝛾 > 0 or β = 𝛾 if 𝛾 = 0. The condition I9 is a particular case of the condition I3 . Relation (10.16) also implies, by the corresponding variant of the continuous the­ orem, the following relation, for every n = 1, . . . , N: d

W ε,n −→ W0,n

as ε → 0 .

(10.17)

Relation (10.17) implies that the condition J4 (a) (its univariate variant) holds. The condition J13 implies that the condition J4 (b) (its univariate variant) holds, since random variables W0,n , n = 1, . . . , N have normal distributions with positive variances and, therefore, possess positive everywhere probability density functions. Thus, all conditions of Theorem 5.3.1 hold. By applying this theorem to the above approximation model, we get the asymptotic relation (10.12). It is worth to note one more time that Theorem 10.1.1 also covers the case of univariate homogeneous in time binomial tree approximation for log-price processes represented by univariate homogeneous in time Gaussian random walk.

10.1.2 Convergence of option rewards for a univariate binomial tree model with asymptotically fitted parameters Alternatively, we can approximate the log-price process Y0,n by the log-price Y ε,n repre­ sented by inhomogeneous in time binomial random walk given, for every ε ∈ (0, ε0 ], by the following stochastic transition dynamic relation: Y ε,n = Y ε,n−1 + W ε,n ,

n = 1, 2, . . . .

(10.18)

10.1 Univariate binomial tree approximation models

| 419

where (a) Y ε,0 is a real-valued random variable with the distribution P ε (A), (b) W ε,n = W ε,n,1 + · · · + W ε,n,r ε,n , n = 1, 2, . . . , where W ε,n,l , n, l = 1, 2, . . . is a family of inde­ pendent binary random binomial variables, which take values δ ε,n,+ and δ ε,n,− with probabilities, respectively, p ε,n,+ and p ε,n,−, and (c) the random variable Y ε,0 and the family of random variables W ε,n,l , n, l = 1, 2, . . . are independent. Also, usual assumptions are (d) δ ε,n,− < δ ε,n,+, n = 1, 2, . . . , (e) 0 ≤ p ε,n,+ = 1 − p ε,n,− ≤ 1, n = 1, 2, . . . , and (f) r ε,n , n = 1, 2, . . . are positive integer numbers. The difference with the previous approximation model is that we admit possibil­ ity for the corresponding binomial random variables W ε,n to have the parameters r ε,n depending on n. In this case, we fit parameters of approximating log-price processes Y ε,n in such a way that the following asymptotic moment fitting relation holds: EW ε,n → μ n ,

VarW ε,n → σ 2n ,

as ε → 0 ,

n = 1, . . . , N .

(10.19)

The solution of this fitting problem is presented in Lemma 9.1.3. According to this lemma relation (10.19) holds, if one can choose the parameters of the log-price process Y ε,n given by the following formulas, for every ε ∈ (0, ε0 ] and n = 1, . . . , N: 8 9 σ2 δ μn 1 , p ε,n,± = , r ε,n = r ε 2n + 1 , (10.20) δ ε,n,± = √ + λ ε,n , λ ε,n = rε r ε,n 2 δ where (i) one can choose for the parameters δ, r ε any real values δ, r ε > 0, (ii) then, to pass parameter r ε → ∞ as ε → 0. The random variables W ε,n , n = 1, . . . , N are binomial random variables. Namely, the random variable W ε,n takes values, δ δ w ε,n,l = l √ + r ε,n λ ε,n = l √ + μ n , rε rε with the corresponding probabilities, ⎧ rε,n +l # r ε,n 1 2 ⎪ ⎪ ⎪ 2 ⎨ C r ε,n p ε,n,l = ⎪0 ⎪ ⎪ ⎩ l = −r ε,n , . . . , r ε,n .

if if

r ε,n + l 2 r ε,n + l 2

l = −r ε,n , . . . , r ε,n

(10.21)

is integer , is not integer ,

(10.22)

Note that we slightly modified the above standard definition of binomial distribu­ tion by including the range of all possible values integer numbers −r ε,n ≤ l ≤ r ε,n , including the values with the corresponding probabilities equal to zero. Let us assume that Y ε,n = y = const ∈ R1 . In this case, trajectories of the log-price process Y ε,n+r, r = 0, 1, . . . generate, for every ε ∈ (0, ε0 ], the unique tree of nodes   (10.23) T ε,(y,n) = y ε,n,n+r,l, n + r , l = −r ε,n,n+r, . . . , r ε,n,n+r, r = 0, 1, . . . , where δ y ε,n,n+r,l = y + l √ + μ n,n+r , rε

−r ε,n,n+r ≤ l ≤ r ε,n,n+r ,

r = 0, 1, . . . .

(10.24)

420 | 10 Convergence of tree-type reward approximations and μ n,n+r =

n

+r m = n +1

μm ,

n

+r

r ε,n,n+r =

r ε,m ,

r = 0, 1, . . . .

(10.25)

m = n +1

The tree T ε,(y,n) contains the total number of nodes after r steps, & & ' ' σ2 σ2 L ε,n,n+r ≤ r ε 2 + 1 r2 + r ε 2 + 2 r + 1 , δ δ

(10.26)

where σ 2 = max1≤n≤N σ 2n . The following lemma is a direct corollary of Lemma 3.3.3. Lemma 10.1.2. Let, for every ε ∈ (0, ε0 ], the approximating log-price process Y ε,n is the binomial random walk given by the stochastic transition dynamic relation (10.18), with parameters defined in relation (10.20). Then the reward functions ϕ ε,n+r(y ε,n,n+r,l), for points y ε,n,n,0 = y and y ε,n,n+r,l, l = −r ε,n,n+r , . . . , r ε,n,n+r, r = 1, . . . N − n, are, for every ε ∈ (0, ε0 ], and y ∈ R1 , n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ ϕ ε,N (y ε,n,N,l) = g ε (N, e y ε,n,N,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = −r ε,n,N , . . . , r ε,n,N , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ y ⎪ ⎪ ϕ ε,n+r(y ε,n,n+r,l) = max g ε (n + r, e ε,n,n+r,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ' ⎨ r ε,n+r+1

(10.27)   ϕ ( y ) p , ⎪ ε,n + r +1 ε,n,n + r +1,l +l ε,n + r +1,l ⎪ ⎪ ⎪  ⎪ l =− r ε,n+r+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l = −r ε,n,n+r , . . . , r ε,n,n+r , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎛ ⎞ ⎪ ⎪ r ε,n+1 ⎪ ⎪

⎪ ⎪ y ⎪ ϕ ε,n (y) = max ⎝g ε (n, e ), ϕ ε,n (y ε,n,n+1,l)p ε,n+1,l⎠ . ⎪ ⎩ l  =− r ε,n+1

The above backward recurrence algorithm for finding reward functions is supported by the following convergence result. Theorem 10.1.2. Let the log-price process Y 0,n be represented by the Gaussian random walk given by the stochastic transition dynamic relation (10.1) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the binomial random walks Y ε,n given by the stochastic transition dynamic relation (10.18), with parameters given by relation (10.20). Let also the conditions B4 [𝛾] hold for some 𝛾 ≥ 0, the conditions I9 and J13 hold, and r ε → ∞ as ε → 0. Then the following relation takes place, for any y ε → y0 ∈ Yn , n = 0, 1, . . . , N: ϕ ε,n (y ε ) → ϕ0,n (y0 )

as ε → 0 .

(10.28)

Proof. It is analogous to the proof of Theorem 10.1.1. The moment-generating func­ tions for the random variable W ε,n have for n = 1, . . . , N and ε ∈ (0, ε0 ] the following

10.1 Univariate binomial tree approximation models |

421

form:

 δ  r ε,n β√ 1 − β √δr ε 1 , Ee βW ε,n = e βμ n e rε + e 2 2 while, for ε = 0, it has the following form: 1

Ee βW0,n = e βμ n + 2 β

2 2 σn

,

β ∈ R1 ,

β ∈ R1 .

(10.29)

(10.30)

It follows from formula (10.29) and the Taylor asymptotic expansion for the ex­ ponential function that the following asymptotic relation takes place, for every n = 1, . . . , N and β ∈ R1 :  δ  r ε,n δ β √rε 1 − β √r ε 1 βW ε,n βμ n =e e +e Ee 2 2 & & ' & ''[r ε σ2n2 ]+1 2 2 δ 1 δ βδ βδ β + o2 − √ = e βμ n 1 + + o2 √ 2 rε rε rε 1

→ e βμ n + 2 β

2 2 σn

= Ee βW0,n

as ε → 0 .

(10.31)

The continuation of the proof repeats the corresponding part of the proof for The­ orem 10.1.1.

10.1.3 Convergence of optimal expected rewards for a univariate binomial tree model (0)

The optimal expected reward Φ0 = Φ0 (Mmax,N ) for the log-price process Y0,n , defined by the stochastic transition dynamic relation (10.1), is given by the following formula:

(10.32) Φ0 = Eϕ0,0 (Y0,0) = ϕ0,0 (y)P0 (dy) . R1

It is natural to choose, for every ε ∈ (0, ε0 ], the initial distribution P ε (A) for the approximating log-price process Y ε,n as the corresponding skeleton approximation for the distribution P0 (A)

P ε (A) = P0 (Iε,l ) , A ∈ B1 , (10.33) y ε,0,0,l ∈ A

where y ε,0,0,l = l √ and

Iε,l

δ + λ ε,0 , r ε,0

l = −r ε,0 , . . . , r ε,0 ,

! ⎧# ⎪ −∞, √ rδε,0 (−r ε,0 + 21 ) + λ ε,0 ⎪ ⎪ ⎨# ! δ 1 δ 1 √ √ = r ε,0 ( l − 2 ) , r ε,0 ( l + 2 ) + λ ε,0 ⎪ ⎪ ⎪ ⎩ √ δ (r − 1 ), ∞) + λ r ε,0

ε,0

2

ε,0

(10.34)

if l = −r ε,0 , if − r ε,0 < l < r ε,0 , if l = r ε,0 ,

(10.35)

422 | 10 Convergence of tree-type reward approximations where real numbers λ ε,0 are positive integer numbers r ε,0 chosen in such a way that the following condition holds: N5 : (a) r ε,0 → ∞ as ε → 0, √ (b) ±δ r ε,0 + λ ε,0 → ±∞ as ε → 0. (ε)

Correspondingly, the optimal expected reward Φ ε = Φ ε (Mmax,N ) for the log-price process Y ε,n , defined by the transition dynamic relation (10.2), is given, for every ε ∈ (0, ε0 ], by the following formula: Φε =

r ε,0

ϕ ε,0(y ε,0,0,l)P0 (Iε,l) .

