Mathematics of the Bond Market: A Lévy Processes Approach (Encyclopedia of Mathematics and its Applications) [1 ed.] 1107101298, 9781107101296

Table of contents :
Contents
Preface
The Field
Levy Modelling
Aims of the Book
Structure of the Book
Acknowledgements
Introduction
I.1 Bonds
I.2 Models
I.3 Content of the Book
PART I BOND MARKET IN DISCRETE TIME
1 Elements of the Bond Market
1.1 Prices and Rates
1.2 Models of the Bond Market
1.3 Portfolios and Strategies
1.4 Contingent Claims
1.5 Arbitrage
2 Arbitrage-Free Bond Markets
2.1 Martingale Modelling
2.2 Martingale Measures for HJM Models
2.2.1 Existence of Martingale Measures
2.2.2 Uniqueness of the Martingale Measure
2.3 Martingale Measures and Martingale Representation Property
2.3.1 Martingale Representation Property
2.3.2 Generalized Martingale Representation Property
2.3.3 Girsanov’s Theorems
2.3.4 Application to HJM Models
2.4 Markovian Models under the Martingale Measure
2.4.1 Models with Markovian Trace
2.4.2 Affine Models
2.4.3 Dynamics of the Short Rate in Affine Models
2.4.4 Shape of Forward Curves in Affine Models
2.4.5 Factor Models
3 Completeness
3.1 Concepts of Completeness
3.2 Necessary Conditions for Completeness
3.3 Sufficient Conditions for Completeness
3.4 Approximate Completeness
3.4.1 General Characterization
3.4.2 Bond Curves in a Finite Dimensional Space
3.4.3 Bond Curves in Hilbert Spaces
3.5 Models with Martingale Prices
3.5.1 HJM Models
3.5.2 Multiplicative Factor Model
3.5.3 Affine Models
3.6 Replication with Finite Portfolios
3.7 Completeness and Martingale Measures
PART II FUNDAMENTALS OF STOCHASTIC ANALYSIS
4 Stochastic Preliminaries
4.1 Generalities
4.2 Doob–Meyer Decomposition
4.2.1 Predictable Quadratic Variation of Square Integrable Martingales
4.2.2 Compensators of Finite Variation Processes
4.3 Semimartingales
4.4 Stochastic Integration
4.4.1 Bounded Variation Integrators
4.4.2 Square Integrable Martingales as Integrators
4.4.3 Integration over Random Measures
4.4.4 Ito’s Formula
5 Levy Processes
5.1 Basics on Levy Processes
5.2 Levy–Ito Decomposition
5.3 Special Classes
5.3.1 Finite Variation Processes
5.3.2 Subordinators
5.3.3 Levy Martingales
5.4 Stochastic Integration
5.4.1 Square Integrable Integrators
5.4.2 Integration over Compensated Jump Measures
5.4.3 Stochastic Fubini’s Theorem
5.4.4 Ito’s Formula for Levy Processes
6 Martingale Representation and Girsanov’s Theorems
6.1 Martingale Representation Theorem
6.2 Girsanov’s Theorem and Equivalent Measures
PART III BOND MARKET IN CONTINUOUS TIME
7 Fundamentals
7.1 Prices and Rates
7.1.1 Bank Account and Discounted Bond Prices
7.1.2 Prices and Rates in Function Spaces
7.2 Portfolios and Strategies
7.2.1 Portfolios
7.2.2 Strategies and the Wealth Process
7.2.3 Wealth Process as Stochastic Integral
7.3 Non-arbitrage, Claims and Their Prices
7.4 HJM Modelling
7.4.1 Bond Prices Formula
7.4.2 Forward Curves in Function Spaces
7.5 Factor Models and the Musiela Parametrization
8 Arbitrage-Free HJM Markets
8.1 Heath–Jarrow–Morton Conditions
8.1.1 Proof of Theorem 8.1.1
8.2 Martingale Measures
8.2.1 Specification of Drift
8.2.2 Models with No Martingale Measures
8.2.3 Invariance of Levy Noise
8.2.4 Volatility-Based Models
8.2.5 Uniqueness of the Martingale Measure
9 Arbitrage-Free Forward Curves Models
9.1 Term Structure Equation
9.1.1 Markov Chain and CIR as Factor Processes
9.1.2 Multiplicative Factor Process
9.1.3 Affine Term Structure Model
9.1.4 Ornstein–Uhlenbeck Factors
10 Arbitrage-Free Affine Term Structure
10.1 Preliminary Model Requirements
10.2 Jump Diffusion Short Rate
10.2.1 Analytical HJM Condition
10.2.2 Generalized CIR Equations
10.2.3 Exploding Short Rates
10.2.4 Multidimensional Noise
10.3 General Markovian Short Rate
10.3.1 Filipovi´ c’s Theorems
10.3.2 Comments on Filipovi´ c’s Theorems
10.3.3 Examples
10.3.4 Back to Short-Rate Equations
11 Completeness
11.1 Problem of Completeness
11.2 Representation of Discounted Bond Prices
11.3 Admissible Strategies
11.4 Hedging Equation
11.5 Completeness for the HJM Model
11.5.1 Levy Measure with Finite Support
11.5.2 Proofs of Theorems 11.5.1–11.5.3
11.5.3 Incomplete Markets
11.6 Completeness for Affine Models
11.7 Completeness for Factor Models
11.8 Approximate Completeness
11.8.1 HJM Model
11.8.2 Factor Model
11.8.3 Affine Model
PART IV STOCHASTIC EQUATIONS IN THE BOND MARKET
12 Stochastic Equations for Forward Rates
12.1 Heath–Jarrow–Morton Equation
12.2 Morton’s Equation
12.3 The Equations in the Musiela Parametrization
13 Analysis of the HJMM Equation
13.1 Existence of Solutions to the HJMM Equation
13.1.1 Local Solutions
13.1.2 Global Solutions
13.1.3 Applications to the Morton–Musiela Equation
14 Analysis of Morton’s Equation
14.1 Results
14.1.1 Comments on Assumptions (A1)–(A3)
14.2 Applications of the Main Theorems
14.3 Proof of Theorem 14.1.1
14.3.1 Outline of the Proof
14.3.2 Equivalence of Equations (14.1.1) and (14.1.9)
14.3.3 Auxiliary Results
14.3.4 Conclusion of the Proof
14.4 Proof of Theorem 14.1.2
15 Analysis of the Morton–Musiela Equation
15.1 Formulation and Comments on the Results
15.1.1 Comments on the Results
15.2 Proofs of Theorems 15.1.1 and 15.1.2
15.2.1 Equivalence Results
15.2.2 Proof of Theorem 15.1.1
15.2.3 Proof of Theorem 15.1.2
Appendix A
A.1 Martingale Representation for Jump Levy Processes
A.1.1 Multiple Chaos Processes
A.1.2 Representation of Chaoses
A.1.3 Chaos Expansion Theorem
A.1.4 Representation of Square Integrable Martingales
A.1.5 Representations of Local Martingales
Appendix B
B.1 Semigroups and Generators
B.1.1 Generators for Equations with Levy Noise
Appendix C
C.1 General Evolution Equations
References
Index

Citation preview

M AT H E M AT I C S O F T H E B O N D M A R K E T Mathematical models of bond markets are of interest to researchers working in applied mathematics, especially in mathematical finance. This book concerns bond market models in which random elements are represented by L´evy processes. These are more flexible than classical models and are well suited to describing prices quoted in a discontinuous fashion. The book’s key aims are to characterize bond markets that are free of arbitrage and to analyze their completeness. Nonlinear stochastic partial differential equations (SPDEs) are an important tool in the analysis. The authors begin with a relatively elementary analysis in discrete time, suitable for readers who are not familiar with finance or continuous time stochastic analysis. The book should be of interest to mathematicians, in particular to probabilists, who wish to learn the theory of the bond market and to be exposed to attractive open mathematical problems.

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Encyclopedia of Mathematics and Its Applications

Mathematics of the Bond Market A L´evy Processes Approach M I C H A Ł BA R S K I Faculty of Mathematics, Informatics and Mechanics, University of Warsaw

JERZY ZABCZYK Institute of Mathematics, Polish Academy of Sciences

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107101296 DOI: 10.1017/9781316181836 c Michał Barski and Jerzy Zabczyk 2020  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2020 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalogue record for this publication is available from the British Library. ISBN 978-1-107-10129-6 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To our wives Anna and Barbara

Contents

Preface The Field L´evy Modelling Aims of the Book Structure of the Book Acknowledgements Introduction I.1 Bonds I.2 Models I.3 Content of the Book PART I BOND MARKET IN DISCRETE TIME

page xiii xiii xiv xv xv xvii 1 1 2 5 7

1

Elements of the Bond Market 1.1 Prices and Rates 1.2 Models of the Bond Market 1.3 Portfolios and Strategies 1.4 Contingent Claims 1.5 Arbitrage

9 9 12 13 16 18

2

Arbitrage-Free Bond Markets 2.1 Martingale Modelling 2.2 Martingale Measures for HJM Models 2.2.1 Existence of Martingale Measures 2.2.2 Uniqueness of the Martingale Measure 2.3 Martingale Measures and Martingale Representation Property 2.3.1 Martingale Representation Property 2.3.2 Generalized Martingale Representation Property

23 23 24 24 27 31 32 35

viii

Contents

2.4

3

2.3.3 Girsanov’s Theorems 2.3.4 Application to HJM Models Markovian Models under the Martingale Measure 2.4.1 Models with Markovian Trace 2.4.2 Affine Models 2.4.3 Dynamics of the Short Rate in Affine Models 2.4.4 Shape of Forward Curves in Affine Models 2.4.5 Factor Models

Completeness 3.1 Concepts of Completeness 3.2 Necessary Conditions for Completeness 3.3 Sufficient Conditions for Completeness 3.4 Approximate Completeness 3.4.1 General Characterization 3.4.2 Bond Curves in a Finite Dimensional Space 3.4.3 Bond Curves in Hilbert Spaces 3.5 Models with Martingale Prices 3.5.1 HJM Models 3.5.2 Multiplicative Factor Model 3.5.3 Affine Models 3.6 Replication with Finite Portfolios 3.7 Completeness and Martingale Measures PART II FUNDAMENTALS OF STOCHASTIC ANALYSIS

4

5

37 41 44 45 48 52 58 61 65 65 68 70 74 75 77 78 82 83 88 92 95 100 105

Stochastic Preliminaries 4.1 Generalities 4.2 Doob–Meyer Decomposition 4.2.1 Predictable Quadratic Variation of Square Integrable Martingales 4.2.2 Compensators of Finite Variation Processes 4.3 Semimartingales 4.4 Stochastic Integration 4.4.1 Bounded Variation Integrators 4.4.2 Square Integrable Martingales as Integrators 4.4.3 Integration over Random Measures 4.4.4 Itˆo’s Formula

107 107 109 111 112 114 117 117 118 121 123

L´evy Processes 5.1 Basics on L´evy Processes 5.2 L´evy–Itˆo Decomposition

126 126 128

Contents 5.3

5.4

6

Special Classes 5.3.1 Finite Variation Processes 5.3.2 Subordinators 5.3.3 L´evy Martingales Stochastic Integration 5.4.1 Square Integrable Integrators 5.4.2 Integration over Compensated Jump Measures 5.4.3 Stochastic Fubini’s Theorem 5.4.4 Ito’s Formula for L´evy Processes

Martingale Representation and Girsanov’s Theorems 6.1 Martingale Representation Theorem 6.2 Girsanov’s Theorem and Equivalent Measures PART III BOND MARKET IN CONTINUOUS TIME

ix 131 131 133 134 136 137 138 140 141 142 142 143 151

7

Fundamentals 7.1 Prices and Rates 7.1.1 Bank Account and Discounted Bond Prices 7.1.2 Prices and Rates in Function Spaces 7.2 Portfolios and Strategies 7.2.1 Portfolios 7.2.2 Strategies and the Wealth Process 7.2.3 Wealth Process as Stochastic Integral 7.3 Non-arbitrage, Claims and Their Prices 7.4 HJM Modelling 7.4.1 Bond Prices Formula 7.4.2 Forward Curves in Function Spaces 7.5 Factor Models and the Musiela Parametrization

153 153 155 157 161 161 162 166 171 173 177 180 182

8

Arbitrage-Free HJM Markets 8.1 Heath–Jarrow–Morton Conditions 8.1.1 Proof of Theorem 8.1.1 8.2 Martingale Measures 8.2.1 Specification of Drift 8.2.2 Models with No Martingale Measures 8.2.3 Invariance of L´evy Noise 8.2.4 Volatility-Based Models 8.2.5 Uniqueness of the Martingale Measure

184 184 188 191 193 194 197 200 203

9

Arbitrage-Free Forward Curves Models 9.1 Term Structure Equation 9.1.1 Markov Chain and CIR as Factor Processes

207 207 210

x

Contents 9.1.2 9.1.3 9.1.4

Multiplicative Factor Process Affine Term Structure Model Ornstein–Uhlenbeck Factors

212 214 216

10

Arbitrage-Free Affine Term Structure 10.1 Preliminary Model Requirements 10.2 Jump Diffusion Short Rate 10.2.1 Analytical HJM Condition 10.2.2 Generalized CIR Equations 10.2.3 Exploding Short Rates 10.2.4 Multidimensional Noise 10.3 General Markovian Short Rate 10.3.1 Filipovi´c’s Theorems 10.3.2 Comments on Filipovi´c’s Theorems 10.3.3 Examples 10.3.4 Back to Short-Rate Equations

220 220 221 222 226 234 236 238 238 240 244 245

11

Completeness 11.1 Problem of Completeness 11.2 Representation of Discounted Bond Prices 11.3 Admissible Strategies 11.4 Hedging Equation 11.5 Completeness for the HJM Model 11.5.1 L´evy Measure with Finite Support 11.5.2 Proofs of Theorems 11.5.1–11.5.3 11.5.3 Incomplete Markets 11.6 Completeness for Affine Models 11.7 Completeness for Factor Models 11.8 Approximate Completeness 11.8.1 HJM Model 11.8.2 Factor Model 11.8.3 Affine Model

252 252 253 257 260 261 261 264 269 275 277 280 283 288 289

PART IV STOCHASTIC EQUATIONS IN THE BOND MARKET

293

12

Stochastic Equations for Forward Rates 12.1 Heath–Jarrow–Morton Equation 12.2 Morton’s Equation 12.3 The Equations in the Musiela Parametrization

295 295 296 297

13

Analysis of the HJMM Equation 13.1 Existence of Solutions to the HJMM Equation 13.1.1 Local Solutions

300 300 302

Contents 13.1.2 Global Solutions 13.1.3 Applications to the Morton–Musiela Equation

xi 307 309

14

Analysis of Morton’s Equation 14.1 Results 14.1.1 Comments on Assumptions (A1)–(A3) 14.2 Applications of the Main Theorems 14.3 Proof of Theorem 14.1.1 14.3.1 Outline of the Proof 14.3.2 Equivalence of Equations (14.1.1) and (14.1.9) 14.3.3 Auxiliary Results 14.3.4 Conclusion of the Proof 14.4 Proof of Theorem 14.1.2

312 312 314 315 322 322 323 324 329 330

15

Analysis of the Morton–Musiela Equation 15.1 Formulation and Comments on the Results 15.1.1 Comments on the Results 15.2 Proofs of Theorems 15.1.1 and 15.1.2 15.2.1 Equivalence Results 15.2.2 Proof of Theorem 15.1.1 15.2.3 Proof of Theorem 15.1.2

332 332 333 334 334 335 337

Appendix A A.1 Martingale Representation for Jump L´evy Processes A.1.1 Multiple Chaos Processes A.1.2 Representation of Chaoses A.1.3 Chaos Expansion Theorem A.1.4 Representation of Square Integrable Martingales A.1.5 Representations of Local Martingales

342 342 343 347 350 352 354

Appendix B B.1 Semigroups and Generators B.1.1 Generators for Equations with L´evy Noise

360 360 361

Appendix C C.1 General Evolution Equations

367 367

References Index

373 379

Preface

The Field The book is devoted to the mathematical theory of the bond market, which is a part of mathematical finance. It is addressed to mathematicians, especially to probabilists who are not necessarily familiar with mathematical finance. In fact, Part I – out of the four parts of this book – treats the subject in discrete time and the knowledge of classical probability, as presented in Feller [51], is sufficient for its understanding. Mathematical finance is today a part of stochastic analysis. Such concepts as stochastic integral and martingales play a fundamental role in finance. For instance, the mathematical theory of stochastic integration is well developed for large classes of integrators and integrands, and general concepts are ideally suited to financial modelling. Integrators are price processes of financial commodities, integrands describe trading strategies and the integrals represent accumulated wealth. Basic objects of the theory are two random fields P(t, T), f (t, T), 0 ≤ t ≤ T, and a stochastic process R(t), t ≥ 0, defined on a filtered probability space (, F, (Ft ), P). They are related to each other by the formulas P(t, T) = e−

T t

f (t,s)ds

,

0 ≤ t ≤ T,

R(t) = f (t, t), t ≥ 0,

and interpreted as, respectively, bond prices, forward rates and short rate. In particular, P(t, T) is the price of a bond at time t that matures at time T, that is, the owner of the bond will receive cash P(T, T) at time T. The theory is relatively young, approximately 40 years old, and poses new mathematical questions. An important one is about the absence of arbitrage. Intuitively, the market should not allow agents to accumulate wealth, by clever investments, without the possibility of facing losses. This property of bond models is mathematically expressed in the concept of non-arbitrage. A related question concerns conditions under which there exists a martingale measure for the bond

xiv

Preface

prices, that is, a probability measure Q equivalent to P such that for each T ≥ 0, the process of discounted bond prices t

ˆ T) = e− P(t,

0

R(s)ds

P(t, T),

t ∈ [0, T]

is a local martingale under Q. Problems of this type have never been asked earlier. Another question is that of completeness of the market. Mathematically it is equivalent to the condition that each, say, bounded FT ∗ -measurable random variable, with T ∗ > 0, can be represented as a sum of a constant and a stochastic integral, over ˆ ·), t ∈ [0, T ∗ ]. the interval [0, T ∗ ], with integrator P(t, The time evolution of bond prices, short rates and forward rates is studied using the theory of L´evy processes and stochastic differential equations. In fact, applications of the theory of stochastic partial differential equations with L´evy noise – a relatively young branch of stochastic processes – are discussed in the book in great detail. For the reader’s convenience the book starts with an extensive treatment of discrete time models. Here the role of L´evy processes is played by random walks.

L´evy Modelling A good model of bond prices should satisfy several conditions and allow easy confrontation with reality. Stochastic processes used in applications are numerically “tractable” if they are of Markov type or, more specifically, if they are solutions of stochastic equations. For them, at least theoretically, one can find finite dimensional distributions by solving parabolic equations of Kolmogorov type. As already mentioned, the book is concerned with models in which random elements are represented through L´evy processes that are natural generalizations of the Wiener process. There are several reasons to go outside the classical paradigm. Models based on L´evy processes allow one to treat situations leading to heavy-tailed distributions. Moreover, they allow exploiting the full strength of Markovian modelling because the most general Markov processes are solutions of stochastic differential equations driven by L´evy processes. Since L´evy processes admit jumps, they are well suited to describing prices quoted on exchanges in a discontinuous fashion. The mathematical theory of the bond market sets a specific area in financial mathematics. Its analysis involves an infinite dimensional setting because basic objects of the theory, bond prices and forward rates, are function-valued processes. Such a framework can hardly be found in classical stock market models. The research literature on the L´evy bond market is very extensive and growing with an increasing speed. The starting point was the seminal 1997 papers by Bj¨ork, Kabanov and Runggaldier [20] and Bj¨ork, Di Masi, Kabanov and Runggaldier [19] that laid down the foundations for the analysis of the bond market in a stochastic model with a general discontinuous noise and prompted further research in that

Preface

xv

direction. Important contributions describing basic properties of the bond market with L´evy noise are due to Eberlein, Jacod and Raible [48], [47]. Interesting results were published in particular by Filipovi´c, Tappe and Teichmann [52], [54], [56], [57]. Several issues were treated by the authors of the present book [5], [3], [7], [6], [8] and [9] and together with Jakubowski [76], [4]. As the results are mathematically rather involved, it seemed that a book on the subject giving solid foundations for future research would be a welcome contribution. There are rather few books containing material on L´evy modelling of the financial market and there is none devoted to the bond market. The well-known book by Cont and Tankov [29] deals with stock markets. Only in the final comments does it indicate L´evy bond markets as a possible direction of research. Similarly Applebaum [2] considers some problems of L´evy stock markets limiting his discussion of the bond market to some far-reaching suggestions. In the book [100] by Peszat and Zabczyk a more extensive treatment is available, but many questions were left for further study. The well-known books of Carmona and Tehranchi [25] and Filipovi´c [52] as well as part of the classical monograph of Bj¨ork [16] are devoted to the bond market, but all deal with models based on the Wiener process.