(10.36)

l =− r ε,0

We are going to apply Theorem 5.3.2, in order to prove convergence of the optimal expected rewards for log-price processes Y ε,n , to the corresponding reward functions for log-price processes Y0,n as ε → 0. Let us assume that the following condition holds for some β ≥ 0: D20 [β ]: Ee β| Y0,0 | < K91 , for some 1 < K91 < ∞, and also that the following condition holds: K18 : P0 (Y0 ) = 0, where Y0 is the set defined in the condition I9 . Theorem 10.1.3. Let the log-price process Y 0,n be represented by the Gaussian random walk given by the stochastic transition dynamic relation (10.1) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the binomial random walks Y ε,n given by the stochastic transition dynamic relation (10.2), with parameters given by relation (10.4). Let also the conditions B4 [𝛾] and D20 [β ] hold for parameters 𝛾 and β such that, either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions N5 , I9 , J13 and K18 hold, and r ε → ∞ as ε → 0. Then the following relation takes place: Φ ε → Φ0

as ε → 0 .

(10.37)

Proof. We shall use Theorem 5.3.2. Let us define skeleton functions h ε (y) =

r ε,0

l =− r ε,0

&

' δ l √ + λ ε,0 I (y ∈ Iε,l ) rε

(10.38)

Obviously, for every ε ∈ (0, ε0 ], P ε (A) = P{h ε (Y0,n ) ∈ A} ,

A ∈ B1 .

(10.39)

10.1 Univariate binomial tree approximation models | 423

The skeleton functions h ε (y) satisfy the following inequalities for y ∈ R1 : # !   |h ε (y)| ≤ |y|I y ∉ −δ r ε,0 + λ ε,0 , δ r ε,0 + λ ε,0 ' & #    δ + |y| + √ I y ∈ [−δ r ε,0 + λ ε,0 , δ r ε,0 + λ ε,0 ] r ε,0 δ ≤ |y| + √ . (10.40) r ε,0 This inequality and the condition D20 [β ] imply the following relation: Ee β| Y ε,0 | = Ee β| h ε (Y0,0 )| ≤ e

√

βδ r ε,0

Ee β| Y0,0 | → Ee β| Y0,0 | < ∞

as ε → 0 .

(10.41)

Relation (10.41) implies that the condition D7 [β¯ ] (its one-dimensional variant) holds. The skeleton functions h ε (y) also satisfy the following inequalities for y ∈ R1 :   |h ε (y) − y| ≤ (−δ r ε,0 + λ ε,0 − y)I (y ≤ −δ r ε,0 + λ ε,0 )   δ I (δ r ε,0 + λ ε,0 < y ≤ δ r ε,0 + λ ε,0 ) r ε,0   + (y − δ r ε,0 − λ ε,0)I (y > δ r ε,0 + λ ε,0 ) . +√

(10.42)

This inequalities and the condition N10 obviously imply that h ε (y) → y as ε → 0, a.s. for every y ∈ R1 , and, therefore, h ε (Y0,0) −→ Y0,0 as ε → 0. Therefore, P ε (·) ⇒ P0 (·)

as ε → 0 .

(10.43)

Relation (10.43) implies that the condition K4 (a) (its one-dimensional variant) holds. The condition K18 is a variant of the condition K4 (b). Other conditions of Theorem 5.3.2 also hold that was shown in the proof of Theo­ rem 10.1.1. By applying Theorem 5.3.2 to the log-price processes Y ε,n we get the asymp­ totic relation (10.37). The following theorem give analogous result for the binomial tree approximation model with asymptotically fitted parameters. Theorem 10.1.4. Let the log-price process Y 0,n be represented by the Gaussian random walk given by the stochastic transition dynamic relation (10.1) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the binomial random walks Y ε,n given by the stochastic transition dynamic relation (10.18), with parameters given by relation (10.20). Let also the conditions B4 [𝛾] and D20 [β ] hold for parameters 𝛾 and β such that, either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions N5 , I9 , J13 and K18 hold, and r ε → ∞ as ε → 0. Then the following relation takes place: Φ ε → Φ0

as ε → 0 .

The proof of this theorem is analogous to the proof of Theorem 10.1.3.

(10.44)

424 | 10 Convergence of tree-type reward approximations Remark 10.1.1. If the model assumption (e) σ 2n > 0, n = 1, 2, . . . may not hold, the condition J13 in Theorems 10.1.1–10.1.4 can be replaced by the assumption that P{y0 + W0,n ∈ Y n } = 0, y0 ∈ Yn−1 , n = 1, . . . , N.

10.2 Multivariate homogeneous in space tree models In this section, we present results about convergence of rewards for American-type options for multivariate binomial and trinomial tree approximations for log-price pro­ cesses represented by multivariate Gaussian random walks. We give the correspond­ ing results for homogeneous in time binomial tree approximations and inhomoge­ neous in time trinomial tree approximations.

10.2.1 Convergence of option rewards for multivariate homogeneous in time binomial tree models Let us consider the model of multivariate homogeneous in time Gaussian random walk, which can be considered as a discrete time analog of the multivariate Wienner 0,n is given by the following process (Brownian motion). In this case, the log-price Y stochastic transition dynamic relation: 0,n−1 + W  0,n , 0,n = Y Y

n = 1, 2, . . . .

(10.45)

0,0 = (Y 0,0,1 , . . . , Y 0,0,k ) is a random vector taking values in the space where (a) Y  0,n = (W0,n,1 , . . . , W0,n,k), n = 1, 2, . . . is a Rk , and with distribution P0 (A), (b) W sequence of i.i.d. k-dimensional normal random vectors with means EW0,n,i = μ i , i = 1, . . . , k, variances VarW0,n,i = σ 2i , i = 1, . . . , k, and covariances E(W0,n,i − 0,0 and the μ i )(W0,n,i − μ i ) = ρ i,j σ i σ j , i, j = 1, . . . , k, and (c) the random vector Y  0,n , n = 1, 2, . . . are independent. sequence of random vectors W Also, usual assumptions are (d) μ i ∈ R1 , i = 1, . . . , (e) σ i > 0, i = 1, . . . , k; (f) |ρ i,j | ≤ 1, ρ i,i = 1, i, j = 1, . . . , k, (g) Σ = σ i,j , where σ i,j = ρ i,j σ i σ j , i, j = 1, . . . , k, is a nonnegatively defined k × k matrix, and (h) det(Σ) ≠ 0. 0,n by the multivariate homogeneous We shall approximate the log-price process Y  ε,n . In this case, the log-price Y ε,n is given, for every in time binomial random walks Y ε ∈ (0, ε0 ], by the following stochastic transition dynamic relation:  ε,n−1 + W  ε,n , ε,n = Y Y

n = 1, 2, . . . .

(10.46)

ε,0 = (Y ε,0,1, . . . , Y ε,0,k) is a random vector taking values in the space Rk , where (a) Y  ε,n,1 + · · · + W  ε,n,r ε ,  ε,n = (W ε,n,1, . . . , W ε,n,k) = W and with distribution P ε (A), (b) W

10.2 Multivariate homogeneous in space tree models | 425

 ε,n,l = (W ε,n,l,1, . . . , W ε,n,l,k), l, n = 1, 2, . . . is a family of i.i.d. n = 1, 2, . . . , where W binary random vectors taking values δ¯ ε,¯ı = (δ ε,1,ı1 , . . . , δ ε,k,ı k ) with the corresponding probabilities p ε,¯ı , for ¯ı ∈ Ik = {¯ı = (ı1 , . . . ı k ) : ı1 , . . . ı k = ±}, and (c) the random ε,0 and the family of random vectors W  ε,n,l , l, n = 1, 2, . . . are independent. vector Y Also, usual assumptions are (d) −∞ < δ ε,i, − < δ ε,i, + < ∞ , i = 1, . . . , k,  (e) p ε,¯ı ≥ 0, ¯ı,∈Ik p ε,¯ı = 1, (f) r ε is a positive integer number. We assume that a pay-off function g(n, ey ) does not depend on the parameter ε and also g(n, ey ) is a real-valued Borel function of the argument  y ∈ Rk , for every n = 0, 1, . . . , N. y) for the log-price process In this case, the corresponding reward function ϕ ε,n (  Y ε,n is also a Borel function of argument  y ∈ Rk , for every n = 0, 1, . . . , N.  ε,n in such a way that the We fit parameters of approximating log-price processes Y following exact moment-fitting relations hold, for every ε ∈ (0, ε0 ]: ⎧ ⎪ EW ε,1,i = μ i , VarW ε,1,i = σ 2i , i = 1, . . . , k , ⎪ ⎪ ⎨ E(W ε,1,i − EW ε,1,i)(W ε,1,i − EW ε,1,j) = ρ i,j σ i σ j , (10.47) ⎪ ⎪ ⎪ ⎩1 ≤ i < j ≤ k .

Lemma 9.2.1 gives two variants for solution of this fitting problem. According to the first variant presented in Lemma 9.2.1, the fitting relations (10.47) hold, if parameters for the log-price process Y ε,n are given, for every ε ∈ (0, ε0 ], by the following formulas: ⎧ σi ⎪ ⎪ δ ε,i, ± = ±δ ε,i + λ ε,i , δ ε,i = √ , i = 1, . . . , k , ⎪ ⎪ rε ⎪ ⎪ ⎨ μi 1 (10.48) , p ε,i, ± = , i = 1, . . . , k , λ ε,i = ⎪ ⎪ rε 2 ⎪ ⎪ ⎪ ⎪ 1 ⎩p 1≤ i < j ≤ k. ε,i,j,+, ± = p ε,i,j, −,∓ = 4 (1 ± ρ i,j ) , where (i) one can choose for the parameter r ε any value r ε = 1, 2, . . . , and (ii) the cor­  ε,1,1 consis­ responding higher order distributions for components of random vectors W tent with the above one- and two-dimensional distributions should be defined using algorithm described in Lemma 9.2.2. According to the second variant also presented in Lemma 9.2.1, the fitting relations (10.47) hold, if parameters of the log-price process Y ε,n are given, for every ε ∈ (0, ε0 ], by the following formulas:  ⎧ ⎪ ı δ2i − σ 2i ⎪ i δ μi ⎪ i ⎪ ⎪ δ = ± δ + λ , δ = , λ = + , √ √ ε,i, ± ε,i ε,i ε,i ε,i ⎪ ⎨ rε rε rε  ⎛ ⎞ (10.49) ⎪ ⎪ ı i δ2i − σ 2i ⎪ 1⎝ ⎪ ⎠ ⎪ , i = 1, . . . , k, ⎪ ⎩ p ε,i, ± = 2 1 ∓ δi

426 | 10 Convergence of tree-type reward approximations and

  ⎧ ⎛ ⎞ 2 2 ⎪ ı j δ2j − σ 2j ⎪ σi σj 1 1 1 ⎝ ıi δi − σi ⎪ ⎪ ⎠ ⎪ p ε,i,j,+,+ = + ρ i,j + − − ⎪ ⎪ ⎪ 4 4 δi δj 4 δi δj ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ ⎪ ı i δ2i − σ 2i ı j δ2j − σ 2j ⎪ 1 ⎪ ⎪ + , 1≤ i 0; (ii) r ε → ∞ as ε → 0. The random variables W ε (y), y ∈ R1 are binomial random variables. Namely, the random variable W ε (y) takes values, δ w ε,l (y) = l √ , rε

l = −r ε ( y) , . . . , r ε ( y) ,

with the corresponding probabilities, ⎧ r ε (y ) +l r ε (y ) +l r ε (y ) +l ⎪ 2 ⎪ ⎪C r ε (y) (p ε, +(y)) 2 (p ε, −(y)) 2 ⎨ p ε,l (y) = ⎪0 ⎪ ⎪ ⎩ l = −r ε ( y) , . . . , r ε ( y) .

if if

r ε (y) +l 2 r ε (y) −l 2

(10.107)

is integer , is not integer , (10.108)

10.3 Univariate inhomogeneous in space tree models |

443

Note that we slightly modified above the standard definition of binomial distribu­ tion by including in the range of possible values all integer numbers −r ε (y) ≤ l ≤ r ε (y) including the values with the corresponding probabilities equal to zero. Let us assume that Y ε,n = y = const ∈ R1 , for some 0 ≤ n ≤ N. In this case, trajectories of the log-price process Y ε,n+r, r = 0, 1, . . . generate, for every ε ∈ (0, ε0 ], the unique tree of nodes, + T ε,(y,n) = {(y ε,n,n+r,l, n + r), r− ε,n,n + r( y ) ≤ l ≤ r ε,n,n + r( y ) , r = 0, 1, . . . } ,

(10.109)

where δ y ε,n,n+r,l = y + l √ , r− ( y ) ≤ l ≤ r+ ε,n,n + r( y ) , r = 0, 1, . . . , r ε ε,n,n+r

(10.110)

and ⎧ ⎪ ⎪ ⎪0 ⎨ ± r ε,n,n+r(y) = ⎪±r ε (y) ⎪ ⎪ ⎩± max −

r ε,n,n+r−1 ( y )

if r = 0 , #

≤l≤r+ ε,n,n+r−1 ( y )

#

± l ± rε y +

δ √ l rε



if r = 1 , (10.111) if r > 1 .

The following lemma is a direct corollary of Lemma 3.4.1. Lemma 10.3.1. Let, for every ε ∈ (0, ε0 ], the approximating log-price process Y ε,n is a homogeneous binomial Markov chain with transition probabilities given by rela­ tion (10.104) and parameters defined in relation (10.106). Then the reward functions + ϕ ε,n+r(y ε,n,n+r,l), for points y ε,n,n,0 = y and y ε,n,n+r,l, l = r− ε,n,n + r( y ) , . . . , r ε,n,n + r( y ), r = 1, . . . N − n, are, for every ε ∈ (0, ε0 ], and y ∈ R1 , n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ ⎪ ϕ ε,N (y ε,n,N,l) = g ε (N, e y ε,n,N,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ l = r− ⎪ ε,n,N ( y ) , . . . , r ε,n,N ( y ) , ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r(y ε,n,n+r,l) = max g ε (n + r, e y ε,n,n+r,l ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ' r ε (y)

(10.112)   ϕ ε,n+r+1(y ε,n,n+r+1,l+l )p ε,l (y ε,n,n+r,l) , ⎪ ⎪ ⎪ ⎪ ⎪ l  =− r ε ( y ) ⎪ ⎪ ⎪ ⎪ − ⎪ ⎪ r = N − n − 1, . . . , 1 , l = r ε,n,n+r(y), . . . , r+ ⎪ ε,n,n + r( y ) , ⎪ ⎪ ⎪ ⎪ ' & ⎪ r ε (y) ⎪

⎪ ⎪ y ⎪ ⎪ ϕ ε,n (y) = max g ε (n, e ), ϕ ε,n (y ε,n,n+1,l)p ε,l (y) . ⎪ ⎩  l =− r ε ( y )

Let us assume that the functions μ (y) and σ 2 (y) are bounded, i.e. the condition G12 holds. This condition implies that the corresponding tree T ε,(y,n) generated by the trajec­  ε,n =  tories of the log-price process Y ε,n+r, r = 0, 1, . . . with the value Y y has the total

444 | 10 Convergence of tree-type reward approximations number of nodes after r steps, L ε,n,n+r ≤ 1 + 2K86 (δ)r ε + 1 + · · · + 2K86 (δ)r ε r + 1 = K86 (δ)r ε r2 + (K86 (δ)r ε + 1)r + 1 .

(10.113)

The explicit formula for K86 (δ) was given in Subsection 9.4.1. The expression on the right-hand side in (10.113) is a polynomial (quadratic) function of r. We are going to apply Theorem 5.2.1, in order to prove convergence of the re­ ward functions for log-price processes Y ε,n to the corresponding reward functions for log-price processes Y0,n as ε → 0. We assume that the condition B4 [𝛾] holds for the pay-off function g(n, e y ), for some 𝛾 ≥ 0. The condition I1 reduces, in this case, to the form of the condition I9 . Let us also assume that the following condition holds: J15 : (a) There exists a set Y ∈ B1 such that functions μ (y) and σ 2 (y) are continuous in points y ∈ Y; (b) L1 (Y) = 0 and L1 (Yn ) = 0, n = 1, . . . , N, where Yn , n = 1, . . . , N are sets penetrating the condition I9 . We shall see that, in this case, the condition J1 (a) (its one-dimensional variant for the model without index component) holds for log-prices Y ε,n with parameters given by relation (10.106), if r ε → ∞ as ε → 0. Also the random variables W0 (y), y ∈ R1 have normal distributions that make it possible to replace the condition J1 (b) (its onedimensional variant for the model without index component) by the condition J15 . The above backward recurrence algorithm for finding reward functions can be now supported by the following convergence result. Theorem 10.3.1. Let the log-price process Y 0,n be represented by the inhomogeneous in space Gaussian random walk with transition probabilities given by relation (10.103) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the inhomogeneous binomial random walk Y ε,n with transition probabilities given by rela­ tion (10.104) and parameters satisfying the condition G12 and given by relation (10.106). Let also conditions B4 [𝛾] holds for some 𝛾 ≥ 0, conditions I9 and J15 hold, and r ε → ∞ as ε → 0. Then the following relations take place for any y ε → y0 ∈ Yn ∩ Y, n = 0, 1, . . . , N: (10.114) ϕ ε,n (y ε ) → ϕ0,n (y0 ) as ε → 0 . Proof. We are going to use Theorem 5.2.1. The moment-generating function for the random variables W ε (y) has, for every ε ∈ (0, ε0 ] and y ∈ R1 , the following form, for β ∈ R1 : &

Ee

βW ε ( y )

=

δ

e

β √r ε

& & √ ' √ '' r ε (y) δ μ(y) r ε μ(y) r ε 1 − β √rε 1 1+ ∗ 1− ∗ , +e 2 r ε (y) δ 2 r ε (y) δ

(10.115)

10.3 Univariate inhomogeneous in space tree models |

445

while, for ε = 0, these moment-generating functions have the following form: 1

Ee βW0 (y) = e βμ(y) + 2 β

σ (y)

2 2

.

(10.116)

Using the condition G12 , we get the following inequalities, for every ε ∈ (0, ε0 ], y ∈ R1 , and any β ∈ R1 : & & √ ' √ ' μ(y) r ε μ(y) r ε β √δr ε 1 − β √δr ε 1 1+ ∗ 1− ∗ 0≤e +e 2 r ε (y) δ 2 r ε (y) δ ' & & ' √ βδ β2 δ2 1 μ(y) r ε + ··· 1+ ∗ = 1+ √ + rε 2!r ε 2 r ε (y) δ & ' & √ ' 2 2 βδ β δ 1 μ(y) r ε + 1− √ + + ··· 1− ∗ rε 2!r ε 2 r ε (y) δ & ' & ' √ 2 2 4 4 β δ β δ βδ β3 δ3 μ(y) r ε = 1+ + + · · · + + + · · · √ √ 2!r ε 4!r2ε rε 3!( r ε )2 r∗ ε (y) δ & ' & ' 2 2 2 2 2 2 β δ β δ β δ βμ (y) = 1 + 1 + 4! + · · · + 1+ + ··· 2r ε rε r∗ rε ε (y) &2 ' 2 2 β2 δ 2 |β ||μ (y) | β δ ≤ 1 + e rε + . (10.117) 2r ε r∗ ε (y) The condition G12 also implies the following inequality, for every ε ∈ (0, ε0 ], and y ∈ R1 : .    4δ 2 μ 2 ( y ) 2 2 4 r ε σ (y) + σ (y) + r ε 1 + 1 + 4δ rε ∗ ≤ r K , (10.118) r ε (y) = ε 85 2δ2 2δ2 where K85 is the constant penetrating condition G12 . Using formula (10.115), inequalities (10.117) and (10.118), and taking into account | μ (y) | δ ≤ √r ε , we get the following inequality, for inequalities r ε (y) ≤ r∗ ε ( y ) + 1 and r ∗ ε (y) every ε ∈ (0, ε0 ], y ∈ R1 , and β ∈ R1 : &

&

''r∗ε (y) +1 |β ||μ (y) | β2 δ2 + 2r ε r∗ ε (y) & '' 2 2 2 2 β δ |β |δ β δ r ε ≤ 1+e + √ 2r ε rε ( & ') 2 2 ∗ 2 2 β δ β δ r ε (y) r × exp e ε + |β ||μ (y) | 2r ε & '' & β2 δ 2 |β |δ β2 δ2 + √ ≤ 1 + e rε 2r ε rε ⎧  ⎛ ⎞⎫ 4δ 2 ⎪ ⎪ ⎨ β2 δ 2 1 ⎬ 1 + 1 + rε 2 ⎝ ⎠ r ε K + |β |K85 . × exp e β 85 ⎪ ⎪ 2 2 ⎩ ⎭

Ee βW ε (y) ≤ 1 + e &

β2 δ 2 rε

(10.119)

446 | 10 Convergence of tree-type reward approximations Also, the condition G12 implies that the following inequality holds, for every y ∈ R1 , and any β ∈ R1 : 1

Ee βW0 (y) = e βμ(y) + 2 β

σ (y)

2 2

1

≤ exp{(|β | + 2 β 2 )K85 } .