Aims of the Book Our first aim is to mathematically characterize those L´evy bond markets that are free of arbitrage. Intuitively, a market is arbitrage free if a trader is not able to generate profit without taking risk. A sufficient condition for that is the existence of the socalled martingale probability measure equivalent to the basic one. The second main concept we analyze is completeness of the market. Again, intuitively, a market is complete if a trader can construct a strategy that reproduces any prespecified financial contract. It turns out that a useful tool to construct arbitrage-free bond market models is provided by stochastic equations. The stochastic equations that appear here are nonlinear and sometimes with partial derivatives. Their analysis is one of the main novelties of the book. The analysis of the mentioned issues is mathematically rather involved. To make the material more accessible we begin by considering a discrete time setting. It is of independent interest, and almost all results from the continuous time framework are proven here in a more direct way.

Structure of the Book The book consists of four parts preceded by an Introduction that, in particular, contains some financial background. Part I deals with discrete time models and

xvi

Preface

it is aimed at those readers who have had no previous contact with mathematical finance. The randomness is generated by a sequence of independent identically distributed random variables, a counterpart of the increments of L´evy processes. The results described in this part suggest what can be obtained in the much more challenging continuous time setting. Part II is an overview of results from stochastic analysis required for the continuous time framework. In Part III we treat in detail bond markets driven by L´evy processes, covering such topics as non-arbitrage conditions including the derivation of the general Heath–Jarrow–Morton conditions as well as the existence of martingale measures and completeness of the models. Special attention is paid to the important class of models with affine term structure and general models with Markovian factors. In Part IV we construct arbitrage-free models with the use of stochastic partial differential equations with L´evy noise. The equations that appear there are of unusual type as their coefficients, both linear and nonlinear, are of nonlocal character.

Acknowledgements

It is a pleasure to thank our colleagues Tomas Bj¨ork, Nikos Frangos, Jacek Jakubowski, Szymon Peszat, Anna Rusinek and Thorsten Schmidt for discussions on the topics of this book. We also thank Jerzy Trzeciak for language consultations. The first author thanks Leipzig University and Warsaw University for good working conditions and the Institute of Mathematics Polish Academy of Sciences for constant support. The second author is grateful to his home institution, the Institute of Mathematics Polish Academy of Sciences for providing a stimulating research environment. Financial support from the Warsaw Center of Mathematical and Computer Sciences is gratefully acknowledged. Any comments and remarks from the readers are welcome and can be sent to [email protected].

Introduction

I.1 Bonds Bonds are financial assets issued by governments, central banks or companies. Their holders receive some fixed payments at future dates. The lifetime of a bond is specified by its maturity – the date when the nominal value of the bond is paid. All previous payments are called coupons and they are usually fixed as fractions of the nominal value of the bond. The payments received by the holder, although fixed, can, however, be influenced by the credit rating of the issuer. This means that in case of the issuer’s bankruptcy the promised payments can be reduced or even cancelled. There are many kinds of bonds depending on the length of maturity, the frequency of coupon dates and the credit rating of the issuer. Bonds with maturities between 2 and 5 years are called short-term bonds or bills, those with maturities between 6 and 12 years are medium-term bonds or notes. Maturities of the long-term bonds exceed 12 years but usually are not longer than 30 years. Perpetual bonds called also consols have infinite maturities, so they pay a stream of coupons forever. The credit rating of the issuer, which describes his/her default probability, is assigned by rating agencies and usually denoted by a combination of letters A, B, C, D corrected by + or −. The highest rank AAA is followed by AA+, AA and so on till D. Coupons of a bond with a high credit rating offer lower payments than those with a low credit rating but the probability that they will be paid without reduction is higher. Real gamblers who have no risk aversion may invest money in junk bonds offering profitable coupons that are, however, biased by a critical rating value. Bonds and related financial contracts constitute an enormous market with trading volume exceeding that of the shares. Instead of going deeper into classifying the variety of bonds, we will now focus on their mathematical description. In this book we consider zero coupon risk-free bonds with nominal value 1, which means that 1 unit of cash is paid to the holder at maturity. There are no coupons and the default probability of the issuer disappears. A bond with maturity T > 0 will also be called a T-bond as it is uniquely characterized by its maturity, and its price at time

2

Introduction

t ∈ [0, T] will be denoted by P(t, T). So, P(0, T) stands for the initial price of the T-bond and P(T, T) = 1 is its nominal value. The set of all maturities will be assumed to be [0, +∞), and by a bond market we mean the family of T-bonds with T ≥ 0. Our model framework with an infinite number of bonds is a kind of mathematical idealization of the real bond market where only a finite number of bonds are traded, but it can be justified by a huge variety of available bonds. Consideration of bonds without coupons is not really restrictive. In fact, every nonzero coupon bond can be represented as a combination of zero coupon bonds no matter what its coupon scheme. A property that does not feature in our study is the default possibility of the issuer. So, the standing assumption in the whole book is that the nominal value of each bond will be paid with probability one. The family of prices P(t, T),

t ∈ [0, T];

T≥0

is called the term structure of zero coupon bond prices. The number P(t, T) can be identified with a risk-free investment with two dates of payment given by the pair (t, T), where 0 ≤ t < T. Indeed, buying the T-bond for P(t, T) units of cash at time t provides the payoff P(T, T) = 1 at T. Since P(t, T) and the nominal value are known at time t, the deal is free of risk. Although P(0, T) and P(T, T) are known at t = 0, the price evolution t → P(t, T) on (0, T) is random and is affected by the state of the economy. One should realize that bonds and stocks represent two competitive parts of the security market that combine investment gain and risk in a different way. In a good economical situation the stock market is developing well and its low investment risk attracts investors. In this situation the bond market, to be competitive, must offer high gains, i.e. the difference between current prices and nominal values of bonds should be high. This means that bond prices are low. Conversely, high bond prices correspond to high uncertainty on the stock market related to economical perturbations.

I.2 Models It is of prime importance to develop stochastic models that describe the evolution of bond price processes in a way that reflects their real behaviour. Now we briefly introduce models investigated in the book. Heath–Jarrow–Morton Models A forward rate is a random function of two variables f (t, T) = f (ω, t, T),

t ∈ [0, T],

T ≥ 0,

such that for each t ≥ 0 the trajectory T → f (t, T) is known at time t. The bond prices are then given by P(t, T) = e−

T t

f (t,u)du

,

t ∈ [0, T],

T ≥ 0.

I.2 Models

3

The previous bond price formula reflects two important properties observed on the real market. The bond price P(t, T) behaves in a regular way in T and is chaotic in t providing that time fluctuations of the forward rate are sufficiently rough. In the seminal paper [67] of Heath, Jarrow and Morton the forward rate dynamics has the form df (t, T) = α(t, T)dt + σ (t, T)dW(t), f (0, T) = f0 (T),

t ∈ [0, T],

T ≥ 0,

T ≥ 0,

(I.2.1)

where W is a Wiener process. In this approach the forward rate is viewed as a family of stochastic processes t → f (t, T) parametrized by T ≥ 0. Then (I.2.1) is a system of separate differential equations with coefficients α(·, T), σ (·, T) and initial condition f0 (T) for each T. Our aim is to extend (I.2.1) by replacing W by an Rd -valued L´evy process Z = (Z1 , . . . , Zd ). Then (I.2.1) boils down to df (t, T) = α(t, T)dt +

d 

σi (t, T)dZi (t),

t ∈ [0, T],

T ≥ 0, (I.2.2)

i=1

f (0, T) = f0 (T),

T ≥ 0.

The equation (I.2.2) extends the previous model framework significantly by admitting a large class of noise distributions and incorporating new path properties of forward rates, like jumps. Factor Models Bond prices and forward rates can also be treated as functions of time to maturity. For a fixed date t we focus now on the functions x → P(t, t + x),

x → f (t, t + x),

x ≥ 0,

where x := T − t with T ≥ t. Modelling the shapes of the preceding functions and their stochastic evolution in time is encompassed by the factor models P(t, T) = F(T − t, X(t)),

f (t, T) = G(T − t, X(t)),

0 ≤ t ≤ T,

(I.2.3)

where F, G are deterministic functions and X is some stochastic process bringing randomness to the model. The process X is called a factor and should be interpreted as consisting of observed economical parameters. In particular, it can be given by the short-rate process R(t). We study models (I.2.3) where X is a Markov process and characterize admissible functions F, G in (I.2.3), in terms of the transition semigroup of X. Of prime interest are factors specified by stochastic equations, like the well-known Cox–Ingersol–Ross short-rate model √  dR(t) = (aR(t) + b)dt + c R(t)dW(t), R(0) = R0 , t > 0,

4

Introduction

or Vasiˇcek, Ho–Lee and Hull–White models (see Bj¨ork [16], Filipovi´c [52] for details). We go, however, beyond the continuous paths framework and deal also with multiplicative factors of the form dX(t) = aX(t)dt + bX(t)dZ(t), X(0) = x, as well as with the Ornstein–Uhlenbeck short-rate process dR(t) = (a + bR(t))dt + dZ(t),

R(0) = R0 ,

t ≥ 0,

where Z is a L´evy process and a, b some constants. Affine Term Structure Models In the affine term structure model the bond prices have the form P(t, T) = e−C(T−t)−D(T−t)R(t) ,

0 ≤ t ≤ T,

(I.2.4)

where C, D are deterministic regular functions and R stands for the short-rate process. In fact, (I.2.4) is a particular case of (I.2.3) with G(u, x) = C (u) + D (u)x

(I.2.5)

and the random factor given by the short-rate process R. The linear dependence over x in (I.2.5) implied by (I.2.4) allows us to characterize L´evy processes Z, which generate short rates of the form dR(t) = F(R(t))dt +

d 

Gi (R(t−))dZi (t),

t ≥ 0, R(0) = x

(I.2.6)

i=1

that are admissible for affine models. Among real valued L´evy martingales, the only ones turn out to be the Wiener process and the α-stable martingale with L´evy measure 1 ν(dy) = 1+α 1[0,+∞) (y)dy, α ∈ (1, 2). y In the multidimensional case the coordinates of Z can be given by the Wiener process, the α-stable martingales with α ∈ (1, 2), α-stable subordinators with α ∈ (0, 1) and an arbitrary subordinator that enters (I.2.6) in the additive way. We also present a general characterization of admissible Markov short rates that generate affine models in terms of their generators. This part of the material is based on the paper [53] of Filipovi´c and also provides a characterization of the functions C, D in (I.2.4). Constructing Models An efficient way to construct arbitrage-free models is by using the theory of partial differential equations for forward rate processes. The no-arbitrage requirement leads

I.3 Content of the Book

5

to equations with nonlocal and nonlinear coefficients. A typical example is the following equation for the forward rate  α  x ∂ r(t, x) = r(t, v)dv r(t, x) dt + r(t, x)dZ(t), x ≥ 0, t ≥ 0, r(t, x) + ∂x 0 where r(t, x) := f (t, t + x),

x ≥ 0, t ≥ 0,

and Z is an α-stable martingale. Equations of this type with the Wiener process Z were introduced by Musiela [97]. The equations prompt interesting mathematical questions about existence and uniqueness of solutions and their positivity, discussed in Part IV.

I.3 Content of the Book The book consists of four parts: (I) “Bond Market in Discrete Time”; (II) “Fundamentals of Stochastic Analysis”; (III) “Bond Market in Continuous Time”; and (IV) “Stochastic Equations in the Bond Market”. The first part has a more elementary character than the remaining three. It uses classical probability concepts and results rather than more advanced stochastic analysis, as in the rest of the book. The book ends with Appendices containing the proof of the martingale representation theorem in the pure jump case, material on generators of equations with L´evy processes and on evolution equations. Special care is devoted to the following models of the bond market: the HJM model in which forward rates are defined by stochastic equations; factor models in which price curves are moved by stochastic processes of economic factors; and affine models in which bond prices are exponential functions of the short rate. Part I starts from preliminaries on the discrete time financial market in Chapter 1. Arbitrage-free models are studied in Chapter 2. We derive, in particular, a discrete time version of the CIR equation of the continuous time theory. Practically, all Markovian short-rate processes of the affine term structure are determined. Completeness of the bond market is studied in Chapter 3. Bond curves are vectors with an infinite number of coordinates and only those models with curves evolving in finite dimensional spaces might be complete. Specific conditions for approximate completeness of the main models are deduced. Part II is divided into three substantial chapters. Chapter 4 recalls concepts and results from stochastic analysis like semimartingales, square integrable martingales and Doob–Meyer decomposition. Stochastic integration with respect to semimartingales and random measures as well as Ito’s formula are treated. They will be of constant use later. Chapter 5 concerns L´evy processes, our basic tool. We first apply the general stochastic analysis theory to this class of processes and describe specific

6

Introduction

subclasses. Then in Chapter 6 we formulate the integral representation theorem for local martingales with respect to the L´evy filtration due to Kunita. Essential, although rather classical elements of the proof, like chaos expansion and multiple Itˆo–Wiener integrals are presented in Appendix A. The second part of this chapter is concerned with Girsanov’s formula for densities of equivalent measures. Part III concerning the continuous time bond market starts with a mathematical description of the models and their elementary properties in Chapter 7. Arbitragefree Heath–Jarrow–Morton models of the bond market are analyzed in Chapter 8. The main results here are general non-arbitrage conditions of the HJM type for an arbitrary physical probability measure. Chapter 9 investigates the non-arbitrage problem when the models are given in the form of forward curves moved by Markovian factor processes. The main result is the term structure equation. Some applications to special factor processes, like multiplicative or Ornstein–Uhlenbeck processes are presented as well. Chapter 10 is devoted to non-arbitrage conditions for affine models of bond prices. It consists of two major sections concerned, respectively, with short rates given as solutions of general stochastic equations and short rates that are general Markov processes. We present results due to Filipovi´c. We also give a generalization to discontinuous short-rate processes of the Cox–Ingersol– Ross theorem. The final Chapter 11 is on completeness of the bond market. The hedging problem is formulated in terms of the solvability of the so-called hedging equation. It is discussed in various settings related to special forms of the L´evy process. Approximate completeness is discussed as well. Part IV focuses on building arbitrage-free markets through stochastic equations. In Chapter 12 the equations are introduced. General equations for the forward curve under the martingale measure are analyzed by the methods of stochastic evolution equations in Chapter 13. Conditions for local and global existence of solutions are established. Some applications to the so-called Morton–Musiela equation are presented as well. Chapter 14 treats the case when volatility in the HJM model is a linear function of the forward curve. Then the forward rate satisfies the so-called Morton’s equation. The equation has a unique solution for a large class of L´evy processes characterized in terms of the logarithmic growth conditions of the L´evy exponent. We develop the method introduced by Morton, who treated the Wiener case and obtained a negative result. The Morton–Musiela equation is treated in Chapter 15.

Part I Bond Market in Discrete Time

1 Elements of the Bond Market

Here we introduce basic concepts of the bond market model in discrete time, like bond prices, rates, portfolios and strategies. Contingent claims and nonarbitrage conditions are discussed as well.

1.1 Prices and Rates A bond market consists of four stochastic processes T ∈ N0 ,

P(t, T), B(t),

t ∈ N0 , T ∈ N0 ,

f (t, T), R(t),

t = 0, 1, . . . , T,

t = 0, 1, . . . , T,

t ∈ N0 ,

called, respectively, bond prices, bank account, forward rates and short rate, defined on a probability space (, F, P), equipped with a filtration {Ft , t ∈ N0 }, N0 := {0, 1, . . .}, and adapted to the filtration. This means that for each t ∈ N0 the random variables P(t, T), T ≥ t;

f (t, T), T ≥ t;

B(t);

R(t);

are Ft -measurable. Often Ft is assumed to be generated by P(t, T), T ≥ t: Ft = σ { P(s, T); s = 0, 1, . . . , t, T ≥ s}.

(1.1.1)

In this case the filtration is the minimal one. The value P(t, T) is interpreted as the price, at time t, of the bond that matures at time T, called maturity of the bond. That means that the owner of this T-bond will receive at time T the so-called nominal value, specified on the bond, which we assume to be 1. Hence it is natural to assume that P(t, T) ≥ 0

and

P(T, T) = 1.

(1.1.2)

10

Elements of the Bond Market

Other elements of the bond market are related to the bond prices in some specific ways and satisfy some natural requirements. The value B(t) is interpreted as the amount of money in the bank account at time t resulting from depositing 1 at time 0. It is convenient to assume that the deposit grows as a result of converting it at each moment t into bonds that mature at t + 1. Thus, if the deposit at time t was D, then one buys D·

1 , P(t, t + 1)

t + 1-bonds and the deposit at time t + 1 is D·

1 1 · P(t + 1, t + 1) = D · . P(t, t + 1) P(t, t + 1)

This leads to the recurrent identity B(t + 1) = B(t)

1 , P(t, t + 1)

(1.1.3)

and to the definition of the short rate R(t) eR(t) :=

1 , P(t, t + 1)

(1.1.4)

which clearly shows that R is an adapted process. It follows also that B(0) = 1,

B(t) =

1 , P(0, 1)P(1, 2) · . . . · P(t − 1, t)

t = 1, 2, . . . ,

(1.1.5)

which implies that B(t) is Ft−1 -measurable for each t ≥ 1, hence B is a predictable process. As a consequence of (1.1.5) and (1.1.4) one obtains

t−1

B(t) = e

s=0 R(s)

,

t = 1, 2, . . . ,

B(0) = 1.

(1.1.6)

ˆ T) defined by Of great importance are the so-called discounted bond prices P(t, ˆ T) := P(t,

P(t, T) , B(t)

t ≤ T.

(1.1.7)

The forward rate process f (t, T) is determined by the identity P(t, T) = e−

T−1 s=t

f (t,s)

,

t ≤ T.

(1.1.8)

Thus, for t ≤ T, e−f (t,T) =

P(t, T + 1) , P(t, T)

and

f (t, T) = ln

P(t, T) . P(t, T + 1)

In particular, f (t, t) = R(t), t ∈ N0 .

(1.1.9)

1.1 Prices and Rates

11

The relations (1.1.2), (1.1.4), (1.1.6), (1.1.8) and (1.1.9) are assumed to be always true. ˆ T) and In fact, the definitions of bond prices P(t, T), discounted bond prices P(t, forward rates f (t, T) can be extended to the case t > T by assuming that nominal values of bonds are transferred into the bank account at their maturity times. Thus, for t > T one defines P(t, T) = P(T, T) · e

t−1 s=T

R(s)

t−1

=e

T−1

s=0 R(s)−

R(s)

s=0

= B(t)/B(T).

Consequently, ˆ T) = P(t, T)/B(t) = 1/B(T), P(t,

t > T.

Defining forward rates f (t, T) for t > T by the same formula as for t ≤ T, i.e. (1.1.8), we obtain f (t, T) = ln

P(t, T) B(t)/B(T) = ln = ln eR(T) = R(T), P(t, T + 1) B(t)/B(T + 1)

t > T. (1.1.10)

Notice that (1.1.10) leads to the following formula for the discounted bond prices ˆ T) = e− P(t, = e−

t−1

s=0 R(s)

T−1 s=0

P(t, T) = e−

f (t,s)

,

t−1

s=0 f (t,s)

· e−

T−1 s=t

f (t,s)

t, T ∈ N0 .

(1.1.11) (1.1.12)

Summarizing, the extended processes are given by P(t, T) = B(t)/B(T),

ˆ T) = 1/B(T), P(t,

f (t, T) = f (T, T) = R(T),

t > T. (1.1.13)

Typically bond prices are bounded, in the sense P(t, T) ≤ 1,

t ≤ T,

(1.1.14)

and monotone with respect to T, i.e. P(t, T) ≥ P(t, T + 1).

(1.1.15)

This means that bonds with longer maturities should be cheaper. If (1.1.14) and (1.1.15) hold, then the market will be called regular. On the real bond market, both properties, (1.1.14) and (1.1.15), can, however, break down when the economy slumps. It is clear that (1.1.15) is equivalent to the positivity of forward rate, i.e. f (t, s) ≥ 0,

t ≤ s.

(1.1.16)

If the forward rate is positive, then (1.1.14) holds as well, so positive forward rates generate regular markets.

12

Elements of the Bond Market

1.2 Models of the Bond Market There are four models of the bond market discussed in the sequel: (a) The discrete time HJM model is a version of the famous Heath, Jarrow and Morton (HJM) model in continuous time introduced in [67]. One stipulates that for each T, the forward rates f (t, T) change stochastically in t and f (t + 1, T) = f (t, T) + α(t, T) + σ (t, T), ξt+1 ,

0 ≤ t < T.

(1.2.1)

The previously mentioned ξ1 , ξ2 , . . . are independent identically distributed random variables taking values in U = Rd . Its partial sum Z(t) = ξ1 + ξ2 + · · · + ξt ,

t = 1, 2, . . . ,

(1.2.2)

can be viewed as a discrete time counterpart of a L´evy process in continuous time. For each T the processes α(·, T) and σ (·, T) are adapted to the filtration {Ft }, which is given by F0 = {, ∅},

Ft := σ {ξ1 , ξ2 , . . . , ξt },

t = 1, 2, . . . ,

and ·, · stands for the scalar product in U. (1.2.1) means that the future value of the forward rate f (t + 1, T) arises from the current one f (t, T) by shifting it by α(t, T), which is known, and perturbing it by σ (t, T), ξt+1 , which is random. This interpretation justifies α(·, ·), σ (·, ·) to be called drift, or volatility of the forward rate. The initial forward curve f (0, T), T = 0, 1, . . ., is regarded as known at time zero. It is often convenient to study the bond market in the moving frame based on the so-called Musiela parametrization and write f (t, ·) in terms of time to maturity T − t. Thus, one defines r(t, j) := f (t, t + j),

j = 0, 1, . . . .