(10.120)

The modulus Δ± β ( Y ε, · , N ) takes for the above log-price processes Y ε,n the following form, for every ε ∈ [0, ε0 ] and β ≥ 0: ± βW ε ( y ) . Δ± β ( Y ε, · , N ) = sup e

(10.121)

y ∈R1

Relations (10.119) and (10.120) imply the following asymptotic relation, which holds for any β ≥ 0: && & '' β2 δ 2 |β |δ β2 δ2 ± r ε lim Δ β (Y ε,· , N ) ≤ lim 1+e + √ ε →0 ε →0 2r ε rε ⎧  ⎛ ⎞⎫⎞ 4δ 2 ⎪ ⎪ ⎨ β2 δ 2 1 ⎬ 1 + 1 + rε ⎟ + βK85 ⎠ ⎠ × exp e rε ⎝ β 2 K85 ⎪ ⎪ 2 2 ⎩ ⎭ #   = exp β + 21 β 2 K85 < ∞ . (10.122) Relations (10.120) and (10.122) imply that the condition E10 [β¯ ] (its univariate vari­ ant for the model without index component) holds. Therefore, by Lemma 5.1.5, the condition C6 [β¯ ] (its one-dimensional variant for the model without index component) 2 holds for any β ≥ 0, for example with the constant K93 = 1 + 2e(| β|+β )K 86 replacing the constant K30,1 penetrating this condition. The condition B4 [𝛾] is a one-dimensional variant of the condition B5 [𝛾¯] (its uni­ variate variant for the model without index component and a pay-off function, which does not depend on parameter ε). Note that one can always take the parameter β > 𝛾 if 𝛾 > 0 or β = 𝛾 if 𝛾 = 0. The condition I1 (its univariate l variant for the model without index component and a pay-off function, which does not depend on the parameter ε) reduces in this case to the form of the condition I9 . Let us take an arbitrary n = 0, 1, . . . , N and y ε → y0 ∈ Yn ∩ Y as ε → 0. By the initial model assumptions, σ 2 (y0 ) > 0 and, by the conditions I9 and J15 , μ (y ε ) → μ (y0 ) and σ 2 (y ε ) → σ 2 (y0 ) as ε → 0, and, therefore, r ε ( y ε ) = [ r∗ ε ( y ε )] + 1 .  r ε σ2 (y ε ) + σ4 (y ε ) + ∼ r∗ ε (y ε ) = 2δ2 σ2 (y0 ) ∼ rε as ε → 0 . δ2

4δ 2 μ 2 ( y ε ) rε



(10.123)

10.3 Univariate inhomogeneous in space tree models

| 447

and, for every β ∈ R1 , && Ee

βW ε ( y ε )

& '' & √ ' βδ β2 δ2 βδ 1 μ(y ε ) r ε = 1+ √ + + o2 √ 1+ ∗ rε 2r ε rε 2 r ε (y ε )δ & '' & & √ '' r ε (y ε ) βδ β2 δ2 βδ 1 μ(y ε ) r ε + 1− √ + + o2 − √ 1− ∗ rε 2r ε rε 2 r ε (y ε )δ & & ' & √ ' 2 2 βμ (y ε ) 1 β δ βδ 1 μ(y ε ) r ε (10.124) + = 1+ ∗ + o2 √ 1+ ∗ r ε (y ε ) 2 rε rε 2 r ε (y ε )δ & ' & √ ''r ε (y ε ) μ(y ε ) r ε βδ 1 1− ∗ +o2 − √ rε 2 r ε (y ε )δ   1 2 2 → exp βμ (y0 ) + 2 β σ (y0 ) = Ee βW0 (y0 ) as ε → 0 ,

since by relation (10.123),  & ' & & ' & √ ' √ '  βδ 1 μ(y ε ) r ε βδ 1 μ(y ε ) r ε     r ε (y ε ) o2 √ + o2 − √ 1 + ∗ 1 − ∗  rε 2 r ε (y ε )δ rε 2 r ε (y ε )δ   &   &  ' '     βδ βδ     ≤ o2 √ r r (10.125) ε ( y ε ) +  o 2 − √ ε ( y ε ) → 0 as ε → 0 .     rε rε

Relation (10.124) implies, by the corresponding variant of the continuous theorem, the following relation, for an arbitrary n = 1, . . . , N and y ε → y0 ∈ Yn−1 ∩ Y as ε → 0: d

y ε + W ε (y ε ) −→ y0 + W0 (y0 )

as ε → 0 .

(10.126)

Relation (10.126) is equivalent to the condition J1 (a) (its univariate variant for the model without index component). The condition J15 (b) implies that the condition J1 (b) (its univariate variant for the model without index component) holds, since random variables W0 (y), y ∈ R1 have normal distributions with positive variances, and, thus, possess positive probability density functions with respect to the Lebesgue measure. Thus, all conditions of Theorem 5.2.1 hold. By applying this theorem to the above approximation model, we get the asymptotic relation (10.114).

10.3.2 Convergence of reward functions for univariate inhomogeneous in time and space trinomial tree approximations Let us consider the model, where a log-price process Y0,n is a univariate inhomoge­ neous in time and space Gaussian random walk, with the phase space R1 , an initial distribution P0 (A), and transition probabilities given for y ∈ R1 , A ∈ B1 , n = 1, 2, . . .

448 | 10 Convergence of tree-type reward approximations by the following relation: P0,n (y, A) = P{Y0,n ∈ A/Y0,n−1 = y} = P{y + W0,n (y) ∈ A} ,

(10.127)

where (a) W0,n (y) is, for every y ∈ R1 , n = 1, 2, . . . , a normal random variable with a mean value EW0,n (y) = μ n (y) and a variance VarW0,n (y) = σ 2n (y). Also, usual assumptions are (b) μ n (y) is, for every n = 1, 2, . . . , a measurable function acting from R1 to R1 , (c) σ 2n (y) is, for every n = 1, 2, . . . , a measurable func­ tion acting from R1 to (0, ∞). We shall approximate the log-price process Y0,n by a log-price process Y ε,n which is, for every ε ∈ (0, ε0 ], a univariate inhomogeneous in time and space trinomial random walk, with the phase space R1 , an initial distribution P ε (A), and transition probabilities given for y ∈ R1 , A ∈ B1 , n = 1, 2, . . . by the following relation: P ε,n (y, A) = P{Y ε,n ∈ A/Y ε,n−1 = y} = P{y + W ε,n (y) ∈ A} ,

(10.128)

where (a) W ε,n (y) is, for every y ∈ R1 , n = 1, 2, . . . , a trinomial random variable, which can be represented in the form W ε,n ( y) = W ε,n,1( y ) + · · · + W ε,n,r ε ( y ), where  ε,n,l( W y), l = 1, 2, . . . are the i.i.d. random variables taking values δ ε,ȷ with the prob­ y ), for ȷ = +, ◦, −. abilities p ε,n,ȷ( Also, usual assumptions are (b) δ ε,− = −δ ε < δ ε,◦ = 0 < δ ε,+ = +δ ε , (c) p ε,n,ȷ(y), ȷ = +, ◦, − , n = 1, 2, . . . are the measurable functions acting from R1 to [0, 1] such  that ȷ=+,◦,− p ε,n,ȷ(y) = 1, y ∈ R1 , n = 1, 2, . . . ; (d) r ε is a positive integer number. We use the log-prices Y ε,n for approximation of the log-price Y0,n asymptotically, as ε → 0. In this case, a usual approach is to fit the corresponding parameters, i.e. to try to choose parameter δ ε , probabilities p ε,n,ȷ(y), and parameter r ε in such a way that the following exact fitting relations hold, for every ε ∈ (0, ε0 ]: ⎧ ⎨ EW ε,n (y) = EW0,n (y) , (10.129) ⎩ VarW ε,n (y) = VarW0,n (y) , y ∈ R1 , n = 1, . . . , N . The solution of this fitting problem is presented in Lemma 9.5.1. According to this lemma, the fitting relations (10.129) hold, if to assume that the conditions G14 , G15 and to choose parameters of the log-price process Y ε,n given by the following formulas, for every ε ∈ (0, ε0 ] and n = 1, . . . , N: ⎧ δ ⎪ ⎪ δε = √ , ⎪ ⎪ ⎪ rε ⎪ ⎪ ⎪ & ' ⎪ ⎨ 1 σ 2n (y) μ 2n (y) μ n (y) + 2 ± √ , p ε,n,±(y) = (10.130) ⎪ 2 δ2 δ rε δ rε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ 2 (y) μ 2n (y) ⎪ ⎪ ⎩ p ε,n, ◦(y) = 1 − n − 2 , y ∈ R1 , n = 1, . . . , N , δ2 δ rε where (i) one can choose for parameters δ and r ε any values such that (a) δ ≥  K 2 δ2 2 2 K89 + K89 , (b) r ε ≥ 89 σ 4 , where σ > 0 and K 89 < ∞ are parameters penetrat­ ing, respectively the conditions G14 and G15 .

10.3 Univariate inhomogeneous in space tree models |

449

The random variable W ε,n (y) is for every y ∈ R1 , n = 1, . . . , N a trinomial ran­ dom variable. Namely, the random variable W ε,n (y) takes values w ε,r ε ,l with the cor­ responding probabilities p ε,n,r ε ,l (y), for −r ε ≤ l ≤ r ε . Here, for every y ∈ R1 , n = 1, . . . , N, r = 1, 2, . . . , δ w ε,r,l = lδ ε = l √ , −r ≤ l ≤ r , rε

(10.131)

and, for −r ≤ l ≤ r, p ε,n,r,l(y) = P{W ε,n (y) = W ε,n,1(y) + · · · + W ε,n,r(y) = w ε,r,l}I (|l| ≤ r) . (10.132) The probabilities p ε,n,r,l(y) can be found using the following recurrent convolu­ tion formulas: ⎧ ⎨p ε,n,ȷ(y) I (l = e ȷ ) if r = 1, −1 ≤ l ≤ 1 , p ε,n,r,l(y) = ⎩1 (10.133)   p ε,n,r−1,l−l (y)p ε,n,1,l (y) if r > 1, −r ≤ l ≤ r .  l =−1

where e ȷ = 1, 0, −1 if ȷ = +, ◦, −. The additional comments concerning algorithms for computing of the probabili­ ties p ε,n,r,l(y) can be found in Subsection 3.2.3. Let us assume that Y ε,n = y = const ∈ R1 . In this case, trajectories of the log-price process Y ε,n+r, r = 0, 1, . . . generate, for every ε ∈ (0, ε0 ], the unique tree of nodes T ε,(y,n) = {(y ε,n,n+r,l, n + r), −r ε r ≤ l ≤ r ε r, r = 0, 1, . . . } ,

(10.134)

where δ y ε,n,n+r,l = y + lδ ε = y + l √ , rε

−r ε r ≤ l ≤ r ε r .