(1.2.3)

In particular, the short rate is given by R(t) = r(t, 0),

t ≥ 0.

(b) In the forward rate model with Markovian trace one defines the γ -trace of the forward rate r by rγ (t) := (r(t, 0), . . . , r(t, γ )),

t = 0, 1, . . . ,

(1.2.4)

where γ ∈ N0 := {0, 1, 2, . . .}, which is simply a random vector consisting of the first γ coordinates of r given by (1.2.3). The model is based on the assumption γ +1 that rγ is a Markov chain in the space R+ and examined with the use of its transition operator P(·, ·). The particular case with γ = 0 corresponds to the Markovian short-rate process R(t) = r0 (t), t ≥ 0.

1.3 Portfolios and Strategies

13

(c) The affine model, called also affine term structure, is a particular case of the model with Markovian trace in which the forward rate is of the form r(t, k) = C(k + 1) − C(k) + (D(k + 1) − D(k)), rγ (t) ,

t≥0

(1.2.5)

for some deterministic functions C, D such that C(0) = 0, D(0) = 0. Thus, the shape of the forward curve k → r(t, k) is given and controlled in a linear way by the trace only. Condition (1.2.5) can be equivalently written with the use of the bond prices, i.e. P(t, T) = e−C(T−t)− D(T−t),r

γ (t)

,

t ≤ T.

(d) In the Markovian factor model, which is a generalization of the affine model, one requires that P(t, T) = F(T − t, X(t)),

t, T = 0, 1, . . . , t ≤ T,

or, equivalently, f (t, T) = G(T − t, X(t)),

t, T = 0, 1, . . . , t ≤ T,

where X(t), t ≥ 0 is a Markov chain on (E, E), called factor, and F(·, x), G(·, x) are deterministic functions describing possible shapes of the bond curves, or forward curves for a fixed value x of the factor. It is clear that F(0, x) = 1, F(k, x) = e−

k−1 j=0

G(j,x)

, k = 1, . . . , x ∈ E.

In particular, the short-rate process may serve as the factor. The basic issue in the factor model is the interplay between the transition operator of the factor process and the shapes of forward or bond curves.

1.3 Portfolios and Strategies A portfolio at time t ≥ 0 is a family of Ft -measurable random variables   b(t), ϕt (j)j=t+2,t+3,..., , where b(t) is the amount of money deposited in the bank account and ϕt (j) stands for the number of bonds that will mature at time j, bought by the investor. Note that at time t one can trade with bonds maturing at t + 2 or later. The reason for that is that the evolution at time t of bonds that have matured or will mature at t + 1 is governed by the short-rate process R(t). Indeed, by (1.1.4), P(t, t + 1) = e−R(t) =

1 , B(t)

and for bonds that have matured earlier than t + 1 we can assume that their nominal values are automatically transferred to the bank. Hence investing in those bonds is

14

Elements of the Bond Market

equivalent to putting money in the savings account. Finite portfolios are based on a finite number of bonds, that is, ϕt (j) = 0 for all j > j(t) and some number j(t). The proper framework for the bond market relies, however, on portfolios involving an infinite number of bonds, i.e. ϕt = {ϕt (j)} may have infinitely many non-vanishing coordinates. Then the corresponding portfolio wealth at time t is given by  X(t) = b(t) + ϕt (j)P(t, j), t = 0, 1, . . . , (1.3.1) j≥t+2

providing that the above sum is finite. To guarantee that, let us assume that forward rates are positive. It follows then from the formula P(t, T) = e−

T−1 s=t

f (t,s)

≤ 1,

t≤T

that the price process P(t) := (P(t, t + 2), (t, t + 3), . . .) takes values in the set of bounded sequences m, i.e. in m := {x = (x1 , x2 , . . .) : sup | xi |< +∞}. i

Consequently, portfolios may take values in

l1 ,

l1 := {x = (x1 , x2 , . . .) :

where

+∞ 

| xn |< +∞},

n=1

that is, ϕt = (ϕt (t + 2), ϕt (t + 3), . . .) ∈ l1 , because then        ϕt (j)P(t, j) ≤ | ϕt (j) |=| ϕt |l1 < +∞,  j≥t+2

j≥t+2

and the portfolio wealth is also well defined. By (1.3.1), portfolio wealth at time t + 1 equals  X(t + 1) = b(t + 1) + ϕt+1 (j)P(t + 1, j). j≥t+3

If X(t + 1) arises from X(t) only through the fluctuations of bond prices and bank account, then we say that the self-financing condition at time t + 1 is preserved. Specifically, this means that  X(t + 1) = b(t)eR(t) + ϕt (j)P(t + 1, j). (1.3.2) j≥t+2

A self-financing strategy is a sequence of portfolios (b(t), ϕt ), t = 0, 1, 2, . . . for which the self-financing condition holds at any time. If this is the case, then from the identity X(t) = X(0) +

t−1  (X(s + 1) − X(s)), s=0

1.3 Portfolios and Strategies

15

and by (1.3.1), (1.3.2), we obtain X(t + 1) − X(t) = (eR(t) − 1)b(t) +



ϕt (j)(P(t + 1, j) − P(t, j)).

j≥t+2

Using more compact notation, we have thus X(t + 1) = (eR(t) − 1)b(t) + ϕt , P(t + 1) , where P(t + 1) := P(t + 1, j) = P(t + 1, j) − P(t, j),

j ≥ t + 2.

So, we see that the portfolio wealth of a self-financing strategy (b(t), ϕt ) evolves according to the formula X(t) = X(0) +

t−1 t−1   (eR(s) − 1)b(s) + ϕs , P(s + 1) . s=0

(1.3.3)

s=0

Since P(s), for each s = 0, . . . , t, takes values in the space of bounded sequences m, the last sum is clearly well defined. A self-financing strategy is thus defined by (1.3.1) and (1.3.3). In the sequel we will use the following alternative characterization of the self-financing condition in terms of the discounted portfolio wealth and the discounted bond prices ˆ := X(t)

X(t) , B(t)

ˆ j) := P(t,

P(t, j) , B(t)

t = 0, 1, . . . , j = 0, 1, . . . .

It is useful to note and easy to check with the use of (1.1.13), that for any s = 0, 1, . . ., ˆ + 1, j) = 0, P(s

j = 0, 1, . . . , s + 1,

where ˆ + 1, j) := P(s ˆ + 1, j) − P(s, ˆ j), s = 0, 1, . . . , j = 0, 1, . . . . P(s Proposition 1.3.1 (a) If (b(t), ϕt ) is a self-financing strategy, then ˆ = X(0) + X(t)

t−1  ˆ + 1) , ϕs , P(s

t = 1, 2, . . . .

(1.3.4)

s=0

(b) For any initial capital x and a strategy ϕt there exists a unique b(t) such that the strategy (b(t), ϕt ) is self-financing and X(0) = x. Let us notice that the sum in (1.3.4) is well defined, because ϕs , for each s, takes ˆ + 1) in m. values in l1 and P(s

16

Elements of the Bond Market

Proof

(a) From (1.3.1) and (1.3.2) we have   b(t) P(t, j) b(t) ˆ = ˆ j), X(t) ϕt (j) ϕt (j)P(t, + = + B(t) B(t) B(t) j≥t+2

ˆ + 1) = eR(t) X(t

j≥t+2

  b(t) P(t + 1, j) b(t) ˆ + 1, j), ϕt (j) ϕt (j)P(t + = + B(t + 1) B(t + 1) B(t) j≥t+2

which implies ˆ + 1) = X(t



j≥t+2

ˆ + 1, j) = ϕt , P(t ˆ + 1) . ϕt (j)P(t

j≥t+2

ˆ Formula (1.3.4) can be obtained by adding the increments of X(t). (b) The sequence b(·) is defined inductively by setting first b(0):  ϕ0 (j)P(0, j), b(0) = x − j≥2

and then using the equation:   ϕt+1 (j)P(t + 1, j) = b(t)eR(t) + ϕt (j)P(t, j). b(t + 1) + j≥t+3

j≥t+2

The strategy (b(t), ϕt ) is self-financing and it is the only self-financing strategy that satisfies X(0) = x. Proposition 1.3.1 is of great importance because it simplifies the problem of construction of self-financing strategies (b(t), ϕt ) to a free choice of the initial capital X(0) = x and investment in bonds, i.e. (ϕt ) only. In the sequel we will often identify any self-financing strategy (b(t), ϕt ) with the pair (X(0), ϕt ).

1.4 Contingent Claims Let us consider a financial contract that obliges its seller to pay at a fixed future date t > 0 some random amount of money X to the buyer of the contract. The contract is formulated at time 0 and X is assumed to be an Ft -measurable random variable. It is called contingent claim at time t for short, claim at t. Typically, claims are some functions of the prices of bonds that are available on the market up to time t. In this case, the filtration is assumed to be the minimal one, i.e. Fs = σ { P(u, T); u = 0, 1, . . . , s, T = 0, 1, 2, . . .}. The goal of the seller might be to find a self-financing strategy (b(u), ϕu ) such that for the corresponding wealth process, X(s) = X(0) +

s−1 s−1   (eR(u) − 1)b(u) + ϕu , P(u + 1) , s=0

u=0

s = 0, 1, . . . , t,

1.4 Contingent Claims

17

one has at time t X(t) = X.

(1.4.1)

Recall that (b(u), ϕu ) are adapted to {Fu } and ϕu lives in l1 . Condition (1.4.1) means that the seller could eliminate the risk arising from paying X at time t by starting from X(0) and following the strategy (b(u), ϕu ). Therefore, X(0) is called a fair price of the contract. A strategy (b(u), ϕu ) for which (1.4.1) holds is called a replicating strategy, or hedging strategy, for X and the claim X is called attainable. It is convenient to reformulate the problem of looking for replicating strategies in ˆ j) = P(s, j)/B(s). terms of discounted values, i.e. Xˆ = X/B(t), P(s, Proposition 1.4.1 Let X be a contingent claim. (a) If (b(s), ϕs ) is a replicating strategy for X, then the discounted claim admits the representation Xˆ = X(0) +

t−1 

ˆ + 1) . ϕs , P(s

(1.4.2)

s=0

(b) If there exists a pair (x, ϕs ) such that Xˆ = x +

t−1  ˆ + 1) ϕs , P(s

(1.4.3)

s=0

then there exists a replicating strategy (b(s), ϕs ) for X with initial wealth X(0) = x. Proof (a) If (b(s), ϕs ) is a replicating strategy for X, then the related wealth process clearly satisfies ˆ X(t)/B(t) = X. By Proposition 1.3.1 we obtain (1.4.2). (b) For the pair (x, ϕs ) satisfying (1.4.3) we can find, by Proposition 1.3.1, b(s) such that (b(s), ϕs ) is self-financing and X(0) = x. Again, by Proposition 1.3.1, the final ˆ discounted wealth process of this strategy X(t) equals the right side of (1.4.3). So, ˆ ˆ which implies that X(t) = X. Thus, (b(s), ϕs ) is a (1.4.3) means that X(t) = X, replicating strategy for X. Proposition 1.4.1 is a key result for the problem of determining replicating strategies. It allows us to forget the self-financing condition and first find any pair (x, ϕt ) satisfying (1.4.3). If we do this, then it is always possible to construct a replicating strategy for X.

18

Elements of the Bond Market

1.5 Arbitrage A self-financing strategy (X(0), ϕt ) is an arbitrage (or arbitrage strategy) if, for some t0 > 0, the corresponding portfolio wealth process satisfies P(X(t0 ) ≥ 0) = 1,

X(0) = 0,

P(X(t0 ) > 0) > 0.

(1.5.1)

The preceding conditions can be interpreted as a risk-free possibility of realizing positive gain starting from zero initial endowment. The possibility of constructing such strategies in the model should be excluded. A bond market that does not allow arbitrage strategies is called arbitrage free. Let us first specify two classes of self-financing strategies that will be used for examining the absence of arbitrage. A self-financing strategy (ϕt ) belongs to A1 if there exist constants M and K such that, for each t, | ϕt (j) |≤ M,

j ≥ t + 2,

and

ϕt (j) = 0

for j ≥ K.

Clearly, any element of A1 is a finite portfolio. The class A2 , by definition, consists of self-financing strategies that are bounded in l1 , i.e. for some M and each t ≥ 0  | ϕt (j) |≤ M. | ϕt |l1 = j≥t+2

One of the main problems of the bond market theory is to find sufficient and necessary conditions under which the market is arbitrage free. We present two basic and rather simple results on arbitrage-free markets. Theorem 1.5.1 Let us assume that, for any T ∈ N0 , the discounted price process of the T-bond is a martingale. Then there are no arbitrage strategies in the class A1 . If, additionally, forward rates are positive, then there are no arbitrage strategies in the class A2 . Proof Let (ϕt ) be a self-financing strategy. The corresponding discounted portfolio ˆ is a martingale if and only if wealth X(t) ˆ + 1) − X(t) ˆ | Ft ) = 0, E(X(t or, by (1.3.4), equivalently, ˆ + 1) | Ft ) = E E( ϕt , P(t

 

t = 0, 1, 2, . . . ,

   ˆ + 1, j) − P(t, ˆ j) | Ft ϕt (j) P(t

j≥t+2

=



  ˆ +1, j) − P(t, ˆ j) | Ft = 0, ϕt (j)E P(t

t = 0, 1, 2, . . . ,

j≥t+2

providing that exchange of summation and conditional expectation is allowed. This is clearly the case when ϕt belongs to A1 . The same is true if forward rates are

1.5 Arbitrage

19

ˆ T) ≤ 1 for all t ≤ T. Since P(t, ˆ j) is positive and ϕt belongs to A2 because then P(t, a martingale for each j,   ˆ + 1, j) − P(t, ˆ j) | Ft = 0, E P(t ˆ is a martingale. and it follows that for both classes A1 and A2 the process X(t) Let us assume that ϕ is an arbitrage strategy. Then, by passing to discounted values in (1.5.1), for some t0 > 0, X(0) = 0,

ˆ 0 ) ≥ 0) = 1, P(X(t

ˆ 0 ) > 0) > 0. P(X(t

ˆ is a martingale, Since X(t) ˆ 0 ) = E[X(t ˆ 0 )1 ˆ 0 = EX(t {X(t0 )>0} ] > 0, which is a contradiction. An important sufficient condition for the absence of arbitrage can be formulated in terms of the existence of the so-called martingale measure. A measure Q on (, F) is a martingale measure for a bond market defined on (, F, {Ft } , P) if Q(A) = 0

⇐⇒

P(A) = 0,

A ∈ F,

and, for each T > 0 the process ˆ T), 0 ≤ t ≤ T, P(t,

is a martingale under Q.

In fact, in Theorem 1.5.1 we assumed that P is a martingale measure. However, each step in the proof remains true if we replace P with an arbitrary martingale measure. Hence, the following generalization of Theorem 1.5.1 is true. Theorem 1.5.2 Let us assume that there exists a martingale measure Q. Then there are no arbitrage strategies in the class A1 . If, additionally, forward rates are positive, then there are no arbitrage strategies in the class A2 . A Counterexample In the classical stock market, where the number of trading assets is finite, the absence of arbitrage implies the existence of a martingale measure. This fact constitutes the well-known First Fundamental Theorem of Asset Pricing. In the bond market setting, where the number of tradeable bonds is infinite, this implication turns out not to be true. We will construct an example of an arbitrage-free regular bond market that admits no martingale measures. Proposition 1.5.3 There exists a one-period regular bond market that does not admit a martingale measure and is arbitrage free in the class of strategies that are only assumed to generate finite initial wealth.

20

Elements of the Bond Market

Proof We adapt here the idea from the example presented in Schachermayer [115] to the bond market setting. The prices at time t = 0 are given by a deterministic sequence P(0, T), T = 0, 1, 2, .. satisfying P(0, 0) = 1,

0 < P(0, T) ≤ 1,

P(0, T) ≥ P(0, T + 1),

T = 0, 1, . . . . (1.5.2)

For regularity we need the random sequence P(1, T), T = 1, 2, . . ., to satisfy P(1, 1) = 1,

0 < P(1, T) ≤ 1,

P(1, T) ≥ P(1, T + 1),

T = 1, 2, . . . . (1.5.3)

In fact, (1.5.3) can be expressed in terms of increments of the discounted prices ˆ T) − P(0, ˆ T) = P(1, T)P(0, 1) − P(0, T), η(T) := P(1,

T = 1, 2, . . . .

Since P(1, T) =

η(T) + P(0, T) , P(0, 1)

T = 1, 2, . . . ,

(1.5.3) is equivalent to η(1) = 0,

η(T) + P(0, T) > 0,

η(T + 1) − η(T) ≤ P(0, T) − P(0, T + 1),

T = 1, 2, . . . .

(1.5.4)

The model is defined on the probability space  = {ω1 , ω2 , . . .} of natural numbers, i.e. ωk = k, k = 1, 2, . . ., with the measure P({ωk }) = 21k , k = 1, . . .. The initial prices are given by P(0, T) = and η(T) by

η(1) ≡ 0,

η(T)(ωk ) :=

1 , 2T

T = 0, 1, . . . ,

⎧ ⎪ ⎪ ⎨

for k = T,

⎪ ⎩ 0

elsewhere,

1 nT −1 ⎪ nT

for k = T + 1,

T = 2, 3, . . . ,

where n > 4. It is clear that (1.5.2) is satisfied. Since η(1) = 1 > −P(0, 1),

η(T) ≥ −

1 1 > − T = −P(0, T), T n 2

T = 2, 3, . . . ,

and η(2) − η(1) ≤ η(T + 1) − η(T) ≤

1 1 < 2 = P(0, 1) − P(0, 2), 2 n 2

1 n+1 1 1 + T = T+1 ≤ T+1 = P(0, T) − P(0, T + 1), n nT+1 n 2 T = 2, 3, . . . ,

1.5 Arbitrage

21

so (1.5.4) is satisfied as well. Let us assume that Q is a martingale measure. Then 1 1 − qT+1 T , T n n

0 = EQ [η(T)] = qT

T = 2, 3, . . . ,

where qk := Q({ωk }), k = 1, 2, . . .. But this implies that qT = qT+1 for T = 2, 3, . . .,

which is impossible because +∞ k=1 qk = 1. Now let us assume that ϕ(T), T = 1, 2, . . . is an arbitrage strategy. The discounted portfolio wealth at time t = 1 satisfies ˆ X(1) =

+∞ 

ϕ(T)η(T) ≥ 0.

T=1

In particular, ˆ X(1)(ω 2 ) = ϕ(2)

1 ≥0 n2

and ˆ X(1)(ω 3 ) = −ϕ(2)

1 1 + ϕ(3) 3 ≥ 0. n2 n

Consequently, ϕ(2) ≥ 0 and ϕ(3) ≥ nϕ(2). By induction, we obtain ϕ(T) ≥ nT−2 ϕ(2), T = 2, 3, . . .. Then the initial wealth equals +∞  T=1

+∞

+∞

T=2

T=2

 nT−2 1  1 ϕ(1) ϕ(T)P(0, T) = ϕ(1) + ϕ(T) T ≥ = +∞. + ϕ(2) 2 2 2 2T

This shows that there are no arbitrage strategies with final initial capital. Remark 1.5.4 In the literature on financial modelling, a more general definition ˆ T) of a martingale measure Q can be found. Very often one only requires that P(t, are local martingales under Q. In fact, in the present setting both definitions are ˆ T) is positive and Proposition 1.5.5. equivalent. This follows from the fact that P(t, Proposition 1.5.5 martingale.

If X(t), t = 0, 1, 2, . . . , is a positive local martingale, then it is a

Proof Let {τn } be a localizing sequence of stopping times, that is, τn ≤ τn+1 , τn ↑ +∞ and X(τn ∧ t) t = 0, 1, . . . , is a martingale for each n. In particular, X0 is integrable. Since X is positive, application of Fatou’s lemma in the formula E[X(τn ∧ t) | Ft−1 ] = X(τn ∧ (t − 1)),

t = 1, 2, . . . , n = 1, 2, . . . ,

yields E[X(t) | Ft−1 ] ≤ X(t − 1),

t = 1, 2, . . . ,

(1.5.5)

22

Elements of the Bond Market

and X(0) ≥ E[X(1)] ≥ . . . . Since | X(τn ∧ t) |≤| X(1) | + · · · + | X(t) | and

  E | X(1) | + · · · + | X(t) | ≤ tX(0) < +∞,

we let again n ↑ +∞ in (1.5.5) and, by dominated convergence, obtain E[X(t) | Ft−1 ] = X(t − 1),

t = 1, 2, . . . .

2 Arbitrage-Free Bond Markets

In this chapter we characterize bond markets that are arbitrage free. Conditions for the existence of martingale measures are studied. Special attention is paid to Markovian models of forward rates, in particular to factor models and affine models.

2.1 Martingale Modelling Economists believe that the market economy does not allow arbitrage and therefore realistic models of bond prices should be arbitrage free (see (1.5.1)). Sometimes models satisfy a stronger condition than the existence of a martingale measure (see Theorem 1.5.2), which we denote (MP) as abbreviation for martingale prices: For arbitrary T = 1, 2, . . . the discounted bond price process ˆ T) = P(t,

P(t, T) , B(t)

t = 0, 1, . . . , T

(MP)

is a martingale. So, (MP) tells us that under the original measure P the discounted bond prices are martingales on the underlying probability space (, F, Ft ; t = 0, 1, . . .). Although (MP) is a strong requirement, theoretical models satisfying (MP) are important for determining prices of contingent claims. Let X be an FT ∗ -measurable, with some T ∗ > 0, contingent claim that is attainable. Then, by Proposition 1.4.1, its discounted value can be represented in the form x+

∗ −1 T

ˆ + 1) − P(s) ˆ ϕ(s), P(s = Xˆ

(2.1.1)

s=0

for some adapted process ϕ(s), and x in (2.1.1) defines the price of X (see Section 1.4). Taking expectations in (2.1.1) yields ˆ = E[Xe− x = E[X]

T ∗ −1 s=0

R(s)

],

which is a simple formula for the fair price of the claim X.

(2.1.2)

24

Arbitrage-Free Bond Markets

Additional Markovian type assumptions under (MP) make the model easier to handle and, in particular, lead to more explicit formulas for financial quantities like x in (2.1.2). If the bond market does not satisfy (MP) but one can modify it by measure change such that (MP) holds under a new measure, then one can still use (2.1.2) for calculating x. Hence we will consider also a weaker condition than (MP), which also precludes arbitrage. We denote it by (MM) as abbreviation for martingale measure property. There exists a measure Q ∼ P such that for each T = 1, 2, . . ., ˆ T), P(t,

t = 0, 1, . . . , T

(MM)

is a Q-martingale. In this chapter we examine conditions (MM) and (MP) in models introduced in Section 1.2. All of them are formulated in terms of forward rates and incorporate some kind of Markovianity concepts.

2.2 Martingale Measures for HJM Models In describing martingale measures in the HJM model it is convenient to extend the definition of the forward rate f (t, T) also for t > T. As described in Section 1.1 this can be done by assuming that nominal values of bonds are kept on the bank account after their maturity times. This boils down to writing the HJM model (see Section 1.2 (a)) in the form f (t + 1, T) = f (t, T) + α(t, T) + σ (t, T), ξt+1 ,

t, T = 0, 1, . . . ,

(2.2.1)

where we put α(t, T) = 0,

σ (t, T) = 0,

t ≥ T.

Consequently, f (s, t) = f (t, t) = R(t), and ˆ T) = e− P(t,

T−1 s=0

f (t,s)

,

t 0. So, we are looking for a measure Q such that each discounted T-bond price, ˆ T), P(t,

t = 0, 1, . . . , T ∧ T ∗ ,

2.2 Martingale Measures for HJM Models

25

is a Q-martingale for T = 0, 1, 2, . . . . Since Q is equivalent to P, its density ρ = ρT ∗ is FT ∗ -measurable and there exists a function ψ such that ρ = ψ(ξ1 , ξ2 , . . . , ξT ∗ ).

(2.2.3)

The corresponding density process ρt :=

dQ   , dP Ft

t = 0, 1, . . . , T ∗

has the form ρt = ψt (ξ1 , ξ2 , . . . , ξt ), where {ψt } is given by ψt (x1 , x2 , . . . , xt ) =

t = 0, 1, . . . , T ∗ ,

(2.2.4)

 ψ(x1 , x2 , . . . , xt , y1 , y2 , . . . , yT ∗ −t )μ(dy1 ) . . . μ(dyT ∗ −t ),

t = 0, 1, . . . , T ∗ − 1 and ψT ∗ = ψ. Above μ stands for the distribution of ξ1 under P. We use also the Laplace exponent ϕξ and the Laplace transform Lξ of ξ1 which are given by Lξ (u) = E[e− u,ξ ] = eϕξ (u) ,

u ∈ U = Rd ,

(2.2.5)

providing that the expectation above is finite. The following result shows how ψ, α and σ should be related to each other to guarantee that ψ corresponds to a martingale measure. Specifically, it provides conditions for the (MM) property to be satisfied. Theorem 2.2.1 Let Q be a measure such that Q ∼ P and ψ be the function describing its density via (2.2.3). Let us define the functions  ψt+1 (x1 , x2 , . . . , xt , y) μ(dy), ψt,t+1 (x1 , x2 , . . . , xt , λ) := e− λ,y · ψt (x1 , x2 , . . . , xt ) U t = 0, 1, . . . , T ∗ − 1. ˆ T), T = 0, 1, 2, . . . are Then the processes of discounted bond prices P(t, Q-martingales if and only if  

ψt,t+1 (ξ1 , . . . , ξt , Ts=t σ (t, s)) α(t, T) = ln , t = 0, 1, . . . , T − 1, P − a.s.

T−1 ψt,t+1 (ξ1 , . . . , ξt , s=t σ (t, s)) (2.2.6) ˆ T), T = 0, 1, 2, . . . are P-martingales if and only if In particular, P(t,    T T−1   α(t, T) = ϕξ σ (t, s) − ϕξ σ (t, s) , t = 0, 1, . . . , T − 1. (2.2.7) s=t

s=t

26

Arbitrage-Free Bond Markets

ˆ T) is a Q-martingale if and only if P(t, ˆ T)ρt is a Proof We use the fact that P(t, P-martingale. By (2.2.2), (2.2.1) and (2.2.4) we have ˆ + 1, T) | Ft ] = E[e− E[ρt+1 P(t = e−

T−1 s=0

T−1 s=0

f (t+1,s) ψ t+1 (ξ1 , ξ2 , . . . , ξt+1 ) | Ft ]



f (t,s) ψ (ξ , ξ , . . . , ξ ) · E e− t 1 2 t

ˆ T)e− = ρt P(t,

T−1 s=0



α(t,s) E e−

T−1 s=0

T−1 s=0

  Ft

{α(t,s)+ σ (t,s),ξt+1 } ψt+1 (ξ1 , ξ2 , . . . , ξt+1 ) 

ψt (ξ1 , ξ2 , . . . , ξt ) 



σ (t,s),ξt+1 ψt+1 (ξ1 , ξ2 , . . . , ξt+1 )  F .  t ψt (ξ1 , ξ2 , . . . , ξt )

T−1 Since the conditional expectation above equals ψt,t+1 (ξ1 , ξ2 , . . . , ξt , s=0 σ (t, s)), ˆ T)ρt is a martingale if and only if we see that P(t,   T−1

 − T−1 α(t,s) s=t e · ψt,t+1 ξ1 , ξ2 , . . . , ξt , σ (t, s) = 1, t = 0, 1, . . . , T − 1, P − a.s. s=t

It follows from the formula above that   T−1 T−1   α(t, s) = ln ψt,t+1 (ξ1 , . . . , ξt , σ (t, s)) , s=t

t = 0, 1, . . . , T − 1,

(2.2.8)

s=t

and repeating the same arguments for the (T + 1)-bond we obtain   T T   α(t, s) = ln ψt,t+1 (ξ1 , . . . , ξt , σ (t, s)) , t = 0, 1, . . . , T. s=t

(2.2.9)

s=t

Subtracting (2.2.8) from (2.2.9) yields (2.2.6). ˆ T) are P-martingales we put ψ ≡ 1. Then ψt ≡ 1 To treat the case when P(t, ∗ for each t = 0, 1, . . . , T and consequently ψt,t+1 (x1 , x2 , . . . , xt , λ) = eϕξ (λ) . The assertion follows directly from (2.2.6). It is of interest to describe models for which the drift α(t, T) is a deterministic function of volatilities and does not depend on the past noise ξ1 , ξ2 , . . . , ξt . With the use of Theorem 2.2.1 we get the following result. Proposition 2.2.2 Let {ht } be a sequence of measurable functions satisfying  ht (y)μ(dy) = 1, t = 1, 2, . . . , T ∗ . ht (y) > 0, μ − a.s., U

Let us define 

e− λ,y ht+1 (y)μ(dy),

gt (λ) := U

λ ∈ U,

t = 0, 1, . . . ., T ∗ − 1.

2.2 Martingale Measures for HJM Models Then the model with drift given by 

 gt ( Ts=1 σ (t, s)) α(t, T) = ln ,

T−1 gt ( s=1 σ (t, s))

t = 0, 1, . . . , T − 1,

T = 0, 1, . . . ,

27

(2.2.10)

satisfies (MM). Moreover, there exists a martingale measure Q such that ξ1 , ξ2 , . . . , ξT ∗ are independent under Q. Proof

Let us notice that for ψ(x1 , x2 , . . . , xT ∗ ) := h1 (x1 )h2 (x2 ) . . . hT ∗ (xT ∗ ),

(2.2.11)

one obtains ψ0 = 1,

ψt (x1 , x2 , . . . , xt ) = h1 (x1 )h2 (x2 ) . . . ht (xt ),

and consequently



ψt,t+1 (x1 , . . . , xt , λ) =

e− λ,y ·

U

 =

t = 1, 2, . . . , T ∗ ,

ψt+1 (x1 , x2 , . . . , xt , y) μ(dy) ψt (x1 , x2 , . . . , xt )

e− λ,y · ht+1 (y)μ(dy)

U

= gt (λ),

t = 0, 1, . . . , T ∗ − 1.

It follows from (2.2.6) that for the model with drift given by (2.2.10) the measure Q with density (2.2.11) is a martingale measure. By (2.2.11) we see that ξ1 , . . . , ξT ∗ are independent also under Q.

2.2.2 Uniqueness of the Martingale Measure With the use of Theorem 2.2.1 we deduce now conditions for the uniqueness of a martingale measure. They are formulated in Theorems 2.2.3 and 2.2.4 below. Theorem 2.2.3 {ξi }. If

Let the forward rate be given by (2.2.1) with real valued factors +∞ 

| σ (t, s) |< +∞,

t = 0, 1, . . . , T ∗ − 1,

s=t

then the arising bond market admits exactly one martingale measure or there are no martingale measures. Theorem 2.2.4 If ξ1 in (2.2.1) takes K < +∞ different values in R and, for any t = 0, 1, . . . , T ∗ − 1, the function T −→

T  s=t

σ (t, s),

T = t, t + 1, . . . ,

28

Arbitrage-Free Bond Markets

takes at least K different values, then the arising bond market admits exactly one martingale measure or there are no martingale measures. Proof of Theorem 2.2.3 We use the notation from Theorem 2.2.1. Recall that a measure with density ψ(ξ1 , ξ2 , . . . , ξT ∗ ) is a martingale measure if and only if the drift α is determined by (2.2.6). So, the model admits a martingale measure if and only if (2.2.6) holds with some function ψ. Let us assume that there exists another ˜ 1 , ξ2 , . . . , ξT ∗ ). Then by (2.2.6), or, equivalently, martingale measure with density ψ(ξ by (2.2.9), for each t = 0, 1, . . . , T ∗ − 1, we obtain ψt,t+1 (ξ1 , . . . , ξt ,

T 

σ (t, s)) = ψ˜ t,t+1 (ξ1 , . . . , ξt ,

T 

s=t

σ (t, s)),

T = t, t + 1, . . . .

s=t

(2.2.12) For the case t = 0 this yields    T  T   ψ0,1 σ (0, s) = ψ˜ 0,1 σ (0, s) , s=0

But

 ψ0,1 (λ) =

R

T = 0, 1, . . . .

s=0

e−λy ψ1 (y)μ(dy),

ψ˜ 0,1 (λ) =

 R

e−λy ψ˜ 1 (y)μ(dy)

are analytic functions which are equal on the convergent sequence λ(T) :=

T s=0 σ (0, s). Hence they are equal in the whole domain. Since they are Laplace transforms of the measures ψ1 (y)μ(dy), ψ˜ 1 (y)μ(dy), also the measures are identical. Hence ψ1 (ξ1 ) = ψ˜ 1 (ξ1 ).

(2.2.13)

Now we use (2.2.12) with t = 1, which yields ψ1,2 (ξ1 ,

T 

σ (1, s)) = ψ˜ 1,2 (ξ1 ,

s=1

T 

σ (1, s)),

T = 1, 2, . . . .

s=1

Since  ψ1,2 (x1 , λ) =

R

e−λy

ψ2 (x1 , y) μ(dy), ψ1 (x1 )

ψ˜ 1,2 (x1 , λ) =

 R

e−λy

ψ˜ 2 (x1 , y) μ(dy) ψ˜ 1 (x1 )

and (2.2.13) holds, we obtain   e−λy ψ2 (ξ1 , y)μ(dy) = e−λy ψ˜ 2 (ξ1 , y)μ(dy) R

R

2.2 Martingale Measures for HJM Models 29

T for each λ = λ(T) = s=1 σ (1, s). Since the last sum converges, again, by the analyticity of the Laplace transform, we obtain that ψ2 (ξ1 , ξ2 ) = ψ˜ 2 (ξ1 , ξ2 ). It is clear that iterative application of the preceding arguments yields ˜ 1 , ξ2 , . . . , ξT ∗ ), ψ(ξ1 , ξ2 , . . . , ξT ∗ ) = ψ(ξ which means that the martingale measure is unique. For the proof of Theorem 2.2.4 we use a corollary from the following auxiliary result. Lemma 2.2.5 The function f (x) := c1 xa1 + c2 xa2 + · · · + cK xaK ,

x > 0,

where ck , ak ∈ R, for k = 1, 2, . . . , K and a1 < a2 < · · · < aK , not all ck are zero, has at most K − 1 positive roots. Proof We follow Gantmacher and Krein [61, p.76] and prove the assertion by induction. The result is clearly true for K = 1. Assuming that it is true for K − 1 we show it for K. Equivalently, we have to show that x−a1 f (x) = c1 + c2 xa2 −a1 + · · · + cK xaK −a1 has at most K − 1 positive roots. But we know from the previous inductive step that  d  −a1 x f (x) = (a2 − a1 )c2 xa2 −a1 −1 + · · · + (aK − a1 )cK xaK −a1 −1 dx has at most K −2 positive roots. However, between two consecutive roots of x−a1 f (x) there is at least one root of its derivative. So, if x−a1 f (x) had at least K roots then its derivative would have at least K − 1 roots, which is a contradiction. Corollary 2.2.6 Let a1 < a2 < · · · < aK be real numbers. Then the vectors ⎛ a ⎞ ⎛ a ⎞ ⎛ a ⎞ x11 x12 x1K ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ v1 := ⎜ ... ⎟ , v2 := ⎜ ... ⎟ , . . . , vK := ⎜ ... ⎟ (2.2.14) ⎝ ⎝ ⎝ ⎠ ⎠ ⎠ a1 a2 aK xK xK xK are linearly independent for any positive reals x1 , x2 , . . . , xK such that xi = xj , i = j. Indeed, let us assume that v1 , v2 , . . . , vK are not linearly independent for some x1 , x2 , . . . , xK . Then there exist constants c1 , c2 , . . . , cK , not all equal 0, such that c1 v1 + c2 v2 + · · · + cK vK = 0.

30

Arbitrage-Free Bond Markets

This, however, means that each xk , k = 1, 2, . . . , K is a root of the function f (x) := c1 xa1 + c2 xa2 + · · · + cK xaK ,

x ≥ 0,

which is impossible by Lemma 2.2.5. ˜ 1 , ξ2 , . . . , ξT ∗) Proof of Theorem 2.2.4 Let us assume that ψ(ξ1 , ξ2 , . . . , ξT ∗) and ψ(ξ are densities of two different martingale measures. As in the proof of Theorem 2.2.3 we show that the condition ψt,t+1 (ξ1 , . . . , ξt ,

T 

σ (t, s)) = ψ˜ t,t+1 (ξ1 , . . . , ξt ,

T 

s=t

σ (t, s)),

T = t, t + 1, . . . ,

s=t

(2.2.15) for each t = 0, 1, . . . , T ∗ − 1, implies that ˜ 1 , ξ2 , . . . , ξT ∗ ). ψ(ξ1 , ξ2 , . . . , ξT ∗ ) = ψ(ξ

(2.2.16)

Let us denote the values of ξ1 by a1 < a2 < · · · < aK and pk := P(ξ1 = ak ) = μ({ak }),

k = 1, 2, . . . , K.

Condition (2.2.15) with t = 0 yields    T  T   ψ0,1 σ (0, s) = ψ˜ 0,1 σ (0, s) , s=0

T = 0, 1, . . . .

(2.2.17)

s=0

But  ψ0,1 (λ) =

R

e−λy ψ1 (y)μ(dy) =

K 

e−λak ψ1 (ak )pk

k=1

and ψ˜ 0,1 (λ) =

 R

e−λy ψ˜ 1 (y)μ(dy) =

K 

e−λak ψ˜ 1 (ak )pk ,

k=1

so (2.2.17) means that K 

e−λ(T)ak ψ1 (ak )pk =

k=1

for each λ(T) := we have K  k=1

T

s=0 σ (0, s).

a

K 

e−λ(T)ak ψ˜ 1 (ak )pk

k=1

Thus, for any K different values λ(T1 ), . . . , λ(TK ),

xi k ψ1 (ak )pk =

K  k=1

xi k ψ˜ 1 (ak )pk , a

i = 1, 2, . . . , K,

2.3 Martingale Measures Using Martingale Representation Property

31

where xk := e−λ(Tk ) > 0, k = 1, 2, . . . , K. By Corollary 2.2.6, ψ1 (ak ) = ψ˜ 1 (ak ), so ψ1 (ξ1 ) = ψ˜ 1 (ξ1 ).

(2.2.18)

The use of (2.2.15) with t = 1 yields ψ1,2 (ξ1 ,

T 

σ (1, s)) = ψ˜ 1,2 (ξ1 ,

s=1

T 

σ (1, s)),

T = 1, 2, . . . .

(2.2.19)

s=1

Since ψ1,2 (x1 , λ) =

K 

e−λak

k=1

ψ2 (x1 , ak ) pk , ψ1 (x1 )

ψ˜ 1,2 (x1 , λ) =

K 

e−λak

k=1

ψ˜ 2 (x1 , ak ) pk ψ˜ 1 (x1 )

and (2.2.18) holds, we obtain from (2.2.19) that K 

  e−λ(T)ak ψ2 (x1 , ak ) − ψ˜ 2 (x1 , ak ) pk = 0

k=1

T

for each λ(T) = s=1 σ (1, s). Taking K different values λ(T1 ), λ(T2 ), . . . , λ(TK ) we conclude again from Corollary 2.2.6 that ψ2 (ξ1 , ξ2 ) = ψ˜ 2 (ξ1 , ξ2 ). Further application of the preceding arguments for t = 2, 3, . . . , T ∗ − 1 yields finally (2.2.16).

2.3 Martingale Measures and Martingale Representation Property In this section we characterize equivalent measures in a way that is specific for continuous time models. It differs from the framework used in Section 2.2, where the density process of an equivalent to P measure Q was represented simply in the form ρt = ψt (ξ1 , ξ2 , . . . , ξt ),

t = 0, 1, . . . , T ∗ ,

where {ξt } is a sequence of Rd -valued independent and identically distributed random variables on (, F, P) and ψt is some function. Our current framework will exploit martingale representation properties related to the martingale Z defined by Z0 := 0,

Zt :=

t 

ξs ,

E[ξt ] = 0,

t = 1, 2, . . . , T ∗ .

s=1

Since the increments of Z are independent and stationary, Z can be viewed as a discrete time counterpart of a L´evy process in continuous time. We say that Z has the martingale representation property if any martingale X adapted to {Ft }, where F0 is

32

Arbitrage-Free Bond Markets

trivial and Ft := σ {ξ1 , ξ2 , . . . , ξt }; t = 1, 2, . . . , T ∗ , can be written as a discrete time stochastic integral over Z, that is, in the form t−1  Ys , Zs+1 − Zs , Xt = X0 +

t = 1, . . . , T ∗ ,

(2.3.1)

s=0

where Y is some adapted process. The martingale representation property is examined in Section 2.3.1, where we show that it requires very restrictive distributional conditions for Z. Therefore, also a generalized martingale representation property of Z will be used to cover the general situation. It turns out that any martingale X can be represented in the following way t−1   ψ(s, y)π˜ ({s + 1}, dy), t = 1, . . . , T ∗ , (2.3.2) Xt = X0 + s=0

U

where ψ(s, y) is an adapted process and π˜ is the so-called compensated jump measure of the process Z. The precise definition of the preceding integral and the proof of (2.3.2) is presented in Section 2.3.2. The concept of jump measure comes from the continuous time setting and is used to prove martingale representation property for L´evy processes. We use both martingale representations, i.e. (2.3.1) and (2.3.2), to describe the density process ρt of an equivalent measure. This leads to the classical and generalized Girsanov’s theorems described in Section 2.3.3, which are used in Section 2.3.4 to characterize martingale measures in the HJM model.

2.3.1 Martingale Representation Property The martingale representation property will be examined in a more general setting than announced previously. Let {Mt }t=0,1,...,T ∗ , M0 = 0 be an U = Rd -valued martingale on a probability space (, F, P) equipped with the filtration Ft := σ (M0 , M1 , . . . , Mt ), t = 0, 1, . . . , T ∗ . The increments of M are not necessarily assumed neither to be independent nor identically distributed. Our goal is to formulate conditions for M such that any Ft -adapted martingale X starting from zero admits the representation Xt =

t−1 

Ys , Ms+1 − Ms ,

t = 1, . . . , T ∗ ,

(2.3.3)

s=0

where Y is some adapted process. Theorem 2.3.1 The martingale M has the martingale representation property if and only if for any t = 0, 1, . . . , T ∗ − 1 the conditional distribution of Mt+1 − Mt given M1 , M2 , . . . , Mt is concentrated on a finite set At in U such that At = n(t) < +∞,

and

dim(span At ) = n(t) − 1.