(10.135)

Note that actual values of points y ε,n,n+r,l do not depend on the parameter n = 0, 1, . . . . The tree T ε,(y,n) contains the total number of nodes after the r steps, L ε,n,n+r ≤ 1 + (2r ε + 1) + · · · + (2r ε r + 1) = r ε r2 + (r ε + 1)r + 1 .

(10.136)

The expression on the right-hand side in (10.136) is a polynomial (quadratic) func­ tion of r. The following lemma is a direct corollary of Lemma 3.4.1. Lemma 10.3.2. Let, for every ε ∈ (0, ε0 ], the approximating log-price process Y ε,n is the multivariate trinomial Markov chain given by the stochastic transition dynamic re­ lation (10.128), with parameters defined in relation (10.130), under the assumption that the conditions G14 and G15 hold. Then the reward functions ϕ ε,n+r(y ε,n+r,l), for points y ε,n,n,0 = y and y ε,n,n+r,l, −r ε r ≤ l ≤ r ε r, r = 1, . . . N − n, are, for every ε ∈ (0, ε0 ],

450 | 10 Convergence of tree-type reward approximations and y ∈ R1 , n = 0, . . . , N, the unique solution for the following recurrence finite system of linear equations: ⎧ ⎪ ϕ ε,N (y ε,n,N,l) = g ε (N, e y ε,n,N,l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − r ε ( N − n) ≤ l ≤ r ε ( N − n) , ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕ ε,n+r(y ε,n,n+r,l) = max g ε (n + r, e y ε,n,n+r,l ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ' rε

(10.137) ⎪ ϕ ε,n+r+1(y ε,n,n+r+1,l+l)p ε,n+r+1,rε,l (y ε,n,n+r,l) , ⎪ ⎪ ⎪  ⎪ l =− r ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − r ε r ≤ l ≤ r ε r , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ & ' ⎪ rε ⎪

⎪ ⎪  y ⎪   ⎪ ϕ ε,n (y) = max g ε (n, e ), ϕ ε,n (y ε,n,n+1,l )p ε,n+1,r ε,l (y) . ⎪ ⎩  l =− r ε

We are going to apply Theorem 5.2.1, in order to prove convergence of the reward func­ tions for log-price processes Y ε,n to the corresponding reward functions for log-price processes Y0,n as ε → 0. First, we assume that the condition B4 [𝛾] holds for the pay-off function g(n, e y ), for some 𝛾 ≥ 0. The condition I1 reduces in this case to the form of the condition I9 . Let us also assume that the following condition holds: J16 : (a) There exist sets Yn ∈ B1 , n = 0, 1, . . . , N such that functions μ n (y) and σ 2n (y) are continuous in points y ∈ Yn−1 , for n = 1, . . . , N;  (b) L1 (Yn ) = 0, L1 (Yn ) = 0, n = 1, . . . , N, where Yn , n = 1, . . . , N are sets penetrating the condition I9 . We shall see that, in this case, the condition J1 (a) (its one-dimensional variant for the model without index component) holds for log-prices Y ε,n with parameters given by relation (10.106), if r ε → ∞ as ε → 0. Also the random variables W0 (y), y ∈ R1 have normal distributions that make it possible to replace the condition J1 (b) (its onedimensional variant for the model without index component) by the condition J15 . The above backward recurrence algorithm for finding reward functions is sup­ ported by the following convergence result. Theorem 10.3.2. Let the log-price process Y 0,n be represented by the inhomogeneous in time and space Gaussian random walk, with transition probabilities given by relation (10.127) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the inhomogeneous trinomial random walk Y ε,n , with transition probabilities given by relation (10.128) and parameters satisfying the conditions G14 , G15 and given by relation (10.130). Let also the conditions B4 [𝛾] holds for some 𝛾 ≥ 0, the conditions I9 and J16 hold, and r ε → ∞ as ε → 0. Then the following relations take place for any y ε → y0 ∈ Yn ∩ Yn , n = 0, 1, . . . , N: ϕ ε,n (y ε ) → ϕ0,n (y0 )

as ε → 0 .

(10.138)

10.3 Univariate inhomogeneous in space tree models

| 451

Proof. We are going to use Theorem 5.2.1. Let the conditions G14 and G15 hold and parameters of the process Y ε,n are given by relation (10.130). In this case, the moment-generating function for the random variables W ε,n (y) has, for every ε ∈ (0, ε0 ] and y ∈ R1 , n = 1, . . . , N, the following form:  δ r ε β√ − β √δr ε p ε,n,−(y) + p ε,n,◦(y) Ee βW ε,n (y) = e rε p ε,n,+(y) + e & ' & δ σ 2n (y) μ 2n (y) μ n (y) β √rε 1 = e + 2 + √ 2 δ2 δ rε δ rε & ' 2 2 δ 1 σ n (y) μ n (y) μ n (y) − β √rε +e + − √ 2 δ2 δ2 r ε δ rε '' r ε & 2 2 σ (y) μ (y) + 1 − n 2 − n2 , β ∈ R1 , (10.139) δ δ rε while, for ε = 0, these moment-generating functions have the following form: 1

Ee βW0,n (y) = e βμ n (y) + 2 β

2 2 σ n (y)

,

β ∈ R1 .

Using the condition G15 , we get the following inequalities, for every ε ∈ y ∈ R1 , n = 1, . . . , N, and any β ∈ R1 : ( ) & ' δ 1 σ 2n (y) μ 2n (y) μ n (y) 0 ≤ exp β √ + + √ rε 2 δ2 δ2 r ε δ rε ( ) & ' δ 1 σ 2n (y) μ 2n (y) μ n (y) + exp −β √ + 2 − √ rε 2 δ2 δ rε δ rε ' & σ2 (y) μ2 (y) + 1 − n 2 − n2 δ δ rε ' & ' & 2 2 βδ β δ 1 σ 2n (y) μ 2n (y) μ n (y) = 1+ √ + +··· + + √ rε 2!r ε 2 δ2 δ2 r ε δ rε ' & ' & 2 2 2 2 βδ β δ 1 σ n (y) μ n (y) μ n (y) + 1− √ + + ··· + 2 − √ rε 2!r ε 2 δ2 δ rε δ rε ' & 2 2 σ (y) μ (y) + 1 − n 2 − n2 δ δ rε ' & '& σ 2n (y) μ 2n (y) β2 δ2 β4 δ4 =1+ + +··· + 2 2!r ε 4!r2ε δ2 δ rε & ' 3 3 βδ β δ μ n (y) + √ + + ··· √ √ rε 3!( r ε )3 δ rε '& ' & ' & 2 K89 β2 δ2 β2 δ2 K89 β2 δ2 |β |δ ≤1+ 1+ + ··· + 2 1+ + ··· + √ 2r ε rε δ2 δ rε rε rε & & ' ' K2 1 β2 δ 2 1 2 = 1 + e rε β K89 + 89 + |β |K89 . rε 2 rε

(10.140) (0, ε0 ],

K89 √ δ rε

(10.141)

452 | 10 Convergence of tree-type reward approximations The relation (10.141) implies that for every ε ∈ (0, ε0 ], y ∈ R1 , n = 1, . . . , N, and any β ∈ R1 , & & & ' ''r ε K2 1 β2 δ 2 1 2 Ee βW ε,n (y) ≤ 1 + e rε β K89 + 89 + |β |K89 rε 2 rε ( & & ' ') 2 2 2 β δ K89 1 2 r ε ≤ exp e . (10.142) + |β |K89 β K89 + 2 rε Also, relation (10.140) implies that for every y ∈ R1 , n = 1, . . . , N, and any β ∈ R1 , 1

Ee βW0,n (y) ≤ e(| β|+ 2 β )K 89 . 2

(10.143)

The modulus Δ± β ( Y ε, · , N ) takes for the above log-price processes Y ε,n the following form, for every ε ∈ [0, ε0 ] and β ≥ 0: ± βW ε,n ( y ) . Δ± β ( Y ε, · , N ) = max sup e

1≤ n ≤ N y ∈R1

(10.144)

Relations (10.142) and (10.143) imply the following asymptotic relation, which holds for any β ≥ 0: ( & & ' ') 2 β2 δ 2 K 1 89 ± 2 + βK89 lim Δ β (Y ε,· , N ) ≤ lim exp e rε β K89 + ε →0 ε →0 2 rε $  % 1 = exp β + β 2 K89 < ∞ . (10.145) 2 Relations (10.142) and (10.145) imply that the condition E10 [β¯ ] (its one-dimensional variant for the model without index component) holds. Therefore, by Lemma 5.1.5, the condition C6 [β¯ ] (its one-dimensional variant for the model without index component) holds for any β ≥ 0, for example with the con­ 2 stant K94 = 1 + 2e(| β|+β )K 89 replacing the constant K30,1 penetrating this condition. The condition B4 [𝛾] is a one-dimensional variant of the condition B5 [𝛾¯] (its onedimensional variant for the model without index component and a pay-off function, which does not depend on the parameter ε). Note that one can always take parameter β > 𝛾, if 𝛾 > 0, or β = 𝛾, if 𝛾 = 0. The condition I1 (its one-dimensional variant for the model without index com­ ponent and a pay-off function, which does not depend on the parameter ε) reduces in this case to the form of condition I9 . Let us take an arbitrary n = 1, . . . , N and y ε → y0 ∈ Yn−1 ∩ Yn−1 as ε → 0. This relation and the conditions I9 and J16 imply that μ n (y ε ) → μ n (y0 )

and

σ 2n (y ε ) → σ n (y0 ) as ε → 0 .