(2.3.4)

2.3 Martingale Measures Using Martingale Representation Property

33

If, additionally, n(t) − 1 = d for each t = 0, 1, . . . , T ∗ − 1, then for any martingale X the representation (2.3.3) is unique. Above At stands for the number of elements of At and span At for the linear space spanned by the set At . Example 2.3.2 A real valued martingale M of the form Mt =

t 

ξs ,

t = 1, 2, . . . ,

s=1

where {ξs } is a sequence of independent random variables with zero mean, has a martingale representation property if and only if the distribution of each ξs is concentrated on two points only. If this is the case, then the representation (2.3.3) is unique. Corollary 2.3.3 The martingale representation property of M implies that any adapted process {Xt } is finite valued. In particular, all moments of Xt are finite. Indeed, it follows from Theorem 2.3.1 that the number of paths of M on a finite time interval is finite. Since Xt is of the form Xt = xt (M0 , M1 , . . . , Mt ), t = 0, 1, . . . , T ∗ − 1, where xt (·) is some function, its values form a finite set. The proof of Theorem 2.3.1 is based on the following auxiliary result. Lemma 2.3.4 Let M be a zero mean U − Rd -valued random variable. Then for any function ψ : U −→ R such that E[ψ(M)] = 0 there exists y ∈ U such that ψ(M) = y, M

P − a.s.

(2.3.5)

if and only if the distribution of M is concentrated on a finite set of vectors m1 , m2 , . . . , mn such that dim(span{m1 , m2 , . . . , mn }) = n − 1. The representation (2.3.5) is unique if n − 1 = d. Recall that the support of a measure μ defined on the Borel subsets of a set I is the smallest closed set A such that μ(A) = μ(I). Proof (Sufficiency) Let us assume that m1 , m2 , . . . , mn−1 are linearly independent. Since (2.3.5) has the form ψ(mi ) = y, mi ,

i = 1, 2, . . . , n,

(2.3.6)

we can find y ∈ U such that the first n − 1 equations are satisfied. The zero mean conditions n  i=1

μi mi = 0,

n  i=1

μi ψ(mi ) = 0,

34

Arbitrage-Free Bond Markets

with μi = P(M = mi ), i = 1, 2, . . . , n, imply that mn = −

n−1 1  μi mi , μn

ψ(mn ) = −

i=1

n−1 1  μi ψ(mi ). μn i=1

This clearly yields that ψ(mn ) = y, mn . (Necessity) If the support of M is infinite then all bounded functions ψ(x),

x ∈ supp{M}

form a linear space of infinite dimension, while the space of linear functions y, x ,

x ∈ supp{M},

y∈U

is of dimension no greater than dim U. It follows that if (2.3.5) holds then M takes a finite number of values only. Thus let us now consider the case when the set of values of M is {m1 , m2 , . . . , mn } and dim(span{m1 , m2 , . . . , mn }) < n − 1. We may also assume that the last two vectors are some linear combinations of m1 , m2 , . . . , mn−2 , i.e. dim(span{m1 , m2 , . . . , mn }) = dim(span{m1 , m2 , . . . , mn−2 }). Let ψ be such that (2.3.6) holds. Then ψ(mn−1 ) and ψ(mn ) are uniquely determined ˜ i) = by ψ(mi ), i = 1, 2, . . . , n − 2. Now let us define the function ψ˜ as follows ψ(m ˜ ˜ ψ(mi ), i = 1, 2, . . . , n − 2 and ψ(mn−1 ) = ψ(mn−1 ), ψ(mn ) = ψ(mn ) be such that ˜ n−1 ) + μn ψ(m ˜ n ) = μn−1 ψ(mn−1 ) + μn ψ(mn ). μn−1 ψ(m ˜ Then E[ψ(M)] = 0 but (2.3.6) is not satisfied. Proof of Theorem 2.3.1 Let X be an arbitrary martingale starting from zero. It can be identified with a sequence of functions ψ1 , ψ2 , . . . , ψT ∗ such that Xt = ψt (M1 , M2 , . . . , Mt ),

t = 1, 2, . . . , T ∗ .

The integrand in (2.3.3) can also be written in that form, i.e. Yt = yt (M0 , M1 , . . . , Mt ),

t = 0, 1, . . . , T ∗ − 1.

We will prove the required distributional property of M inductively. In the first period the problem has the form ψ1 (M1 ) = y0 (M0 ), M1 . By Lemma 2.3.4 we see that necessary and sufficient condition for the representation to hold is that the values of M1 form a set A0 satisfying (2.3.4). Now let us assume that (2.3.4) holds for some t − 1 and show that (2.3.4) holds for t if and only if M has the martingale representation property. We know that ψt (M1 , M2 , . . . , Mt ) =

t−1  ys (M0 , M1 , . . . , Ms ), Ms+1 − Ms s=0

2.3 Martingale Measures Using Martingale Representation Property

35

for some functions y0 , y1 , . . . , yt−1 . For any function ψt+1 such that E[ψt+1 (M1 , . . . , Mt+1 ) | Ft ] = ψt (M0 , M1 , . . . , Mt ), we are looking for a function yt such that ψt+1 (M1 , M2 , . . . , Mt+1 ) =

t  ys (M0 , M1 , . . . , Ms ), Ms+1 − Ms s=0

= ψt (M1 , M2 , . . . , Mt ) + yt (M0 , M1 , . . . , Mt ), Mt+1 − Mt . (2.3.7) Let us consider an arbitrary path of M up to time t, i.e. A := {M1 = m1 , M2 = m2 , . . . , Mt = mt }. Condition (2.3.7) implies the following [ψt+1 (m1 , . . . , mt , Mt+1 ) − ψt (m1 , . . . , mt )]1A = [ yt (m0 , m1 , . . . , mt ), Mt+1 −mt ]1A . (2.3.8) From the martingale properties of M and X we obtain   E [ψt+1 (m1 , . . . , mt , Mt+1 ) − ψt (m1 , . . . , mt )]1A = 0,

  E (Mt+1 − mt )1A = 0,

hence we can again apply Lemma 2.3.4. This tells that yt (m0 , m1 , . . . , mt ) satisfying (2.3.8) exists if and only if the values of (Mt+1 − mt )1A satisfy (2.3.4). This proves the assertion. Uniqueness follows also from Lemma 2.3.4.

2.3.2 Generalized Martingale Representation Property To prove the generalized martingale representation property of the martingale Z0 := 0,

Zt :=

t 

ξs ,

E[ξt ] = 0,

t = 1, 2, . . . , T ∗ ,

(2.3.9)

s=1

where {ξi } is an i.i.d. sequence living in U = Rd we need to extend the concept of stochastic integral related to Z. Recall that the classical integral of an adapted process Y over Z is given by I01 := 0,

It1 :=

t−1 t−1   Ys , Zs+1 − Zs = Ys , ξs+1 , s=0

t = 1, . . . , T ∗ .

(2.3.10)

s=0

For the definition of the extended stochastic integral we need the concept of random measure. Let π : {1, 2, . . .} × U −→ {0, 1, . . .} be given by π({s}, A) = 1{Zs −Zs−1 ∈A} = 1{ξs ∈A} ,

36

Arbitrage-Free Bond Markets

where A ⊆ U. Then π({1, 2, . . . , t}, A) =

t 

1{ξs ∈A}

s=1

gives the number of increments of Z on the interval [0, t], which take values in the set A, and therefore π is called a jump measure or random measure of Z. Since E[π({s}, A)] = E[1{ξs ∈A} ] = P(ξs ∈ A) = μ(A), where μ stands for the law of ξ1 , we see that the process π({1, ˜ 2, . . . , t}, A) :=

t 

1{ξs ∈A} − tμ(A),

t = 1, 2, . . . , T ∗

s=1

is a martingale. The measure π˜ is called a compensated random measure of Z. An extended stochastic integral we obtain by integrating a process ψ(·, y) over the measure π˜ in the following way t−1   2 2 ψ(s, y)π˜ ({s + 1}, dy), I0 = 0, It := s=0

:=

U

t−1  

 ψ(s, y)μ(dy)

 ψ(s, ξs+1 ) −

t=

1, 2, . . . , T ∗ . (2.3.11)

U

s=0

For each y the process ψ(·, y) is assumed to be adapted, i.e. ψ(s, y) is Fs -measurable such that  | ψ(s, y) | μ(dy) < +∞, s = 0, 1, . . . , T ∗ − 1. U

If ψ is of the form ψ(s, x) = ψ(s), x then    t−1  t−1  2 It = ψ(s), ξs+1 , ψ(s), ξs+1 − ψ(s) yμ(dy) = U

s=0

because

t = 1, 2, . . . , T ∗

s=0

 yμ(dy) = 0. U

So, in this case I 2 equals I 1 in (2.3.10) with Ys = ψ(s). With the use of the concept of extended stochastic integral one can easily prove the generalized martingale representation property of the process Z. Proposition 2.3.5 Any martingale N can be represented in the form t−1   Nt = N0 + ψ(s, y)π˜ ({s + 1}, dy), t = 1, 2, . . . , T ∗ s=0

U

(2.3.12)

2.3 Martingale Measures Using Martingale Representation Property for some adapted process ψ(s, y) satisfying  | ψ(s, y) | μ(dy) < +∞,

s = 0, 1, . . . , T ∗ − 1.

37

(2.3.13)

U

Proof Since Ns = h(s, ξ1 , . . . , ξs ), s = 1, 2, . . . , T ∗ for some function h, by the martingale property we obtain  h(s, ξ1 , . . . , ξs , y)μ(dy), s = 0, 1, . . . , T ∗ − 1. Ns = E(Ns+1 | Fs ) = U

Consequently, Nt = N0 +

t−1 t−1   (Ns+1 − Ns ) = N0 + (Ns+1 − E(Ns+1 | Fs )) s=0

= N0 +

t−1 

s=0

h(s, ξ1 , . . . , ξs , ξs+1 ) −

s=0

= N0 +

s=0

t−1   s=0

t−1  

ψ(s, y)π({s ˜ + 1}, dy)

h(s, ξ1 , . . . , ξs , y)μ(dy) U

t = 1, 2, . . . , T ∗ ,

U

where ψ(s, y) := h(s, ξ1 , . . . , ξs , y). (2.3.13) follows from the definition of ψ.

2.3.3 Girsanov’s Theorems Let Q be a measure that is equivalent to P. Girsanov’s theorem provides a description of the corresponding density process dQ ρt := |Ft , t = 0, 1, . . . , T ∗ , dP where {Ft } is the filtration generated by a sequence of zero mean, independent and identically distributed random variables {ξi }, i = 1, 2, . . . , T ∗ . In the classical version of Girsanov’s theorem (see Theorem 2.3.6), ρ has an exponential form involving some martingale. Such representation holds, in particular, in the case when the process Zt = ξ1 + . . . ξt has the martingale representation property. This case corresponds to the Wiener process in the continuous time setting. In the general version of Girsanov’s theorem (see Theorem 2.3.8) ρ has a different form that can be proven under weaker assumptions. Let ϕξ be the Laplace exponent of ξ1 , i.e. E[e− u,ξ ] = eϕξ (u) ,

u ∈ U,

and let us define the set  := {λ ∈ U : ϕξ (λ) < +∞}. Recall that μ stands for the distribution of ξ1 .

(2.3.14)

38

Arbitrage-Free Bond Markets

Theorem 2.3.6 [Girsanov’s theorem – classical version] Let Q ∼ P be a measure with density process ρ satisfying: ! E | ln(ρt ) | < +∞, t = 0, 1, 2, . . . , T ∗ . (2.3.15) (a) Then there exists an adapted process ψ(s, y) satisfying   | ψ(s, y) | μ(dy) < +∞, eψ(s,y) μ(dy) < +∞ U

s = 0, 1, . . . , T ∗ − 1,

U

(2.3.16) such that ρ0 = 1,

t−1 

ρt = e

s=0 U

 ψ(s,y)  

ψ(s,y)π˜ ({s+1},dy)− t−1 μ(dy) − U ψ(s,y)μ(dy)} s=0 {ln U e

,

∗

t = 1, 2, . . . , T . (2.3.17) Conversely, if ρ is of the form (2.3.17) with ψ satisfying (2.3.16) then it is a density process of some measure that is equivalent to P. (b) If the process Z0 := 0,

Zt :=

t 

t = 1, 2, . . . , T ∗

ξs ,

s=1

has the martingale representation property, then there exists an adapted process δ that P-a.s. takes values in the set  such that ρ0 = 1,

t−1

ρt = e

t−1

s=0 δs ,ξs+1 −

s=0 ϕξ (−δs )

,

t = 1, 2, . . . , T ∗ .

(2.3.18)

Conversely, if δ is an adapted process taking values P-a.s. in , then ρ given by (2.3.18) defines the density process of some measure that is equivalent to P. Remark 2.3.7 It follows from Corollary 2.3.3 that (2.3.15) is satisfied if Z has the martingale representation property. Theorem 2.3.8 [Girsanov’s theorem – general version] Let Q ∼ P be a measure with density process ρ0 = 1, ρt , t = 1, 2, . . . , T ∗ . (a) Then there exists an adapted process ψ(s, y), s = 0, 1, . . . , T ∗ − 1, y ∈ U such that  eψ(s,y) μ(dy) < +∞, P − a.s., s = 0, 1, . . . , T ∗ − 1 (2.3.19) U

and for t = 1, 2, . . . , T ∗ ,

t−1 

ρt = e

s=0 U

 ψ(s,y)

ψ(s,y)π({s+1},dy)− t−1 μ(dy) s=0 ln U e

.

(2.3.20)

(b) Conversely, if ρ is of the form (2.3.20) with a predictable process ψ satisfying (2.3.19) then ρ is the density process of some measure Q that is equivalent to P.

2.3 Martingale Measures Using Martingale Representation Property Proof of Theorem 2.3.6 A0 = 0,

39

By (2.3.15) the process t−1 

At :=

E[ln ρs+1 − ln ρs | Fs ],

t = 1, 2, . . . , T ∗

s=0

is well defined and At is Ft−1 -measurable for each t = 1, 2, . . . , T ∗ . Since E[ln ρt+1 − At+1 | Ft ] = E[ln ρt+1 | Ft ] −

t 

E[ln ρs+1 − ln ρs | Fs ]

s=0 t−1 

= ln ρt −

E[ln ρs+1 − ln ρs | Fs ]

s=0

t = 0, 1, . . . , T ∗ − 1,

= ln ρt − At , we see that ln ρ − A is a martingale.

(a) By Proposition 2.3.5 there exists an adapted process ψ(s, y) satisfying the first condition in (2.3.16) such that ln ρt − At =

t−1  

ψ(s, y)π({s ˜ + 1}, dy),

t = 1, 2, . . . , T ∗ .

U

s=0

Hence

t−1 

ρt = e

ψ(s,y)π˜ ({s+1},dy)+At

s=0 U

,

t = 1, 2, . . . , T ∗

(2.3.21)

and the martingale property of ρ yields ρt = E[ρt+1 | Ft ]

t−1 

=e

s=0 U

ψ(s,y)π˜ ({s+1},dy)+At 

· E[eψ(t,ξt+1 )−

U

ψ(t,y)μ(dy)



= ρt E[eψ(t,ξt+1 )−

U

| Ft ] · eAt+1 −At | Ft ] · eAt+1 −At ,

ψ(t,y)μ(dy)

t = 0, 1, . . . , T ∗ − 1.

It follows that 

e−(At+1 −At ) = E[eψ(t,ξt+1 )− = e−

 U

U

ψ(t,y)μ(dy)



ψ(t,y)μ(dy)

| Ft ]

eψ(t,y) μ(dy), U

t = 0, 1, . . . , T ∗ − 1,

40

Arbitrage-Free Bond Markets so the second condition in (2.3.16) is satisfied. Thus we can represent A in the form t−1 t−1    At = (As+1 − As ) = ψ(s, y)μ(dy) s=0

s=0

U

 eψ(s,y) μ(dy) ,



− ln

t = 1, 2, . . . , T ∗ .

U

Consequently, from (2.3.21) we obtain (2.3.17). If ρ is given by (2.3.17), then (2.3.16) ensures that it is a martingale, hence a density process. (b) By the martingale representation property of Z we have t−1  δs , ξs+1 , ln ρt − At =

t = 1, 2, . . . , T ∗

s=0

for some adapted process δ. This yields

t−1

s=0 δs ,ξs+1 +At

ρt = e

,

t = 1, 2, . . . , T ∗ .

It follows from the martingale property of ρ that

t

ρt = E[ρt+1 | Ft ] = E e

t−1

= eAt+1 e

s=0 δs ,ξs+1

s=0 δs ,ξs+1 +At+1

E e δt ,ξt+1 | Ft

! = ρt eAt+1 −At E e δt ,ξt+1 | Ft , Thus

| Ft

!

!

t = 0, 1, . . . , T ∗ − 1.

! e−At+1 +At = E e δt ,ξt+1 | Ft = eϕξ (−δt ) ,

t = 0, 1, . . . , T ∗ − 1,

which shows that δ takes values in  and allows determining the process A by

summing its increments. This yields At = − t−1 s=0 ϕξ (−δs ) and, consequently, (2.3.18). To see the converse let us notice that ρ given by (2.3.18) is a strictly positive martingale, so the assertion follows. Proof of Theorem 2.3.8 If Q ∼ P then there exists a positive measurable function h(x1 , . . . , xT ∗ ), xi ∈ U, i = 1, 2, . . . , T ∗ such that dQ = h(ξ1 , . . . , ξT ∗ ), dP Note that   E h(ξ1 , . . . , ξT ∗ ) =

P − a.s.

 UT

∗

h(x1 , . . . , xT ∗ )μ(dx1 ) . . . μ(dxT ∗ ) = 1.

2.3 Martingale Measures Using Martingale Representation Property Let us define for t = 1, 2, . . . , T ∗ , (x1 , . . . , xt ) ∈ U t  ht (x1 , . . . , xt ) := h(x1 , . . . , xt , y1 , . . . , yT ∗ −t )μ(dy1 ) . . . μ(dyT ∗ −t ), UT

Then

∗ −t

 ht (x1 , . . . , xt−1 , y)μ(dy) = ht−1 (x1 , . . . , xt−1 ),

41

h0 = 1.

t = 2, . . . , T ∗ ,



U

h1 (y)μ(dy) = h0 = 1. U

Let ρt be given by (2.3.20) with ψ given by the formula: ψ(s, y) = ln hs+1 (ξ1 , . . . , ξs , y), s = 0, 1, . . . , T ∗ − 1, y ∈ U. We have to show that ρt = ρt , t = 1, . . . , T ∗ where ρt :=

T ∗ −1

ρT ∗ = e

s=0

T ∗ −1

ln hs+1 (ξ1 ,...,ξs ,ξs+1 )−

"T ∗ −1

= "T ∗ −1s=0  s=0

=

s=0

ln

dQ dP |Ft .

(2.3.22)

Since



U hs+1 (ξ1 ,...,ξs ,y)μ(dy)

hs+1 (ξ1 , . . . , ξs+1 )

U hs+1 (ξ1 , . . . , ξs , y)μ(dy)

"T ∗ −1

s=0 hs+1 (ξ1 , . . . , ξs+1 ) "T ∗ −1 s=0 hs (ξ1 , . . . , ξs )

= hT ∗ (ξ1 , . . . , ξT ∗ ) = ρT ∗ . It is easy to check that ρt , t = 1, 2, . . . , T ∗ is an Ft -martingale. Since ρt , t = 1, 2, . . . , T ∗ is an Ft -martingale we get that ρt = ρt , t = 1, 2, . . . , T ∗ as required. By the latter argument one can prove also (b).

2.3.4 Application to HJM Models Now we present results characterizing martingale measures in the HJM model f (t + 1, T) = f (t, T) + α(t, T) + σ (t, T), ξt+1 ,

t, T = 0, 1, . . . .

(2.3.23)

This problem was already studied in Section 2.2, but now we do this in an alternative manner involving martingale representation properties of the process Z0 = 0,

Zt = ξ1 + ξ2 + · · · + ξt ,

t = 1, 2, . . . , T ∗ .

(2.3.24)

This kind of describing martingale measures is commonly used in continuous time HJM models driven by a L´evy process and therefore discussed here in detail. In particular, we derive the drift conditions that correspond to the HJM conditions in continuous time. Recall that in Section 2.3.1 we showed that Z has the martingale representation property only under very strong distributional restrictions. Therefore the following

42

Arbitrage-Free Bond Markets

Theorem 2.3.9 can be viewed as an alternative characterization of martingale measures to that given by Theorem 2.2.1. This particular situation can be compared to the HJM model in continuous time based on the Wiener process. Recall that the set  is given by  := {λ ∈ U : ϕξ (λ) < +∞}, where ϕξ stands for the Laplace exponent of ξ1 (see (2.3.14)). Theorem 2.3.9 Assume that the martingale Z given by (2.3.24) has the martingale representation property. Then the model (2.3.23) satisfies (MM) if and only if there exists an adapted process δ that P-a.s. takes values in the set  and such that for each T = 0, 1, . . .,  T  T−1    σ (t, s) − δt − ϕξ σ (t, s) − δt , t = 0, 1, 2, . . . , T − 1. α(t, T) = ϕξ s=0

s=0

(2.3.25) ˆ T), T = 0, 1, . . . are martingales under P if and only if In particular, P(t,    T T−1   σ (t, s) − ϕξ σ (t, s) , t = 0, 1, 2, . . . , T − 1. (2.3.26) α(t, T) = ϕξ s=0

s=0

If Z does not necessarily have the martingale representation property, then we can characterize martingale measures using its generalized martingale representation property. This leads to the following Theorem 2.3.10, which is, of course, equivalent to Theorem 2.2.1. Theorem 2.3.10 The model (2.3.23) satisfies (MM) if and only if there exists an adapted process ψ(t, y) satisfying  eψ(s,y) μ(dy) < +∞, s = 0, 1, . . . , T ∗ − 1, U

such that for each T = 0, 1, . . .  

ψ(t,y)− Ts=0 σ (t,s),y μ(dy) Ue α(t, T) = ln  ,

ψ(t,y)− T−1 s=0 σ (t,s),y μ(dy) Ue

t = 0, 1, 2, . . . , T − 1. (2.3.27)

ˆ T), T = 0, 1, . . . are martingales under P if and only if In particular, P(t,    T T−1   σ (t, s) − ϕξ σ (t, s) , t = 0, 1, 2, . . . , T − 1. (2.3.28) α(t, T) = ϕξ s=0

Remark 2.3.11 Theorem 2.2.1.

s=0

The formulae (2.3.26) and (2.3.28) are the same as (2.2.7) in

2.3 Martingale Measures Using Martingale Representation Property

43

Example 2.3.12 We show that for a given volatility processes σ (·, ·) there are models such that the discounted bond prices are martingales under some equivalent measure Q. Let gt be a function such that gt (σ (t, ·)) := gt (σ (t, t), σ (t, t + 1), . . . , σ (t, T ∗ )) satisfies ! E e− gt (σ (t,·)),ξt+1 < +∞. If the drift is given by  α(t, T) = ϕξ

T 

 σ (t, s) − gt (σ (t, ·))

s=0

− ϕξ

T−1 

 σ (t, s) − gt (σ (t, ·)) ,

t = 0, 1, 2, . . . , T − 1

s=0

for T = 0, 1, 2, . . . , T ∗ , then the resulting model admits a martingale measure. This follows from Theorem 2.3.9 because δt = gt (σ (t, ·)) satisfies the required assumptions. Proof of Theorem 2.3.9 By Remark 2.3.7 and Theorem 2.3.6 (b) any measure Q ∼ P can be identified with an adapted process δ taking values in . Since the density process has the form (2.3.18), for any T = 0, 1, 2, . . . , T ∗ and t = 0, 1, . . . , T − 1, we have ˆ + 1, T) | Ft ] = E[e E[ρt+1 P(t

t

t

s=0 ϕξ (−δs ) e−

s=0 δs ,ξs+1 −

= E[ρt e δt ,ξt+1 −ϕξ (−δt ) e−

T−1

T−1

ˆ T)e−ϕξ (−δt )− = ρt P(t,

s=0

s=0

T−1 s=0

f (t+1,s) | F ] t

f (t,s)+α(t,s)+ σ (t,s),ξt+1 | F ] t

α(t,s) · E[e δt ,ξt+1 − T−1 s=0 σ (t,s),ξt+1 | Ft ].

Taking into account that

T−1

E[e δt ,ξt+1 −

s=0

σ (t,s),ξt+1

| Ft ] = E[e−

T−1 s=0

σ (t,s)−δt ,ξt+1

T−1

| Ft ] = eϕξ (

s=0

we obtain that ˆ T)e− ρt P(t,

T−1 s=0

α(t,s)

T−1

· e−ϕξ (−δt ) · eϕξ (

s=0

σ (t,s)−δt )

ˆ T) = ρt P(t,

if and only if T−1  s=0

α(t, s) + ϕξ (−δt ) = ϕξ

T−1 

 σ (t, s) − δt .

s=0

From that condition we arrive at (2.3.25). Condition (2.3.26) we obtain by setting δt ≡ 0 in (2.3.25).

σ (t,s)−δt )

,

44

Arbitrage-Free Bond Markets

Proof of Theorem 2.3.10 In view of the general Girsanov theorem (see Theorem 2.3.8) we can write the density of an equivalent to P measure Q in the form  ψ(s,y)

ψ(s,y)π({s+1},dy)− t−1 μ(dy) s=0 ln U e

t−1 

ρt = e

s=0 U

,

t = 1, 2, . . . , T ∗ .

Then ˆ + 1, T) | Ft ] = E[e E[ρt+1 P(t 

= E[ρt e

s=0 U ψ(s,y)π({s+1},dy)−



ψ(t,y) μ(dy) − U ψ(t,y)π({t+1},dy)−ln U e e

ˆ T)e− ln = ρt P(t,

Since



t



ψ(t,y) μ(dy) − Ue e



E[e

U

T−1

s=0 α(t,s)

ψ(t,y)π({t+1},dy) −

e



=E

T−1

ψ(t,y)−

e

s=0

T−1

s=0 f (t,s)+α(t,s)+ σ (t,s),ξt+1

· E[e

T−1 s=0



ψ(s,y) μ(dy) − T−1 f (t+1,s) s=0 ln U e s=0 e

t



U ψ(t,y)π({t+1},dy) e−

σ (t,s),ξt+1

σ (t,s),y

| Ft ]

T−1

| Ft ]

| Ft ]

s=0 σ (t,s),ξt+1

| Ft ].



π({t + 1}, dy) | Ft

U



T−1

=

eψ(t,y)−

s=0

σ (t,s),y

μ(dy),

U

we see that Q is a martingale measure if and only if 

T−1

1 − α(t,s) ψ(t,y)− T−1 s=0 s=0 σ (t,s),y μ(dy) = 1.  e e ψ(t,y) μ(dy) U Ue It follows that

T−1

e

s=0



α(t,s)

T−1

eψ(t,y)− s=0 σ (t,s),y μ(dy)  = U ψ(t,y) μ(dy) Ue

and one easily comes to (2.3.27). Condition (2.3.28) follows from (2.3.27) by setting ψ ≡ 0. Remark 2.3.13 One can also prove Theorem 2.3.9 with the use of Theorem 2.2.1. From the form of the density (2.3.18) we deduce that ! ρt+1 | Ft = eϕξ (x−δt )−ϕξ (−δt ) . ψt,t+1 (ξ1 , . . . , ξt , x) = E e− x,ξt+1 ρt Then (2.2.6) yields (2.3.25).

2.4 Markovian Models under the Martingale Measure In this section we formulate conditions for the discounted bond prices to be martingales under the original measure. They are based on Markovian properties of the forward rate models introduced in Section 1.2. We follow here, with some modification, the paper of Filipovi´c and Zabczyk [59] and its first version [58].

2.4 Markovian Models under the Martingale Measure

45

2.4.1 Models with Markovian Trace Writing forward rates r(t) = (r(t, 0), r(t, 1), . . .), t = 0, 1, 2, . . . in the Musiela parametrization leads to the following form of bond prices P(t, T) = e−

T−t−1 s=0

r(t,s)

, P(T, T) = 1,

t = 0, 1, . . . , T − 1.

(2.4.1)

Since R(t) = r(t, 0) determines the short rate, the evolution of a savings account is given by B(0) = 1,

t−1

B(t) = e

s=0 R(s)

t−1

=e

s=0 r(s,0)

,

t = 1, 2, . . . .

(2.4.2)

Here we will assume that, for given γ ∈ N0 , the γ -trace rγ (t) = (r(t, 0), . . . , r(t, γ )),

t = 0, 1, . . .

of the forward curve is, under the original measure P, a Markov chain on the γ +1 state space R+ with some transition operator P(·, ·). Our goal is to characterize models satisfying (MP) in terms of P. It turns out that if γ = 0 then arbitrary R1+ -valued Markov chain generates some bond market model satisfying (MP). This case corresponds to the Markovian short rate. In the case when the trace is multidimensional, i.e. γ > 0, the transition semigroup must satisfy some additional conditions. To formulate solution of the problem let us set ϕ(x) := e−x0 ,

x = (x0 , x1 , . . . , xγ ),

and define inductively functions ϕ0 , ϕ1 , . . . as follows  ϕ0 (x) := 1, ϕk+1 (x) := γ +1 e−y0 ϕk (y0 , . . . , yγ )P(x, dy), R+

γ +1

x ∈ R+ , (2.4.3)

where y := (y0 , y1 , . . . , yγ ). In other words, ϕk+1 (x) = P[ϕ · ϕk ](x),

k = 1, 2, . . . ,

γ +1

x ∈ R+ . γ +1

Theorem 2.4.1 Assume that the trace rγ is a Markov chain in R+ transition operator P. (a) If (MP) holds then the forward rate has the representation  ϕk (rγ (t)), t, k ∈ N0 , r(t, k + 1) = ln ϕk+1

with the

(2.4.4)

where {ϕk }, k = 0, 1, 2, . . . are given by (2.4.3). Moreover, if γ ≥ 1 and (MP) γ +1 holds for rγ starting from an arbitrary state in R+ then ϕk (x0 , x1 , . . . , xγ ) = e−(x1 +...+xk ) ,

k = 1, . . . , γ ,

γ +1

x = (x0 , x1 , . . . , xγ ) ∈ R+ , (2.4.5)

46

Arbitrage-Free Bond Markets and thus the transition function P(·, ·) satisfies the conditions:  γ +1 e−(y0 +...+yk ) P(x, dy) = e−(x1 +...+xk+1 ) , k = 0, . . . , γ − 1, x ∈ R+ . γ +1 R+

(2.4.6) (b) If the forward rate is given by (2.4.4) and for γ ≥ 1 the transition function satisfies (2.4.6) then the bond market with prices P(T, T) = 1,

P(t, T) = e−

T−t−1 s=0

r(t,s)

t = 0, 1, . . . , T − 1

,

satisfies (MP). Proof

(a) From (MP) we have

 P(t, T) 1 =E | Ft , B(t) B(T)

which, in view of (2.4.2), yields   T−1 P(t, T) = E e− s=t R(s) | Ft ,

t ≤ T,

t ≤ T − 1.

Consequently, for t ≤ T − 2,  T−2  P(t, T) = E e− s=t R(s) E(e−R(T−1) | FT−2 ) | Ft and, by the Markov property,

 T−2  P(t, T) = E e− s=t R(s) Pϕ(rγ (T − 2)) | Ft .

Similarly, taking into account that Pϕ = P(ϕϕ0 ) = ϕ1 , we obtain for t ≤ T − 3,  T−3  P(t, T) = E e− s=t R(s) E(e−R(T−2) ϕ1 (rγ (T − 2)) | FT−3 ) | Ft  T−3  = E e− s=t R(s) ϕ2 (rγ (T − 3)) | Ft . Finally, by induction, we obtain P(t, T) = e−R(t) ϕT−t−1 (rγ (t)),

t ≤ T − 1.

In view of (2.4.1), for t, k = 0, 1, . . ., P(t, t + k + 1) = e−

k

j=0 r(t,j)

= e−R(t) ϕk (rγ (t)).

Since R(t) = r(t, 0), so finally k  j=1

r(t, j) = − ln ϕk (rγ (t)), k ≥ 1,

(2.4.7)

2.4 Markovian Models under the Martingale Measure

47

which yields (2.4.4). Now we show (2.4.5). It follows from (2.4.7) that ϕk (r(t, 0), . . . , r(t, γ )) = e−

k

j=1 r(t,j)

k = 0, 1, . . . , γ − 1.

,

γ +1

Taking the initial trace rγ (0) = (x0 , x1 , . . . , xγ ) ∈ R+ we get the required representation (2.4.5). The restrictions on the transition function follows now from the definition of ϕk and (2.4.5). (b) To prove the converse we show that for a fixed T ∈ N0 the process P(t, T)/B(t), t ≤ T is a martingale. We can assume that T ≥ 1. Then

t−1 P(t, T) = e−[r(t,0)+···+r(t,T−t−1)] e− s=0 r(s,0) B(t)  − r(t,0)+ ln

ϕ0 ϕ1 +···+ln

=e

ϕT−t−2 ϕT−t−1

 !

(rγ (t)) − t−1 s=0 r(s,0)

,

and consequently 

P(t, T) − tu=0 r(u,0)− =e B(t)

ln

ϕ0 ϕT−t−1



(rγ (t))

= e−

t

u=0 r(u,0)

ϕT−t−1 (rγ (t)).

(2.4.8)

Since   t E e− u=0 r(u,0) ϕT−t−1 (rγ (t)) | Ft−1 

− t−1 r(u,0) u=0 =e e−y0 ϕT−t−1 (y)P(rγ (t − 1), dy) γ +1 R+

and  γ +1

R+

e−y0 ϕT−t−1 (y)P(x, dy) = ϕT−t (x),

by (2.4.8) we obtain 

P(t, T) | Ft−1 E B(t)



= e−

t−1

u=0 r(u,0)

ϕT−t (rγ (t − 1)) =

P(t − 1, T) . B(t − 1)

Remark 2.4.2 Let us stress that if γ = 0, i.e. the short-rate process is Markovian, then (2.4.4) determines forward rates for which the corresponding bond market satisfies (MP). There are no requirements for the transition operator, and the Markov chain can be chosen freely in this case. If γ > 0, then the transition function P must satisfy additional conditions (2.4.6) implied by the fact that ϕk , k = 1, 2, . . . , γ are of the form (2.4.5) and, on the other hand, are defined inductively by (2.4.3).

48

Arbitrage-Free Bond Markets

2.4.2 Affine Models γ +1

Let rγ (t), t = 0, 1, . . . with γ ≥ 0 be an R+ -valued Markov chain with transition operator P describing trace of the forward rate r(t) = r(t, k); t, k = 0, 1, 2, . . .. If there exist deterministic functions C(t), D(t), t = 0, 1, . . . with values in R+ , and γ +1 R+ respectively, such that P(t, T) = e−C(T−t)− D(T−t),r

γ (t)

t ≤ T < +∞,

,

(2.4.9)

then the bond market with such prices is called affine. The positivity of C(t) and D(t) ensures that the model is regular. We find conditions characterizing (MP) in terms of the transition operator P and exponential functions of the form fλ (x) := e− λ,x

for

1+γ

λ, x ∈ R+ .

Proposition 2.4.3 A Markovian model of the trace process rγ (t), t = 0, 1, . . . governed by the transition function P together with functions C, D form an affine model (2.4.9) satisfying (MP) if and only if γ +1

PfD(T) (x) = e−(C(T+1)−C(T))− D(T+1)−e0 ,x , T = 0, 1, . . . , x ∈ R+ ,

(2.4.10)

where e0 := (1, 0, . . . , 0) ∈ R1+γ . Proof

ˆ T + 1) = E(P(1, ˆ T + 1)) is satisfied if and only if Note that P(0, P(0, T + 1) = e−C(T+1)− D(T+1),r (0)   P(1, T + 1)  =E  F0 er(0,0)   γ = E e−r(0,0) e−C(T)− D(T),r (1) |F0 γ

= e−r(0,0)−C(T) · PfD(T) (rγ (0)). 1+γ

Consequently, for all x = (x0 , x1 , . . . , xγ ) ∈ R+

we have

PfD(T) (x) = e−C(T+1)− D(T+1),x ex0 +C(T) = e−C(T+1)− D(T+1)−e0 ,x +C(T) = e−(C(T+1)−C(T))− D(T+1)−e0 ,x .

(2.4.11)

ˆ T + 1) = The Markovian property of rγ ensures that, for any t, the condition P(t, ˆ E(P(t + 1, T + 1) | Ft ) is also equivalent to (2.4.11). It is of interest to give a complete characterization of all affine models satisfying (MP). Taking into account Proposition 2.4.3 we pose the following open problem.

2.4 Markovian Models under the Martingale Measure γ +1

49

γ +1

Problem: Describe all transition functions P(x, ·), x ∈ R+ , on R+ , for which there exist a nondecreasing sequence C(T), T = 0, 1, . . . , C(0) = 0, and a sequence γ +1 of vectors D(T) ∈ R+ , T = 0, 1, . . . , D(0) = 0, such that  γ +1 e− D(T),y P(x, dy) = e−(C(T+1)−C(T))− D(T+1)−e0 ,x , x ∈ R+ , T = 0, 1, . . . . γ +1 R+

(2.4.12) Now we introduce some subclass of transition functions solving the problem. Note that the identity (2.4.12) implies that necessarily the transition function P transforms some exponential functions fλ onto multiple of exponential functions, that is, Pfλ (x) = e− ψ(λ),x −ϕ(λ) ,

(2.4.13) 1+γ

where ψ and ϕ are some functions of λ. We say that a family {μx }, x ∈ R+ 1+γ probability measures on R+ such that weakly

μ0 = δ0 ,

μx −→ δ0 , x→0

of

μx ∗ μy = μx+y

is called infinitely divisible family or convolution semigroup of measures. Let 1+γ e0 , . . . , eγ be the standard basis in R+ . Since μ(x0 ,x1 ,...,xγ ) = μx0 e0 ∗ μx1 e1 ∗ . . . ∗ μxγ eγ , 1+γ

and for each k = 0, . . . , γ the family μyek , y ≥ 0, is infinite divisible on R+ . The Laplace transform of μx defined by  fλ (y)μx (dy) = e− ψ(λ),x 1+γ R+

is characterized by the formula ψk (λ) = βk , λ +

 1+γ R+

(1 − e− λ,y )mk (dy),

(2.4.14) 1+γ

where ψk , k = 0, 1, . . . , γ stand for the components of ψ. Above βk ∈ R+ and mk 1+γ are nonnegative measures on R+ , without atoms at 0 and such that  (1 ∧ |y|)mk (dy) < +∞, k = 0, . . . , γ , (2.4.15) 1+γ R+

see, e.g. Kallenberg [79, p. 291], for the case γ = 0, which can be extended easily to arbitrary γ . In what follows we restrict our considerations to transition functions of the form P(x, dy) = (μx ∗ ν)(dy),

(2.4.16)

50

Arbitrage-Free Bond Markets 1+γ

where {μx } is an infinitely divisible family and ν is a probability measure on R+ with the Laplace transform 

e− λ,y ν(dy) = e−ϕ(λ) , λ ∈ R+ . 1+γ

1+γ

R+

(2.4.17)

1+γ

It is clear that in this case (2.4.13) holds for each λ ∈ R+ . The infinite divisible families of measures will reappear in the part of the book devoted to L´evy processes, see Chapter 5 and Section 5.3.2. Remark 2.4.4 The family of Markov chains for which (2.4.13) holds is larger than that for which (2.4.16) is satisfied. In the example constructed by Hubalek [68] the condition (2.4.13) is satisfied but μx (A) < 0, for some x and a set A, so μx is a signed measure. Let us remark, however, that the left side of (2.4.13) is analytic in λ, so it is determined uniquely by its values on some convergent sequence. So, the requirement 1+γ that (2.4.13) holds for each λ ∈ R+ is not very restrictive. Taking into account Proposition 2.4.3 we arrive at the following solution of the preceding problem (see (2.4.12)) for the transition function (2.4.16). Theorem 2.4.5 (a) The transition function given by (2.4.16) satisfies the constraints (2.4.6) if and only if ϕ(e0 + · · · + ek−1 ) = 0, ψ(e0 + · · · + ek−1 ) = e0 + · · · + ek−1 , k = 0, . . . , γ , (2.4.18) where e0 , . . . eγ is the standard basis in R1+γ . (b) Assume that the transition function of the Markov chain rγ is of the form (2.4.16) and condition (2.4.18) holds. Then (MP) is satisfied if and only if the functions C and D are given by C(T) − C(T − 1) = ϕ(D(T − 1)), D(T) = ψ(D(T − 1)) + e0 ,

T ≥ 1,

(2.4.19)

T ≥ 1.

(2.4.20)

(c) If γ = 0, that is, R(t) = rγ (t), and the transition function of the process R(t) has the form (2.4.16) and the functions C(T), D(T) are given by (2.4.19), (2.4.20), then (MP) is satisfied. Proof Once we prove (a) the part (b) follows directly from (2.4.10). In the present situation the constraint conditions are equivalent to the identity  1+γ

R+

P(x, dy)e− e0 +...+ek−1 ,y = e− e1 +...+ek ,x ,

(2.4.21)

2.4 Markovian Models under the Martingale Measure

51

valid for k = 1, . . . , γ . It follows from the definition of the convolution of measures that the left side of (2.4.21) is equal to   e− e0 +...+ek−1 ,y+z μx (dy)ν(dz), 1+γ 1+γ R+

R+

and therefore to the product   − e0 +...+ek−1 ,y e μx (dy) 1+γ R+

1+γ

R+

e− e0 +...+ek−1 ,z ν(dz).