(10.146)

10.3 Univariate inhomogeneous in space tree models |

Therefore, for every n = 1, . . . , N and β ∈ R1 & '' && βδ β2 δ2 βδ βW ε,n ( y ε ) Ee = 1+ √ + + o2 √ rε 2r ε rε & ' 2 2 1 σ n (y ε ) μ n (y ε ) μ n (y ε ) × + + √ 2 δ2 δ2 r ε δ rε & & '' 2 2 βδ β δ βδ + 1− √ + + o2 − √ rε 2r ε rε & ' 2 2 1 σ n (y ε ) μ n (y ε ) μ n (y ε ) × + − √ 2 δ2 δ2 r ε δ rε ''r ε & σ2 (y ε ) μ2 (y ε ) + 1 − n 2 − n2 δ δ rε & σ2 (y ε ) μ2 (y ε ) μ n (y ε ) = 1+β + β2 n + β2 n 2 rε 2r ε 2r ε & ' & ' 2 2 βδ 1 σ n (y ε ) μ n (y ε ) μ n (y ε ) + o2 √ + + √ rε 2 δ2 δ2 r ε δ rε & '' & ''r ε βδ 1 σ 2n (y ε ) μ 2n (y ε ) μ n (y ε ) +o2 − √ + − √ rε 2 δ2 δ2 r ε δ rε $ % 1 2 2 → exp βμ (y0 ) + β σ (y0 ) = Ee βW0 (y0 ) as ε → 0 , 2 since,  & ' & '  μ2 (y ) βδ 1 σ 2n (y ε ) μ 2n (y ε ) μ n (y ε )  2 n ε β + o + + √ 2 √  2r2ε rε 2 δ2 δ2 r ε δ rε & ' & ' βδ 1 σ 2n (y ε ) μ 2n (y ε ) μ n (y ε )    rε +o2 − √ + − √ rε 2 δ2 δ2 r ε δ rε    & '   & '    μ2 (y ε )  βδ βδ     ≤ β2 n + o2 √ r r ε  + o2 − √ ε  → 0 as ε → 0 .     2r ε rε rε

453

(10.147)

(10.148)

Relation (10.147) implies, by the corresponding variant of the continuous theorem, the following relation, for an arbitrary n = 1, . . . , N and y ε → y0 ∈ Yn−1 ∩ Yn−1 as ε → 0: d y ε + W ε,n (y ε ) −→ y0 + W0,n (y0 ) as ε → 0 . (10.149) Relation (10.149) is equivalent to the condition J1 (a) (its univariate variant for the model without index component). The condition J16 (b) implies condition J1 (b) (its univariate variant for the model without index component) to hold, since random variables W0,n (y), y ∈ R1 , n = 1, . . . , N have normal distributions with positive variances, and, thus, possess positive probability density functions with respect to Lebesgue measure. Thus, all conditions of Theorem 5.2.1 hold. By applying this theorem to the above approximation model, we get the asymptotic relation (10.138).

454 | 10 Convergence of tree-type reward approximations

10.3.3 Convergence of optimal expected rewards for inhomogeneous in space binomial and trinomial tree models Let us first present the corresponding results for the univariate inhomogeneous in space binomial tree model considered in Subsection 10.3.1. (0) In this case, the optimal expected reward Φ0 = Φ0 (Mmax,N ) for the log-price process Y 0,n is given by the following formula:

Φ0 = Eϕ0,0 (Y0,0 ) = ϕ0,0 (y)P0 (dy) . (10.150) R1

It is natural to choose, for every ε ∈ (0, ε0 ], the initial distribution P ε (A) for the approximating log-price process Y ε,n as the corresponding skeleton approximation for the distribution P0 (A),

P ε (A) = P0 (Iε,l) , A ∈ B1 , (10.151) y ε,0,0,l ∈ A

where y ε,0,0,l = l √ and Iε,l

δ , r ε,0

l = −r ε,0 , . . . , r ε,0 ,

# ! ⎧# δ 1 ⎪ √ −∞ , r + − ⎪ ε,0 ⎪# ⎨ # r ε,0  #2 ! δ 1 δ 1 √ √ = l − , l + r ε,0 2 r ε,0 2 ⎪ ⎪ ⎪ ⎩ √δ (r − 1 ), ∞) rε

ε,0

2

(10.152)

if l = −r ε,0 , if − r ε,0 < l < r ε,0 ,

(10.153)

if l = r ε,0 ,

where r ε,0 is a positive integer number. (ε) Correspondingly, the optimal expected reward Φ ε = Φ ε (Mmax,N ) for the log-price process Y ε,n is defined, for every ε ∈ (0, ε0 ], by the following formula: Φε =

r ε,0

ϕ ε,0(y ε,0,0,l)P0 (Iε,l) .

(10.154)

l =− r ε,0

Let us assume that the following condition holds for some β ≥ 0: D22 [β ]: Ee β| Y0,0 | < K95 , for some 1 < K95 < ∞, and also that the following condition holds: K20 : P0 (Y0 ) = 0, where Y0 is the set penetrating condition I9 . Theorem 10.3.3. Let the log-price process Y 0,n be represented by the inhomogeneous in space Gaussian random walk with transition probabilities given by relation (10.103) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the in­ homogeneous binomial random walk Y ε,n with transition probabilities given by relation

10.3 Univariate inhomogeneous in space tree models |

455

(10.104) and parameters satisfying the condition G12 , and given by relation (10.106). Let also conditions B4 [𝛾] and D22 [β ] hold for parameters 𝛾 and β such that, either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions I9 , J15 , and K20 hold, and r ε,0 , r ε → ∞ as ε → 0. Then the following relation takes place: Φ ε → Φ0

as ε → 0 .

(10.155)

The proof of this theorem is analogous to the proof of Theorem 10.1.3. Let us now present the corresponding results for the univariate inhomogeneous in space and time trinomial tree model considered in Subsection 10.3.2 In fact, formulas (10.150)–(10.154) defining the optimal expected reward as well as the condition D22 [β ] and K20 remain the same. The only difference is that the reward functions ϕ0,0 ( y) for the trinomial model considered in Subsection 10.3.2 should be y ) for the used in formulas (10.150) and (10.154) instead of the reward functions ϕ0,0 ( binomial model considered in Subsection 10.3.1. Theorem 10.3.4. Let the log-price process Y 0,n be represented by the inhomogeneous in time and space Gaussian random walk with transition probabilities given by relation (10.127) and the approximating log-price process is represented, for every ε ∈ (0, ε0 ], by the inhomogeneous trinomial random walk Y ε,n with transition probabilities given by relation (10.128) and parameters satisfying the conditions G14 , G15 and given by relation (10.130). Let also the conditions B4 [𝛾] and D22 [β ] hold for the parameters 𝛾 and β such that, either β > 𝛾 > 0 or β = 𝛾 = 0, and also the conditions I9 , J16 , and K20 hold, and r ε,0 , r ε → ∞ as ε → 0. Then the following relation takes place: Φ ε → Φ0

as ε → 0 .

(10.156)

The proof of this theorem is analogous to the proof of Theorem 10.1.3. Remark 10.3.1. If the model assumption (c) σ 2 (y) > 0, y ∈ R1 (in the case of homoge­ neous in time model) may not hold, the condition J15 (a) can be replaced in Theorems 10.3.1 and 10.3.3 by the assumption that there exists a set Y ∈ B1 such that the func­ tions μ (y) and σ 2 (y) are continuous in points y ∈ Y and σ 2 (y) > 0 for y ∈ Y (see relation (10.123)) and the condition J15 (b) by the assumption that P{y0 + W0,n (y0 ) ∈ Y ∪ Yn } = 0, for y0 ∈ Y ∩ Yn−1 , n = 1, . . . , N. Remark 10.3.2. If the model assumption (c) σ 2n (y) > 0, y ∈ R1 , n = 1, 2, . . . (in the case of inhomogeneous in time model) may not hold, the condition J16 (b) can be replaced in Theorems 10.3.2 and 10.3.4 by the assumption that P{y0 + W0,n (y0 ) ∈  Yn ∪ Yn } = 0, for y0 ∈ Yn−1 ∩ Yn−1 , n = 1, . . . , N.

456 | 10 Convergence of tree-type reward approximations

10.4 Multivariate inhomogeneous in space tree models In this section, we present results about convergence of rewards for American-type options for multivariate trinomial tree approximations for log-price processes repre­ sented by multivariate inhomogeneous in time and space Gaussian random walks.

10.4.1 Convergence of reward functions for multivariate inhomogeneous in time and space trinomial tree approximations 0,n is an inhomogeneous in time and space Let us assume that a log-price process Y Gaussian Markov chain with the phase space Rk , an initial distribution P0 (A), and y ∈ Rk , A ∈ Bk , n = 1, 2, . . . by the following rela­ transition probabilities given for  tion:     0,n ∈ A/Y 0,n−1 =   0,n ( P0,n ( y , A) = P Y y =P  y+W y) ∈ A , (10.157)  0,n ( where (a) W y ) = (W0,n,1( y), . . . , W0,n,k( y )) is, for every n = 1, 2, . . . ,  y ∈ Rk , a normal random variable with mean values EW0,n,i( y) = μ n,i ( y), i = 1, . . . , k, vari­ ances VarW0,n,i ( y) = σ 2n,i ( y), i = 1, . . . , k, and covariances E(W0,n,i ( y) − μ n,i ( y)) (W0,n,i( y) − μ n,j ( y))) = ρ n,i,j( y)σ n,i ( y)σ n,j( y), i, j = 1, . . . , k. Also, usual assumptions are (b) μ n,i ( y), i = 1, . . . , k, n = 1, 2, . . . are the mea­ surable functions acting from Rk to R1 , (c) σ 2n,i( y ), i = 1, . . . , k, n = 1, 2, . . . is a y), i, j = 1, . . . , k, n = 1, 2, . . . measurable function acting from R1 to (0, ∞), (d) ρ n,i,j( are the measurable functions acting from Rk to [−1, 1], and ρ n,i,i( y) = 1,  y ∈ Rk , i = 1, . . . , k, n = 1, 2, . . . , and (e) Σ n ( y) = σ n,i,j( y), ,  y ∈ Rk , n = 1, 2, . . . , where y) = ρ n,i,j( y)σ n,i ( y)σ n,j( y ),  y ∈ Rk , i, j = 1, . . . , k, n = 1, 2, . . . , are nonnega­ σ n,i,j( tively defined k × k matrices; (f) det(Σ n ( y)) ≠ 0,  y ∈ Rk , n = 1, 2, . . . . 0,n by the log-price processes Y  ε,n, We shall approximate the log-price process Y which is, for every ε ∈ (0, ε0 ], an inhomogeneous in time and space trinomial Markov chain, with the phase space R1 , an initial distribution P ε (A), and transition probabil­ ities given for  y ∈ Rk , A ∈ Bk , n = 1, 2, . . . by the following relation:     ε,n−1 =   ε,n (  ε,n ∈ A/Y P ε,n ( y , A) = P Y y =P  y+W y) ∈ A , (10.158)  ε,n ( y ) = (W ε,n,1( y), . . . , W ε,n,k( y)) is, for every  y ∈ Rk , n = 1, 2, . . . , a where (a) W  ε,n,1(  ε,n ( trinomial random vector, which can be represented in the form W y) = W y) +   y ), where W ε,n,l( y ) = (W ε,n,l,1( y), . . . , W ε,n,l,k( y )), l = 1, 2, . . . are the · · · + W ε,n,r ε ( i.i.d. random vectors taking values δ¯ ε,¯ȷ = (δ ε,1,ȷ1 , . . . , δ ε,k,ȷ k ) with the probabilities p ε,n,¯ȷ( y) = p ε,n,ȷ1,...ȷ k ( y ), for ¯ȷ ∈ Jk = {¯ȷ =  ȷ1 , . . . ȷ k  : ȷ j , . . . ȷ k = +, ◦, −}. Also, usual assumptions are (b) δ εi,− = −δ εi < δ εi,◦ = 0 < δ εi,+ = +δ εi , i = 1, . . . , k, (c) p ε,n,¯ȷ( y), ¯ȷ ∈ Jk , n = 1, 2, . . . are the measurable functions acting from Rk