Now the definitions of the functions ψ and ϕ easily lead to the required identities. (c) follows from (a) and (b). Example 2.4.6 Assume that γ = 1. Then the constraints on P are of the form: ϕ(e0 ) = 0, ψ(e0 ) = (ψ0 (e0 ), ψ1 (e0 )) = (0, 1). Thus the measure ν should be supported by the set {(y0 , y1 ); y1 = 0} , β00 = 0, m0 = 0 and  0 (1 − e−y0 )m1 (dy0 , dy1 ) = 1. β1 + R2+

For more detailed analysis of the constraints for general γ we refer to Filipovi´c and Zabczyk [59].  It is instructive to derive (2.4.19) and (2.4.20) using Theorem 2.4.1 from the previous section. If (2.4.16) holds then the functions ϕk , k = 0, 1, . . . from (2.4.3) are of the following form ϕ0 = 1,

ϕk (x) = e−Ck − Dk ,x ,

k = 1, 2, . . . ,

where (Ck ), (Dk ) satisfy the recursive relations C0 = 0,

Ck+1 = Ck + ϕ(Dk + e0 ),

(2.4.22)

D0 = 0,

Dk+1 = ψ(Dk + e0 ).

(2.4.23)

In fact, by (2.4.3), we have   ϕk+1 (x) = 1+γ e−y0 ϕk (y)P(x, dy) = R+

−Ck

=e

= e−Ck



 1+γ

R+



1+γ

R+

1+γ

R+

1+γ R+

e−y0 e−Ck − Dk ,y (μx ∗ ν)(dy)

e− Dk +e0 ,y+z μx (dy) ν(dz)

e− Dk +e0 ,y μx (dy) ·

= e−Ck e− ψ(Dk +e0 ),x e−ϕ(Dk +e0 ) ,

 1+γ

R+

e− Dk +e0 ,z ν(dz)

52

Arbitrage-Free Bond Markets

which leads to (2.4.22) and (2.4.23). Thus, by Theorem 2.4.1, formula (2.4.4) r(t, k + 1) = ln e−Ck − Dk ,r

γ (t) +C

k+1 + Dk+1 ,r

γ (t)

= Ck+1 − Ck + Dk+1 − Dk , rγ (t) ,

t, k ≥ 0.

(2.4.24)

Taking into account that (Ck ), (Dk ) satisfy (2.4.22), (2.4.23), we define C(0) := 0,

D(0) := 0,

C(k) := Ck−1 ,

D(k) := Dk−1 + e0 ,

k = 1, 2, . . . .

Then (C(k)), (D(k)) satisfy (2.4.19) and (2.4.20). In fact this is true for k = 0. Assume that the result is true for k = 0, 1, . . . , n. Then C(n + 1) = Cn = Cn−1 + ϕ(Dn−1 + e0 ) = C(n) + ϕ(D(n)) and D(n + 1) = Dn + e0 = ψ(Dn−1 + e0 ) + e0 = ψ(D(n)) + e0 .

2.4.3 Dynamics of the Short Rate in Affine Models It is interesting to find equations of the form t = 0, 1, . . . ,

R(t + 1) = F(R(t)) + G(R(t))ξt+1 ,

(2.4.25)

with a sequence of independent identically distributed random variables ξ1 , ξ2 , . . . such that the transition function admits the representation P(x, dy) = (μx ∗ ν)(dy),

x > 0,

(2.4.26)

where (μx ) is an infinitely divisible family on [0, +∞) and ν is a probability measure on [0, +∞). Recall that in view of Theorem 2.4.5 (c), short rates with the transition function (2.4.26) generate affine models satisfying (MP). The answer is given by the following theorem, which provides a counterpart to CIR equations in the continuous time setting (see Section 10.2.2). We say that a nonnegative random variable ξ has the standard α-stable distribution with α ∈ (0, 1) if E(e−λξ ) = e−λ , λ ≥ 0. α

Theorem 2.4.7

If either

˜ α1 ξt+1 , R(t + 1) = (aR(t) + a˜ ) + (bR(t) + b)

t = 0, 1, . . . ,

(2.4.27)

where ξt is a sequence of independent standard α-stable distributions and a, a˜ , b, b˜ are nonnegative numbers, or R(t + 1) = aR(t) + ξt+1 ,

t = 0, 1, . . . ,

(2.4.28)

2.4 Markovian Models under the Martingale Measure

53

where (ξt ) is an arbitrary sequence of independent, identically distributed nonnegative random variables and a ≥ 0, then the transition function of the Markov chain R(t) is of the form (2.4.26). Conversely, if a Markov chain R(t) of the form (2.4.25) with F, G continuous on [0, +∞) and twice differentiable on (0, +∞) has the transition function (2.4.26) then it is either of the form (2.4.27) or (2.4.28). Only the converse of the theorem requires a proof. It will follow easily from the following two propositions of independent interest covering separately the cases G(0) = 0 and G(0) > 0. The case G(0) < 0 can be treated in a similar way. We use the standard notation for the Laplace transforms of the measures ν and μx ,  +∞  +∞ e−λy ν(dy) = e−ϕ(λ) , e−λy μx (dy) = e−xψ(λ) , 0

0

and the Laplace transform of the noise, i.e. E(e−λξ1 ) = e−ϕξ (λ) . Proposition 2.4.8 The short-rate process with the transition function (2.4.26) has the representation (2.4.25) with differentiable on (0, +∞) functions F and G such that G(0) = 0 and G(x0 ) > 0 for some x0 > 0, if and only if ψ(λ) = aλ + bλα , 1

G(x) = dx α ,

ϕ(λ) = cλ, 1

F(x) = c + ax + ex α ,

e ϕξ (λ) = − λ + bd−α λα , d with constants a, b, c ≥ 0, d > 0, 0 < α < 1 and e ∈ R. Proof

It follows from (2.4.26) that  +∞ e−λy P(x, dy) = e−xψ(λ)−ϕ(λ) .

(2.4.29)

0

Setting R(0) = x in (2.4.25) we obtain E[e−λR(1) ] = E[e−λF(x)−λG(x)ξ1 ] = e−λF(x) e−ϕξ (λG(x)) .

(2.4.30)

The equality of (2.4.29) and (2.4.30) yields the following basic relation ϕξ (λG(x)) + λF(x) = xψ(λ) + ϕ(λ),

x ≥ 0,

λ ≥ 0.

(2.4.31)

Since G(0) = 0, (2.4.31) yields ϕ(λ) = cλ with c := F(0) = R(1) ≥ 0. Consequently, ϕξ (λG(x)) = xψ(λ) − λ(F(x) − c).

(2.4.32)

54

Arbitrage-Free Bond Markets

Differentiation of (2.4.32) over λ and x yields ϕξ (G(x)λ)G(x) = xψ  (λ) − (F(x) − c), ϕξ (G(x)λ)G (x)λ = ψ(λ) − λF  (x). Dividing the identities by G(x) and G (x)λ, respectively, we obtain two formulas for the value ϕξ (G(x)λ). Comparison of them yields ψ  (λ) =

ψ(λ) G(x) ·  + λ G (x)x



F(x) − c G(x)F  (x) − , x xG (x)

(2.4.33)

which hold for λ, x > 0. Choosing a particular x we obtain ψ  (λ) =

ψ(λ) A + B, λ

λ>0

(2.4.34)

for some constants A, B. Using Lemma 2.4.10 from the sequel we obtain the solution   A  λ0 B B λ +λ (2.4.35) ψ(λ0 ) − ψ(λ) = λ0 1−A 1−A when A = 1 and

 ψ(λ) =

ψ(λ0 ) − B ln λ0 λ + Bλ ln λ λ0

when A = 1, with some λ0 > 0. In the case A = 1, the positivity of ψ implies that B = 0 and, consequently that ψ is linear. So, in each case ψ has the form ψ(λ) = aλ + bλα for α = A and relevant constants a, b. Therefore, in view of (2.4.14) we obtain  α aλ + bλ = βλ + (1 − e−λy )m(dy), λ > 0 R+

for some β > 0 and a measure m(dy), which integrates (1 ∧ y). One can justify that the preceding right side is differentiable and obtain  +∞ α−1 =β+ e−λy y m(dy), λ > 0. a + αbλ 0

We see that for α > 1 the preceding left side is an increasing function of λ while the right side decreases. The case α = 1 can also be excluded because then G(0) = 0. Thus α ∈ (0, 1) and a, b ≥ 0. Since ψ satisfies (2.4.33) for all x > 0 and it is given by (2.4.35), we conclude that for all x > 0, G(x) = A = α, G (x)x

F(x) − c G(x)F  (x) − = B. x xG (x)

2.4 Markovian Models under the Martingale Measure

55

Applying Lemma 2.4.10 again, with x0 > 0 such that G(x0 ) > 0, we obtain 1

G(x) = (x/x0 )1/α G(x0 ) =: dx α ,  F(x) = (x/x0 )1/α F(x0 ) − c + x0

 1 B B −x + c =: ex α + ax + c. α−1 α−1

To determine the formula for ϕξ we set z := λG(x) in (2.4.32). This yields  z z − (F(x) − c) ϕξ (z) = xψ G(x) G(x)   z α  1 z e z  =x a 1 +b − 1 ax + ex α = bd−α zα − z. 1 d dx α dx α dx α The fact that F, G and ϕξ given by the theorem satisfy really the basic identity (2.4.31) one shows by direct calculation. Proposition 2.4.9 The short-rate process with the transition semigroup (2.4.26) has the representation (2.4.25) with continuous on [0, +∞) and twice differentiable on (0, +∞) functions F and G such that G(0) > 0, if and only if ˜ α, ϕ(λ) = a˜ λ + bλ

ψ(λ) = aλ + bλα ,

1  1/α α G(0)  b ˜ G(x) = , F(x) = ax + a˜ − x + 1 (˜a − F(0)), bx + b ˜b1/α ˜b  α a˜ − F(0) ˜ λ ϕξ (λ) = λ +b G(0) G(0) with constants a, b, a˜ , b˜ ≥ 0, 0 < α < 1, or ψ(λ) = aλ, G(x) ≡ 1,

ϕ(λ) − arbitrary, F(x) = ax,

ϕξ (λ) = ϕ(λ), where a ≥ 0. Proof We sketch the proof because it is similar to the proof of Proposition 2.4.8. We start from (2.4.31) by putting into the relation ϕξ (λG(x)) + λF(x) = xψ(λ) + ϕ(λ),

x ≥ 0,

λ≥0

the value x = 0. This yields ϕξ (λG(0)) + λF(0) = ϕ(λ),

λ ≥ 0,

(2.4.36)

56

Arbitrage-Free Bond Markets

and, since G(0) > 0,  ϕξ (λ) = ϕ

λ G(0)

−λ

F(0) , G(0)

λ ≥ 0.

(2.4.37)

Using this formula we put ϕξ (λG(x)) into (2.4.36), which yields  λG(x) F(0) λF(x) + ϕ − λG(x) = xψ(λ) + ϕ(λ), λ, x ≥ 0. G(0) G(0)

(2.4.38)

Now one can differentiate (2.4.38) over x and over λ and determine the following two formulas  F  (x)G(0) ψ(λ)G(0) λG(x) ϕ = F(0) − + , λ, x ≥ 0, (2.4.39) G(0) G (x) λG (x)  F(x)G(0) xψ  (λ)G(0) ϕ  (λ)G(0) λG(x) ϕ = F(0) − + + , λ, x ≥ 0. G(0) G(x) G(x) G(x) (2.4.40) Comparison of them yields F(x) − F  (x)

G(x) ψ(λ) G(x) + = xψ  (λ) + ϕ  (λ),  G (x) λ G (x)

λ, x ≥ 0.

It follows by differentiation over x that ψ  (λ) = A where

  d G(x)  A := , dx G (x) x¯

ψ(λ) + B, λ

  d G(x)   B := F(x) − F (x)   dx G (x) x¯

and x¯ is some point from [0, +∞). So, ψ satisfies the same equation as in the previous proof (see (2.4.34)) and one concludes that ψ(λ) = aλ + bλα ,

λ≥0

with α = A ∈ (0, 1] and a, b ≥ 0. By (2.4.39), ˜ α, ϕ(λ) = a˜ λ + bλ

λ ≥ 0,

where a˜ , b˜ ≥ 0. Putting the preceding formulas into (2.4.38) yields   G(x)˜a G(x)F(0) G(x) α ˜ α ˜ λ F(x) + b − xb − b = 0, − − ax − a˜ + λ G(0) G(0) G(0) λ, x ≥ 0.

2.4 Markovian Models under the Martingale Measure

57

It follows that the coefficients standing by λ and by λα disappear, which enables us to determine the functions F and G. Finally, we obtain 1/α G(0)  , x ≥ 0, bx + b˜ G(x) = b˜ 1/α 1  α b F(x) = ax + a˜ − x + 1 (˜a − F(0)). ˜b Let us consider the case α ∈ (0, 1). Let ξ˜ be a random variable with distribution ν. Then ˜

˜

E(e−λξ ) = e−ϕ(λ) = e−˜aλ−bλ , α

which means that ξ˜ − a˜ is α-stable distributed. It follows from (2.4.37) that in this case the distribution of the noise in (2.4.25) is given by ξ1 =

ξ˜ F(0) − . G(0) G(0)

One can check that the condition for positivity of the short rate F(x) + G(x)ξ1 ≥ 0,

x≥0

is satisfied. If α = 1 then ψ(λ) = aλ,

λ ≥ 0,

which means that μx = δ{xa} . Let ζ have now an arbitrary distribution ν on R+ . Then P(x, dy) = μx ∗ ν if and only if R(t + 1) = aR(t) + ζ ,

t ≥ 0.

Proof of Theorem 2.4.7

If a random variable ξ is such that e ϕξ (λ) = − λ + bd−α λα , α ∈ (0, 1), d then it can be represented in the form e b1/α ξ =− + ξ˜ , d d where ξ˜ has a standard α-stable distribution. Similarly, if  α a˜ − F(0) ˜ λ , +b ϕξ (λ) = λ G(0) G(0)

58

Arbitrage-Free Bond Markets

then ξ=

a˜ − F(0) b˜ 1/α + ξ˜ . G(0) G(0)

Taking into account the formulae for F and G in the cases G(0) = 0 and G(0) > 0, respectively, one can obtain, after some calculations, the required formula (2.4.27) for R involving the standard α-stable random variables. If ϕξ is arbitrary, which is the second case in Proposition 2.4.9, one obtains (2.4.28). Lemma 2.4.10

Solution of the equation

y(x) α + γ , x > x0 , x with given y(x0 ) for x0 > 0, is of the form  α   x x0 γ γ y(x) = +x y(x0 ) − x0 1−α 1−α y (x) =

for α = 1 and

 y(x) =

y(x0 ) − γ ln x0 x + γ x ln x x0

for α = 1.

2.4.4 Shape of Forward Curves in Affine Models We consider now the shapes of forward curves k → r(t, k) for affine models satisfying (MP) in the case when γ = 0. The transition function P has the form introduced earlier, i.e. P(x, dy) = (μx ∗ ν)(dy),

x, y ∈ R+ .

To exclude trivial case we assume that ν = δ{0} and μx ((0, +∞)) > 0, for x > 0. Since μx is a convolution semigroup on R+ , the function ψ given by  e−λy μx (dy) = e−ψ(λ)x , λ ≥ 0 (2.4.41) R+

must have the form ψ(λ) = βλ + ψ0 (λ),

 with ψ0 (λ) :=

R+

(1 − e−λy )m(dy),

where β ≥ 0 and the measure m satisfies  (1 ∧ y)m(dy) < +∞ R+

λ ≥ 0, (2.4.42)

2.4 Markovian Models under the Martingale Measure (compare (2.4.14), (2.4.15)). Recall that ϕ is given by  e−λy ν(dy) = e−ϕ(λ) , λ ≥ 0.

59

(2.4.43)

R+

Using (2.4.22), (2.4.23) and (2.4.24), we can write forward rates in the form r(t, k) = ck + dk R(t), where ck :=

# 0

for k = 0,

Ck − Ck−1

for k ≥ 1,

k ∈ N0 , #

dk :=

(2.4.44)

1

for k = 0,

Dk − Dk−1

for k ≥ 1.

The following result describes the behaviour of the sequences (ck ), (dk ), which determine the shapes of forward curves. Theorem 2.4.11

The sequence (ck ) is strictly increasing with # = +∞ if β ≥ 1, lim ck k→+∞ < +∞ if β < 1.

(a) If β > 1 then {dk } is strictly increasing and limk dk = +∞.  +∞ (b) If β = 1 then {dk } is nondecreasing and limk dk = 1 + 0 m(dy). (c) If β < 1 then limk dk = 0 and there exists k∗ ∈ N0 such that dk , k ≥ k∗ is strictly decreasing. If ψ  (1 + ψ(1)) ≥ 1 then {dk } has a hump. Proof First we show that {Dk } is increasing. Since ψ(·) is increasing and, by (2.4.23) we get Dk+1 − Dk = ψ(Dk + 1) − ψ(Dk−1 + 1),

(2.4.45)

we see that Dk−1 ≤ Dk implies that Dk ≤ Dk+1 . So, the monotonicity of {Dk } follows by induction. Since ck = ϕ(Dk−1 + 1) and ϕ(·) is also increasing, we obtain that ck ↑ +∞ if and only if Dk ↑ +∞. We show now that {Dk }

is bounded

⇐⇒

β < 1.

(2.4.46)

Let us assume that β ≥ 1 and {Dk } is bounded. There exists D such that Dk ↑ D and passing to the limit in the condition Dk+1 = ψ(Dk + 1) we obtain D = ψ(D + 1). Consequently, D = ψ(D + 1) = β(1 + D) + ψ0 (1 + D) ≥ 1 + D + ψ0 (1 + D), which is impossible because ψ0 (·) ≥ 0. Now let us assume that β < 1 and {Dk } is unbounded. By (2.4.45) and the Lagrange theorem there exists ηk such that Dk+1 − Dk = ψ  (ηk )(Dk − Dk−1 ),

Dk−1 + 1 ≤ ηk ≤ Dk + 1.

(2.4.47)

60

Arbitrage-Free Bond Markets

It follows that ηk −→ +∞ and, since 



+∞

ψ (λ) = β +

ye−λy m(dy),

0

there exists k∗ and γ such that ψ  (ηk ) ≤ γ < 1 for k > k∗ . Then Dk+1 − Dk ≤ γ (ψ(Dk +1)−ψ(Dk−1 +1)), k > k∗ , which means that {Dk } converges geometrically to zero, which is a contradiction. Now we examine the sequence {dk }. If β < 1 then, by (2.4.46), dk+1 = Dk+1 − Dk ≤ D − Dk −→ 0, where D stands for the limit of Dk . Writing (2.4.47) in the form dk+1 = ψ  (ηk )dk and using the fact that ψ  (ηk ) is decreasing we see that a sufficient condition for {dk } to have a hump is that ψ  (η1 ) ≥ 1. Since 1 ≤ η1 ≤ D1 + 1 and D1 = ψ(1), we need ψ  (1 + ψ(1)) ≥ 1. If β = 1 then dk+1 = Dk+1 − Dk = [(Dk + 1) + ψ0 (Dk + 1)] − Dk  +∞ = 1 + ψ0 (Dk + 1) −→ 1 + m(dy), 0

where the latter convergence holds because Dk ↑ +∞. Since ψ  (·) > β, (2.4.47) yields Dk+1 − Dk ≥ β(Dk − Dk−1 ). Hence, dk −→ +∞ geometrically fast if β > 1. Example 2.4.12 We examine the sufficient condition ψ  (1 + ψ(1)) ≥ 1, from c dy, where 0 < δ < 1 Theorem 2.4.11, for {dk } to have a hump in the case m(dy) = y1+δ and c is a constant. Then ψ(λ) = βλ + c¯ cλδ , where c¯ is another constant, which is independent of c, see Example 5.3.3 in the sequel for detailed calculations. Since 1 and ψ(1) = β + c¯c, it follows that ψ  (λ) = β + c¯ cδ λ1−δ ψ  (1 + ψ(1)) = β + c¯cδ

1 . (β + c¯c)1−δ

Consequently, ψ  (1 + ψ(1)) ≥ 1

⇐⇒

δc¯c(β + c¯c)δ ≥ 1 − β. β + c¯c

It is clear now that we can find c such that the preceding right side is satisfied.

2.4 Markovian Models under the Martingale Measure

61

2.4.5 Factor Models In the Markovian setting one can incorporate into the model concrete shapes of bond and forward curves. Let us consider a factor model in which forward rate is of the form t, T = 0, 1, . . . , T ≥ t,

f (t, T) = G(T − t, Xt ),

(2.4.48)

where G is some deterministic positive function and X a Markov chain on (E, E), with transition operator P(·, ·), defined on a probability space (, F, {Ft }, P). For a fixed value x of the factor X(t), the function v → G(v, x) describes thus the shape of the forward curve in terms of time to maturity v := T − t. Since P(t, T) = e−

T−1 s=t

f (t,s)

= e−

T−1 s=t

G(s−t,Xt )

= e−

T−t−1 u=0

G(u,Xt )

, t, T = 0, 1, . . . , T ≥ t,

it follows that P(t, T) = F(T − t, X(t)),

(2.4.49)

where F is given by F(0, x) = 1,

F(v, x) := e−

v−1 s=0

G(s,x)

,

v = 1, 2, . . . .