10.4 Multivariate inhomogeneous in space tree models

| 457

 y) = 1,  y ∈ Rk , n = 1, 2, . . . ; (d) r ε is a positive integer to [0, 1] such that ¯ȷ∈Jk p ε,n,¯ȷ( number. ε,n for approximation of the log-price process Y 0,n We use the log-price processes Y as ε → 0. In this case, a usual approach is to fit the corresponding parameters, i.e. to try to choose parameter δ ε,i , probabilities p ε,n,¯ȷ( y), and parameter r ε in such a way that the following exact fitting equalities hold, for every ε ∈ (0, ε0 ]: ⎧ ⎪ EW ε,n,i( y ) = EW0,n,i ( y) , i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ VarW ε,n,i( y ) = VarW0,1,i ( y) , i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N , ⎪ ⎪ ⎨ E(W ε,n,i( y ) − EW ε,n,i( y))(W ε,n,i( y) − EW ε,n,j( y)) (10.159) ⎪ ⎪ ⎪ ⎪ ⎪ y) − EW0,n,i ( y))(W0,n,i( y ) − EW0,n,j( y)) , = E(W0,n,i ( ⎪ ⎪ ⎪ ⎪ ⎩ 1≤i < j ≤ k,  y ∈ Rk , n = 1, . . . , N .

The solution of this fitting problem is presented in Lemma 9.5.2. According to this lemma, the fitting relations (10.159) hold, if to assume that the conditions G16 , G17 , ε,n given by the fol­ and R3 hold, and to choose parameters of the log-price process Y lowing formulas, for every ε ∈ (0, ε0 ] and n = 1, . . . , N: ⎧ δi ⎪ ⎪ ⎪ δ ε,i, ± = ± √ , δ ε,i, ◦ = 0 , ⎪ ⎪ rε ⎪ ⎪ ⎪ & 2 ' ⎪ ⎪ ⎨ y) μ 2n,i ( y) μ n,i ( 1 σ n,i ( y) y) = + ± √ , p ε,n,i, ±( (10.160) ⎪ 2 δi rε δ2i δ2i r ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ 2n,i( y) μ 2n,i ( y) ⎪ ⎪ ⎪ y) = 1 − − , i = 1, . . . , k ,  y ∈ Rk , n = 1, . . . , N , ⎩ p ε,n,i,◦( 2 2 δi δi rε and

& ⎧ ⎪ y) σ n,i ( y) σ n,j ( y) μ n,i ( y) μ n,j ( y) 1 ρ n,i,j ( ⎪ ⎪ y) = + ⎪ p ε,n,i,j,±,± ( ⎪ ⎪ 4 δ δ δ δ r ⎪ i j i j ε ⎪ ⎪ & 2 ' & 2 '' & ' ⎪ ⎪ ⎪ σ n,j ( y) μ2n,j ( y) ⎪ σ n,i ( y) μ2n,i ( y) y) 1 μ n,i ( y) μ n,j ( ⎪ ⎪ + + + + ∧ ± ⎪ √ √ ⎪ ⎪ 2 δi rε δj rε δ2i δ2i r ε δ2j δ2j r ε ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ρ n,i,j ( y) σ n,i ( y) σ n,j ( y) μ n,i ( y) μ n,j ( y) 1 ⎪ ⎪ ⎪ y) = − − ⎪ p ε,n,i,j,±,∓ ( ⎪ 4 δi δj δi δj rε ⎪ ⎪ ⎪ ⎪ & 2 ' & 2 '' & ' ⎪ ⎪ ⎪ σ n,j ( y) μ2n,j ( y) σ n,i ( y) μ2n,i ( y) y) y) μ n,j ( 1 μ n,i ( ⎪ ⎪ ⎨ + + + − ∧ ± √ √ 2 δi rε δj rε δ2i δ2i r ε δ2j δ2j r ε ⎪ & ' & ' ⎪ 2 ⎪ 2 2 2 2 ⎪ σ ( y ) μ2n,j ( y) σ ( y ) μ ( y ) σ ( y ) μ ( y ) ⎪ n,j n,i n,i n,i n,i ⎪ ⎪ p ( y ) = 1 − − − 1 − − 1 − − , ∧ ⎪ ε,n,i,j, ◦ , ± ⎪ ⎪ δ2i δ2i r ε δ2i δ2i r ε δ2j δ2j r ε ⎪ ⎪ ⎪ ⎪ & ' & ' ⎪ ⎪ σ2n,j ( y) μ2n,j ( y) σ2n,j ( y) μ2n,j ( y) ⎪ σ2n,i ( y) μ2n,i ( y) ⎪ ⎪ p ( y ) = 1 − − − 1 − − 1 − − , ∧ ⎪ ε,n,i,j, ±, ◦ ⎪ ⎪ δ2j δ2j r ε δ2i δ2i r ε δ2j δ2j r ε ⎪ ⎪ ⎪ ⎪ & ' & ' ⎪ ⎪ ⎪ σ2n,j ( y) μ2n,j ( y) σ2n,i ( y) μ2n,i ( y) ⎪ ⎪ ⎪ p ( y ) = 1 − − 1 − − , ∧ ε,n,i,j, ◦, ◦ ⎪ ⎪ ⎪ δ2i δ2i r ε δ2j δ2j r ε ⎪ ⎪ ⎪ ⎪ ⎩ y ∈ Rk , n = 1, . . . , N , 1≤ i < j ≤ k,  (10.161)

458 | 10 Convergence of tree-type reward approximations where (i)one can choose for the parameters δ i , i = 1, . . . , k and r ε any values such that

2 + K90 , i = 1, . . . , k, (b) ρ < κ ≤ 1, (c) r ε ≥ (a) δ i ≥ K90

2 K 90 σ 2 (κ− ρ)

+

2 2 16K 90 δ σ 4 ( κ − ρ)2 , where

δ=

max1≤i≤k δ i , κ = min1≤i,j≤k δδ ij , while σ 2 > 0, K90 < ∞, and 0 ≤ ρ < 1 are parameters penetrating, respectively, the conditions G16 , G17 , and R3 ; (ii) the corresponding higher  ε,n,1 consistent with the above order distributions for components of random vectors W one- and two-dimensional distributions can be defined using algorithm described in Lemma 9.3.2. Let us denote, Lk,r = {¯l = (l1 , . . . , l k ) : −r ≤ l1 , . . . , l k ≤ r}, r = 1, 2, . . . .

(10.162)

 ε,n ( y) is for every  y ∈ Rk , n = 1, . . . , N a trinomial random The random vector W  ε,n ( vector. Namely, the random vector W y) takes values w  ε,n,r ε ,¯l with the correspond­ ing probabilities p ε,n,r ε (¯l), for ¯l = (l1 , . . . , l k ) ∈ Lk,r ε . Here, for every ¯l = (l1 , . . . , l k ) ∈ Lk,r , r = 1, 2, . . . ,

w  ε,n,r,¯l = (l1 δ ε,1 , . . . , l k δ ε,k ) ,

(10.163)

and   ε,n (  ε,n,1( p ε,n,r,¯l( y) = P W y) = W y)

  ε,n,r( +···+ W y) = w  ε,n,r,¯l I (¯l ∈ Lk,r ) .

(10.164)

The probabilities p ε,n,r,¯l( y) can be found using the following recurrent convolu­ tion formulas: ⎧ ⎨ p ε,n,¯ȷ( y)I (¯l = e¯¯ȷ ) if r = 1, ¯l ∈ Lk,1 , p ε,n,r,¯l( y ) = ⎩ (10.165) y)p ε,n,1,¯l( y) if r > 1, ¯l ∈ Lk,r . ¯l  ∈L p ε,n,r −1,¯l−¯l  ( k,1

where the vectors e¯¯ȷ = (e ȷ1 , . . . , e ȷ k ), ¯ȷ = (ȷ1 , . . . , ȷ k ) ∈ Jk , e ȷ = 1, 0, −1 if, respectively, ȷ = +, ◦, −. The additional comments concerning algorithms for computing of probabilities y ) can be found in Subsection 3.2.3. p ε,n,r,¯l( ε,n =  Let us assume that Y y = (y1 , . . . , y k ) = const ∈ Rk . In this case, trajectories ε,n+r, r = 0, 1, . . . generate, for every ε ∈ (0, ε0 ], the unique of the log-price process Y tree of nodes T ε,(y,n) = {( y ε,n,n+r,¯l, n + r), ¯l = (l1 , . . . , l k ) ∈ Lk,r ε r , r = 0, 1, . . . } ,

(10.166)

where  y ε,n,n+r,¯l = (y ε,n,n+r,1,l1 , . . . , y ε,n,n+r,k,lk ) = (y1 + l1 δ ε,1 , . . . , y k + l k δ ε,k ), ¯l = (l1 , . . . , l k ) ∈ Lk,r ε r , r = 0, 1, . . . .

(10.167)

10.4 Multivariate inhomogeneous in space tree models |

459

The tree T ε,(y,n) contains the total number of nodes after r steps, L ε,n,n+r ≤ 1 + (2r ε + 1)k + · · · + (2r ε r + 1)k r +1

(2r ε x + 1)k dx =

≤ 0

(2r ε (r + 1) + 1)k+1 − 1 . 2r ε (k + 1)

(10.168)

The expression on the right-hand side in (10.168) is a polynomial function of r of the order k + 1. The following lemma is a direct corollary of Lemma 3.4.1.  ε,n is the Lemma 10.4.1. Let, for every ε ∈ (0, ε0 ], the approximating log-price process Y multivariate trinomial random walk given by the stochastic transition dynamic relation (10.158), with parameters defined in relations (10.160) and (10.161), under the assump­ tion that conditions G16 , G17 , and R3 hold. Then the reward functions ϕ ε,n+r( y ε,n,n+r,¯l), ¯ for the points  y ε,n,n,0 =  y and  y ε,n,n+r,¯l, l ∈ Lk,r ε r , r = 1, . . . N − n, are, for every ε ∈ (0, ε0 ], and  y ∈ Rk , n = 0, . . . , N, the unique solution for the following recur­ rence finite system of linear equations: ⎧ ⎪ ϕ ε,N ( y ε,n,N,¯l) = g ε (N, ey ε,n,N,¯l ) , ⎪ ⎪ ⎪ ⎪ ⎪ ¯l ∈ Lk,r ε(N −n) , ⎪ ⎪ ⎪ ⎪ & ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ε,n,n+r,¯l) = max g ε (n + r, ey ε,n,n+r,¯l ), ⎪ ⎪ ϕ ε,n+r( ⎪ ⎪ ⎪ ⎪ ' ⎨

ϕ ( y  ) p  ( y ) , (10.169) ⎪ ε,n + r +1 ε,n,n + r +1,¯l+¯l ε,n + r +1,r ε,¯l ε,n,n + r,¯l ⎪ ⎪ ⎪ ¯l  ∈Lk,r ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯l ∈ Lk,r ε r , r = N − n − 1, . . . , 1 , ⎪ ⎪ ⎪ ⎪ & ' ⎪ ⎪ ⎪

⎪ ⎪  y ⎪ ϕ ε,n ( y ) = max g ε (n, e ), ϕ ε,n ( y ε,n,n+1,¯l)p ε,n+1,r ε,¯l ( y) . ⎪ ⎪ ⎩ ¯ l ∈Lk,r ε

We are going to apply Theorem 5.2.1 in order to prove convergence of the reward func­  ε,n to the corresponding reward functions for the tions for the log-price processes Y 0,n as ε → 0. log-price process Y We assume that the condition B2 [𝛾¯] holds for the pay-off function g(n, ey ), for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components. The condition I1 reduces in this case to the condition I10 . Let us also assume that the following condition holds: J17 : (a) There exists set Yn ∈ Bk , n = 0, 1, . . . , N such that functions μ n,i ( y ), y), ρ 2n,i,j( y), i, j = 1, . . . , k are continuous in points y ∈ Yn−1, for ev­ σ 2n,i ( ery n = 1, . . . , N;  (b) L k (Yn ) = 0, L k (Yn ) = 0, n = 1, . . . , N, where Yn , n = 1, . . . , N are sets penetrating condition I10 .

460 | 10 Convergence of tree-type reward approximations ε,n with pa­ We shall see that, in this case, the condition J1 (a) holds for log-prices Y rameters given by relation (10.160), if r ε → ∞ as ε → 0. Also, the random vectors  0,n ( W y),  y ∈ Rk , n = 1, . . . , N have normal distributions that make it possible to re­ place the condition J1 (b) by the condition J17 . The above backward recurrence algorithm for finding reward functions is sup­ ported by the following convergence result. 0,n be represented by the multivariate in­ Theorem 10.4.1. Let the log-price process Y homogeneous in space and time Gaussian random walk with transition probabilities given by relation (10.157) and the approximating log-price processes are represented, for every ε ∈ (0, ε0 ], by the multivariate inhomogeneous in space and time binomial random walks Y ε,n with transition probabilities given by relation (10.158) and parame­ ters satisfying the conditions G16 , G17 , R3 and given by relation (10.160) and (10.161). Let also conditions B2 [𝛾¯] holds for some vector parameter 𝛾¯ = (𝛾1 , . . . , 𝛾k ) with nonnegative components, the conditions I10 and J17 hold, and r ε → ∞ as ε → 0. Then the following yε →  y0 ∈ Yn ∩ Yn , n = 0, 1, . . . , N: relations take place for any 

ϕ ε,n ( y ε ) → ϕ0,n ( y0 )

as ε → 0 .

(10.170)

Proof. We are going to use Theorem 5.2.1. Let the conditions G16 , G17 hold and parameters of the process Y ε,n are given by relations (10.160) and (10.161). In this case, for ε ∈ (0, ε0 ], the moment-generating functions for the random  ε,n ( y) = (W ε,n,1( y ), . . . , W ε,n,k( y)),  y ∈ Rk , n = 1, . . . , N have the following vectors W ¯ form, for β = (β 1 , . . . , β k ) ∈ Rk : ⎛ ⎞r ε k

k β δ√i eȷ i Ee i=1 β i W ε,n,i (y) = ⎝ e i=1 i rε p ε,n,¯ȷ( y )⎠ , (10.171) ¯ȷ∈Jk

 0,n ( y) = while, for ε = 0, the moment-generating functions for the random vectors W y), . . . , W0,n,k( y)),  y ∈ Rk , n = 1, . . . , N have the following form, for β¯ = (W0,n,1( ( β 1 , . . . , β k ) ∈ Rk :

Ee

k

i =1

( β i W 0,n,i ( y)

= exp

k

i =1

+



1 2 2 β σ ( y) 2 i=1 i n,i ) k

β i μ n,i( y) +

β i β j ρ n,i,j( y)σ n,i ( y)σ n,j( y) .

(10.172)

1≤ i < j ≤ k

Relations (10.160) and (10.171) yield, for ε ∈ (0, ε0 ], the following formulas y ), for the marginal moment-generating functions for the random variables W ε,n,i(

10.4 Multivariate inhomogeneous in space tree models | 461

y ∈ Rk , n = 1, . . . , N, for β ∈ R1 : i = 1, . . . , k,   δ r ε δ β √i − β √riε y) + e p ε,n,i, −( y) + p ε,n,i,◦( y) Ee βW ε,n,i (y) = e rε p ε,n,i,+( & & 2 ' δ σ n,i( y ) μ 2n,i ( y) μ n,i ( y) β √i 1 = e rε + + √ 2 δi rε δ2i δ2i r ε & 2 ' δ σ n,i ( y) μ 2n,i ( y) μ n,i( y) − β √riε 1 + − √ +e 2 δi rε δ2i δ2i r ε '' & r ε σ 2n,i( y) μ 2n,i ( y) + 1− − , 2 2 δi δi rε

(10.173)

while, for ε = 0, these moment-generating functions have the following form: 1

Ee βW0,n,i (y) = e βμ n,i (y)+ 2 β

2 2 σ n,i ( y)

.

(10.174)

Relations (10.173)–(10.174) are analogous, respectively, to relations (10.139)– (10.140) used in the proof of Theorem 10.3.2. This makes it possible to repeat cal­ culations given in the proof of this theorem and to get a relation analogous to relation (10.142), i.e. to get, under the assumption that the following condition holds, for every ε ∈ (0, ε0 ], i = 1, . . . , k,  y ∈ Rk , n = 1, . . . , N, and any β ∈ R1 : & & ' ''r ε & 2 K90 1 β2r δ2i 1 2 βW ε,n,i ( y) + |β |K90 ≤ 1+ e ε Ee β K90 + rε 2 rε ( 22 & & ' ') 2 β δi K 1 90 2 + |β |K90 ≤ exp e rε , (10.175) β K90 + 2 rε where K90 is the constant penetrating condition G17 . Also the condition G17 and relation (10.174) imply that, for every ε ∈ (0, ε0 ], i = 1, . . . , k,  y ∈ Rk , n = 1, . . . , N, and any β ∈ R1 , 1

Ee βW0,n,i (y) ≤ e(| β|+ 2 β )K 90 . 2

(10.176)

The modulus Δ± β ( Y ε,i, ·, N ) takes for the above log-price processes Y ε,n the follow­ ing form, for every ε ∈ [0, ε0 ], i = 1, . . . , k and β ≥ 0: ± βW ε,n,i ( y) . Δ± β ( Y ε,i, ·, N ) = max sup e

1≤ n ≤ N  y∈R

(10.177)

k

Relations (10.175) and (10.176) imply the following asymptotic relation, which holds for any i = 1, . . . , k and β ≥ 0: ( 22 & & ' ') 2 β δi K90 1 2 ± r ε lim Δ β (Y ε,i, ·, N ) ≤ lim exp e + βK90 β K90 + ε →0 ε →0 2 rε $  % 1 β + β 2 K90 < ∞ . (10.178) = exp 2 Relations (10.176) and (10.178) imply that the condition E10 [β¯ ] (its variant for the model without index component) holds. Therefore, by Lemma 5.1.5, the condi­ tion C6 [β¯ ] (its variant for the model without index component) holds for any vector

462 | 10 Convergence of tree-type reward approximations parameter β¯ = (β 1 , . . . , β k ) with nonnegative components, for example, with the 2 constant K96 = 1 + 2e(| β|+β )K 90 replacing the constants K30,i , i = 1, . . . , k penetrating this condition. The condition B2 [𝛾¯] is a variant of the condition B5 [𝛾¯] (its variant for the model without index component and a pay-off function, which does not depend on the pa­ rameter ε). Note that one can always choose the parameter β¯ such that, for every i = 1, . . . , k, either β i > 𝛾i , if 𝛾i > 0, or β i = 𝛾i , if 𝛾i = 0. The condition I1 (its variant for the model without index component and a pay-off function, which does not depend on the parameter ε) reduces in this case to the form of the condition I10 . Let us take an arbitrary n = 0, 1, . . . , N and  yε →  y0 ∈ Yn ∩ Yn as ε → 0. By 2 y ε ) → μ n,i ( y0 ), σ n,i ( y ε ) → σ 2n,i ( y0 ) and ρ n,i,j( yε ) → the conditions I10 and J17 , μ n,i ( ρ n,i,j( y0 ) as ε → 0, for i, j = 1, . . . , k, and, therefore, for every n = 1, . . . , N and β¯ = (β 1 , . . . , β k ) ∈ Rk , ⎛

Ee

k

i=1

β i W ε,n,i ( yε )

⎞r ε ⎛ ⎛ ⎞2 ⎛ ⎞⎞ k k k





δ e δ e δ e 1 i ȷ i ȷ i ȷ ⎜ ⎟ ⎟ ⎜ =⎝ β i √ i + ⎝ β i √ i ⎠ + o2 ⎝ β i √ i ⎠⎠ p ε,n,¯ȷ ( y ε )⎠ ⎝1 + r 2 r r ε ε ε i = 1 i = 1 i = 1 ¯ȷ ∈Jk ⎛ k

δi  = ⎝1 + βi √ p ε,n,i,+ ( y ε ) − p ε,n,i,− ( yε ) r ε i =1 k 1 2 δ 2i  β p ε,n,i,+ ( y ε ) + p ε,n,i,− ( yε ) 2 i =1 i r ε

δi δj # + βi βj p ε,n,i,j,+,+ ( y ε ) + p ε,n,i,j,− ,− ( yε ) rε 1≤i