(2.4.50)

In particular, F(1, x) = e−G(0,x) , x ∈ E. Similarly, for a given value x of X(t) the function v → F(v, x) describes the bond curve. In the case when X(t) is equal to the short rate R(t) on E = [0, +∞), we have R(t) = f (t, t) = G(0, R(t)), and consequently G must be such that G(0, x) = x,

x ≥ 0.

More generally, if X(t) is the γ -trace of the forward rate rγ (t) = (r(t, 0), r(t, 1), . . . , r(t, γ )) with γ > 0, then r(t, k) = f (t, t + k) = G(k, r(t, 0), . . . r(t, k), . . . , r(t, γ )),

k = 0, 1, . . . , γ ,

so G must satisfy G(k, x0 , . . . , xγ ) = xk ,

x0 , x1 , . . . , xγ ≥ 0,

k = 0, 1, . . . , γ .

In factor models satisfying (MM) or (MP) the function G, or equivalently, F, and the transition operator P must satisfy certain conditions that we deduce in the following. If the underlying filtration {Ft } is generated by a sequence ξ1 , ξ2 , . . . of

62

Arbitrage-Free Bond Markets

independent identically distributed random variables, then the Markov process X can be represented in the form X(t + 1) = K(X(t), ξt+1 ),

t = 0, 1, . . .

(2.4.51)

for some function K. The existence of the representation (2.4.51) for an arbitrary Markov chain is a direct conclusion from Theorem 1.2 in Peszat and Zabczyk [100]. Let μ stand for the distribution of ξ1 . Recall that for a measure Q ∼ P one can write the density process in the form ρ0 = 1,

t = 1, 2, . . . T ∗

ρt = ψt (ξ1 , . . . ξt ),

(2.4.52)

for some functions ψt . Theorem 2.4.13 Let Q ∼ P be a measure with density (2.4.52). Then Q is a martingale measure if and only if  ψt+1 (ξ1 , . . . , ξt , y)F(T − t − 1, K(X(t), y))μ(dy) U

= ψt (ξ1 , . . . , ξt )eG(0,X(t)) F(T − t, X(t))

(2.4.53)

for each T = 1, 2, . . . and t = 0, 1, . . . , T − 1. Proof

ˆ T) is a Q-martingale if and only if The process P(t, ˆ + 1, T) | Ft ) = ρt P(t, ˆ T), E(ρt+1 P(t

t = 0, 1, . . . , T − 1.

Since the bank account is given by B(t) = e−

t−1

s=0 R(s)

= e−

t−1

s=0 G(0,X(s))

,

by (2.4.49) we obtain ˆ T) = ψt (ξ1 , . . . , ξt )e− ρt P(t,

t−1

s=0 G(0,X(s))

F(T − t, X(t)).

(2.4.54)

In view of (2.4.51) we have ˆ + 1, T) | Ft ) E(ρt+1 P(t = e− = e−

t

s=0 G(0,X(s))

E (ψt+1 (ξ1 , . . . , ξt+1 )F(T − t − 1, X(t + 1)) | Ft )

t

E (ψt+1 (ξ1 , . . . , ξt+1 )F(T − t − 1, K(X(t), ξt+1 )) | Ft ) 

t ψt+1 (ξ1 , . . . , ξt , y)F(T − t − 1, K(X(t), y))μ(dy). = e− s=0 G(0,X(s)) s=0 G(0,X(s))

U

(2.4.55) From the equality of (2.4.55) and (2.4.54) we obtain the required formula.

2.4 Markovian Models under the Martingale Measure

63

It follows from (2.4.51) that the transition operator of X is given by  Ph(x) = E(h(K(x, ξ1 ))) = h(K(x, y))μ(dy). U

This implies, that in the case when ψ = ψt ≡ 1, (2.4.53) boils down to the condition  F(T − t − 1, K(X(t), y))μ(dy) = P(F(T − t − 1, X(t))) = eG(0,X(t)) F(T − t, X(t)). U

This allows formulating conditions for models to satisfy (MP). As a consequence of Theorem 2.4.13, we obtain the following result. Theorem 2.4.14

The factor model satisfies (MP) if and only if

F(k + 1, x) = F(1, x)P(F(k, ·))(x),

F(0, x) = 1, x ∈ E

k = 0, 1, 2, . . . .

In particular, any factor model satisfying (MP) is determined by the function F(1, x) and the transition operator P of the Markov process X. With the use of Theorem 2.4.14 one can characterize models satisfying (MP) with multiplicative factor X. Proposition 2.4.15 Let the factor X be given by Xt+1 = aXt + bXt ξt+1 ,

X0 = x > 0,

t = 1, 2, . . . ,

where {ξt } is an i.i.d. sequence and a, b are constants. Then the model (2.4.48) with F(1, x) := xγ , x > 0, γ ∈ R, satisfies (MP) if and only if F(k, x) = cγ c2γ . . . c(k−1)γ xkγ ,

k = 1, 2, 3, . . . ,

(2.4.56)

where cγ := E[(a + bξ1 )γ ]. Consequently, under (MP), the bond prices are given by (T−t)γ

P(t, T) = F(T − t, X(t)) = cγ c2γ . . . c(T−t−1)γ Xt

.

If, additionally, γ > 0, E = (0, 1),

0 < a + bξ1 < 1,

P − a.s.

or γ < 0, E = (1, +∞),

a + bξ1 > 1,

P − a.s.,

then the forward rates are positive. In particular, the model is then regular. Proof Let us notice that the class of power functions hα (x) = xα , x > 0, is invariant for the transition operator P. That is Phα (x) = xα E[(a + bξ1 )α ] = cα xα ,

α = 0, x > 0.

64

Arbitrage-Free Bond Markets

It follows from Theorem 2.4.14 that for F(1, x) = xγ , x > 0, F(2, x) = F(1, x)cγ xγ = cγ x2γ and by the inductive argument F(k, x) =

 k−1 $

 cjγ xkγ ,

j=1

which is the required formula (2.4.56). Let us formulate conditions for the positivity of f (t, T) = G(T − t, Xt ). Since e−G(k,x) =

cγ c2γ . . . ckγ x(k+1)γ F(k + 1, x) = F(k, x) cγ c2γ . . . c(k−1)γ xkγ

= ckγ xγ ,

k = 1, 2, . . . ,

we obtain G(0, x) = −γ ln x,

G(k, x) = − ln ckγ − γ ln x,

k = 1, 2, . . . .

It follows that forward rates are positive if X evolves in E = (0, 1) for γ > 0 or in E = (1, +∞) for γ < 0 and, in both cases, ckγ ∈ (0, 1) for k = 1, 2, . . .. This leads to the required conditions.

3 Completeness

In this chapter we study the completeness problem. It turns out that the majority of market models are not complete but, nevertheless, may still be approximately complete. The crucial role here is played by portfolios with an infinite number of bonds. Without them, even approximate completeness may fail. The first sections discuss completeness with respect to the minimal filtration. Models with martingale discounted bond prices are covered next. The final section is about the interplay between completeness and martingale measures.

3.1 Concepts of Completeness Let X be an Ft -measurable claim, with t > 0, on a probability space (, F, P) with filtration {Fs }, s = 0, 1, . . .. Recall that X is attainable at t if there exists a selffinancing strategy {(b(s), ϕs ), s = 0, 1, . . . , t}, where ϕs takes values in l1 , such that the corresponding portfolio wealth X(s) = b(s) + ϕs , P(s) ,

s = 0, 1, . . . , t

satisfies X(t) = X. As explained in Section 1.4 (see Proposition 1.4.1), X is attainable if and only if the discounted claim admits the representation Xˆ = x +

t−1  ˆ + 1) , ϕs , P(s

P − a.s.

(3.1.1)

s=0

ˆ + 1) = for some x ∈ R and some l1 -valued adapted strategy ϕs . Here P(s ˆ P(s + 1, j), j = s + 2, s + 3, . . . stands for the increments of discounted bond prices. Loosely speaking, the market is called complete if, for any t > 0, each claim is attainable. A precise definition of completeness requires specific conditions for the class of claims that are to be replicated. Typically X or Xˆ is supposed to be bounded or p-th power integrable with p ≥ 1. A specification of the underlying filtration is also important. Often, but not always, {Fs } is assumed to be the minimal filtration, i.e.

66

Completeness Fs = σ { P(u, T); u = 0, 1, . . . , s, T = 0, 1, 2, . . .},

s = 0, 1, . . . .

In this case each claim X at time t > 0, can be written as some function of bond prices observed up to time t, i.e. X = X(P(0), P(1), . . . , P(t)),

(3.1.2)

where P(s) = (P(s, j), j = 0, 1, 2, . . .) stands for the bond curve at time s = 0, 1, . . . , t. The class of claims of the preceding form is wide enough to comprise financial contracts that are traded in practice. If {Fs } is required to be the minimal filtration, one can write the replicating condition (3.1.1) in the more convenient form, which will be used in the sequel. Let us notice that (3.1.1) involves increments of the discounted bond prices. Therefore it is natural to expect that one can replace in (3.1.1) the minimal filtration {Fs } by the filtration generated by the discounted bond prices, i.e. ˆ

ˆ T); u ≤ s, T = 0, 1, . . .}, FsP := σ {P(u,

s = 0, 1, . . .

The following result confirms this conjecture. Proposition 3.1.1 For every t = 0, 1, . . ., ˆ

f

Ft = FtP = Ft , where f

Ft := σ {f (s, T); s ≤ t, T = 0, 1, . . .}. f

The filtration {Ft } allows us to deal also with claims depending on forward curves. Proof

ˆ Let us recall basic relations between f , P and P: ⎧ T−1 − s=t f (t,s) for t = 0, 1, . . . , T − 1; ⎪ ⎪ ⎨e P(t, T) := 1 for t = T; ⎪

⎪ ⎩e t−1 f (s,s) s=T for t = T + 1, T + 2, . . . ; f (t, s) = f (s, s), # ˆ T) := P(t,

e−

1

T−1 s=0

f (t,s)

s = 0, 1, . . . , t;

for t = 0, 1, . . . ; T = 1, 2, . . . ; for t = 0, 1, . . . ; T = 0.

If s ≥ t then, by (3.1.3), f (t, s) = ln P(t, s) − ln P(t, s + 1),

(3.1.3)

(3.1.4)

(3.1.5)

3.1 Concepts of Completeness

67

and consequently σ {f (t, s)} ⊆ Ft . If s < t, then, by (3.1.4), σ {f (t, s)} = σ {f (s, s)} and therefore σ {f (t, s)} ⊆ Ft . Consequently, σ {f (t, s)} ⊆ Ft for all s = 0, 1, . . . . Thus f

Ft ⊆ Ft . Since σ {P(t, T); T = 0, 1, . . .} ⊆ σ {f (s, T); s = 0, 1, . . . , t; T = 0, 1, . . .}, we have that f

Ft ⊆ Ft , and consequently f

Ft = Ft . By (3.1.5), ˆ

f

FtP ⊆ Ft . On the other hand, for t = 0, 1, . . . , s = 0, 1, . . . ˆ s) − ln P(t, ˆ s + 1), f (t, s) = ln P(t, so f

ˆ

Ft = FtP . The proof is complete. Let us denote by ηs := (ηs (s + 1), ηs (s + 2), . . .) the sequence describing increments of the discounted bond prices, i.e. ˆ j) − P(s ˆ − 1, j), ηs (j) := P(s,

j = s + 1, s + 2, . . .

and η0 := (P(0, 1), P(0, 2), . . .). Then, obviously, ˆ

FsP = σ {η0 , η1 , . . . , ηs },

s = 0, 1, . . . .

(3.1.6)

In view of (3.1.1), (3.1.6) and Proposition 3.1.1 the completeness problem with the ˆ 0 , η1 , . . . , η t ) minimal filtration boils down to representing each bounded claim X(η in the form ˆ 0 , η1 , . . . , η t ) = x + X(η

t−1  ϕs (η0 , η1 , . . . , ηs ), ηs+1 ,

P − a.s.

(3.1.7)

s=0

As we will see in the sequel, completeness is rather a rare feature of the market as it enforces very restrictive conditions for the bond price process. Therefore one considers also weaker concepts of completeness, where claims are approximated in

68

Completeness

a certain sense. The market is weakly complete at time t > 0 if for each bounded Ft -measurable claim Xˆ and an arbitrary level of accuracy ε > 0, there exists a strategy such that ˆ − Xˆ |< ε, | X(t)

P − a.s.

The interpretation of the preceding condition is that the replication error can be made arbitrarily close to zero. The market is Lp -approximately complete a time t, where p ≥ 1, if for any t > 0, Xˆ ∈ Lp (, Ft , P) and ε > 0 one can find a strategy such that   ˆ − Xˆ |p < ε. E | X(t) If the properties above hold for all t = 1, 2, . . ., then one says that the markets are, respectively, weakly complete or Lp -approximately complete. In the sequel we characterize complete, weakly complete and approximately complete models and discuss the link with the existence and uniqueness of the martingale measure. In Section 3.5 and Section 3.6 we study the completeness problem with other filtrations than the minimal one. To avoid ambiguities, in the formulation of each result we will indicate the class of claims in which the completeness problem is considered. Recall that L∞ (, Ft , P) stands for the set of all bounded Ft -measurable random variables.

3.2 Necessary Conditions for Completeness The representation (3.1.7) required for completeness under the minimal filtration {Ft } depends on properties of the process (ηt ). First we show that weak completeness, like completeness, fails if at least one ηt takes an infinite number of values with positive probability. If the distribution of a random variable η is concentrated on a finite set, one says that η takes a finite number of values. In the opposite case η is said to take an infinite number of values. Theorem 3.2.1 If, for some t ≥ 1, ηt takes an infinite number of values, then the market is not weakly complete at t in the class Xˆ ∈ L∞ (, Ft , P). Proof Assume to the contrary, that ηt takes an infinite number of values and the market is weakly complete at time t. Without loss of generality we can require that the random variables η1 , . . . , ηt−1 take a finite number of values, say ηs takes values as1 , as2 , . . . , asKs , s = 1, 2, . . . , t − 1 and K1 , . . . , Ks are some natural numbers. Let U be the space of infinite sequences x = {xm } taking values in [−1, 1] equipped with the metric ∞  1 | xm − x¯ m | · , x = {xm }, x¯ = {¯xm }. ρ(x, x¯ ) = m 2 1+ | xm − x¯ m | m=1

3.2 Necessary Conditions for Completeness

69

With the use of this metric one can decompose U into the form U=

∞ %

Uj ,

j=1

where Uj are disjoint Borel subsets of U such that P(ηt ∈ Uj ) > 0,

j = 1, 2, . . . .

If h, h¯ are two different functions on U taking values 0 or 1 and constant on each of the sets Uj , then ¯ t ) ||:= ess sup | h(ηt (ω)) − h(η ¯ t (ω)) |= 1. || h(ηt ) − h(η Weak completeness at time t implies that for any function h taking values 0 or 1 on the sets Uj , j = 1, 2, . . ., there exist x and portfolios ϕ0 , ϕ1 , . . . , ϕt−1 such that ϕs = ϕs (η1 , . . . , ηs ) = {ϕsj (η1 , . . . , ηs ), j = 1, 2, . . .}, | ϕs |l1 =

+∞ 

| ϕsj |< +∞,

s = 0, 1, . . . , t − 1,

j=1

and t−1    ϕs , ηs+1 ) |< ε = 1. P | h(ηt ) − (x + s=0

Note that ϕs , ηs+1 =

+∞ 

j

ϕsj (η1 , . . . , ηs )ηs+1 ,

s = 0, 1, . . . , t − 1,

(3.2.1)

j=1 j

j

where ηs+1 are coordinates of ηs+1 and satisfy | ηs+1 |≤ 1. The series in (3.2.1) converges uniformly. Moreover, +∞ 

j

ϕsj (η1 , . . . , ηs )ηs+1 =

+∞ 



j=1 k1 =1,2,...,K1 ··· ks =1,2,...,Ks

j=1

j,k ,...,ks

1 ϕsj (a1k1 , . . . , asks )ηs+1

where j,k ,...,ks

1 ηs+1

:= 1{η1 =a1

s k1 ,...,ηs =aks }

j

ηs+1

and +∞ 



j=1 k1 =1,2,...,K1 ··· ks =1,2,...,Ks

| ϕsj (a1k1 , . . . , asks ) |< +∞,

j,k ,...,ks

1 | ηs+1

|≤ 1.

,

70

Completeness j,k ,...,k

s 1 For each s, the family {ηs+1 ; j = 1, 2, . . . , ki = 1, 2, . . . , Ki , i = 1, 2, . . . , s} is countable and the series is convergent in the essential supremum-norm. Thus each random variable h(ηt ) can be approximated in this norm by linear combinations of j,k1 ,...,ks } which is a contradiction with the fact that finite sums of random variables {ηs+1 for a continuum of random variables h(ηt ) the mutual distances are equal to 1. Assume that for each h(ηt ) there exist an approximating linear finite combination with rational coefficients of elements of the set

j,k ,...,ks

1 {ηs+1

,

j = 1, 2, . . . ; ki = 1, 2, . . . , Ki ; i = 1, . . . , s; s = 1, 2, . . . , t − 1},

with the distance to h(ηt ) less than 1/2. Then for different random variables h(ηt ) the approximating sequnces are different. This is, however, impossible because the set of random variables h(ηt ) is uncountable.

3.3 Sufficient Conditions for Completeness It follows from Theorem 3.2.1 that a necessary condition for the market to be complete under the natural filtration {Ft } is that ηt takes a finite number of values for each t = 1, 2, . . . . Now we formulate sufficient conditions in this case. We begin by showing that the concepts of completeness and weak completeness coincide. Since each claim Xˆ takes a finite number of values, we obviously have that Xˆ ∈ L∞ (, Ft , P) with some t ≥ 1. Proposition 3.3.1 Let {Ft } be the minimal filtration. If, for each t > 0, ηt takes a finite number of values then the market is complete if and only if it is weakly complete. Proof Let dt < +∞ stand for the number of values taken by ηt , where t = 1, 2, . . . and d0 = 1 by definition. The set of all portfolios at time t ≥ 0 of the form ϕt (η0 , η1 , . . . , ηt ) can be identified with the space (l1 )d1 ·d2 ·...·dt . Consequently, the discounted wealth process at time t corresponding to the pair (x, (ηt )) given by Kt (x, ϕ0 , ϕ1 , . . . , ϕt−1 ) := x +

t−1  ϕs , ηs+1 s=0

is a linear transformation from R × l1 × (l1 )d1 × (l1 )d1 ·d2 × · · · × (l1 )d1 ·d2 ·...·dt−1

into Rd1 ·d2 ·...·dt .

Since Rd1 ·d2 ·...·dt describes the set of all discounted claims paid at time t of the form ˆ 0 , η1 , . . . , ηt ), Xˆ = X(η

3.3 Sufficient Conditions for Completeness

71

the market is weakly complete if and only if, for each t > 0, the image of Kt truncated to the ball B := {y ∈ Rd1 ·d2 ·...·dt :| y |≤ 1} is dense in B. This is, however, possible if and only if Im(Kt ) = Rd1 ·d2 ·...·dt ,

t > 0,

which means that the market is complete. Now we formulate specific conditions for the market to be complete. For a finite sequence of vectors {a1 , a2 , . . . , ad } in m and their linear span Gd := span{a1 , a2 , . . . , ad }, let us introduce the following nondegeneracy conditions: • (ND1) dim Gd = d; • (ND2) dim Gd = d − 1 and the d-dimensional vector (1, 1, . . . , 1) does not belong to the image of the operator A : l1 −→ Rd defined by ⎛ ⎞ b, a1 ⎜ b, a2 ⎟ ⎜ ⎟ Ab := ⎜ (3.3.1) ⎟ , b ∈ l1 . .. ⎝ ⎠ . b, ad Theorem 3.3.2 Let {Ft } be the minimal filtration and the number of values of ηt be finite for any t > 0. Then the market is complete if and only if, for each t > 0, the conditional distribution of ηt with respect to η0 , η1 , . . . , ηt−1 is concentrated on the set {at1 , at2 , . . . , atdt }, with dt < +∞, of vectors in m which satisfies (ND1) or (ND2). If this is the case then each replicating strategy can be replaced by a replicating strategy with finite portfolios at any time. In the preceding formulation the set {at1 , at2 , . . . , atdt } and dt depend, of course, on η0 , η1 , . . . , ηt−1 . The second part of Theorem 3.3.2 follows from the fact that any ˜ so that the final wealth l1 -valued strategy ϕs can be reduced to some finite one ϕ, remains the same, i.e. ˆ =x+ X(t)

t−1 

ϕs , ηs+1 = x +

s=0

t−1  ϕ˜s , ηs+1 . s=0

This is possible because ηt takes a finite number of values in m for each t ≥ 0. We prove this in Proposition 3.3.4. The proof of the first part of Theorem 3.3.2 is based on the following auxiliary result. Proposition 3.3.3 Let a1 , a2 , . . . , ad be a finite set of vectors in m. Any sequence of reals γ1 , γ2 , . . . , γd can be represented in the form γi = a + b, ai ,

i = 1, 2, . . . , d,

(3.3.2)

72

Completeness

where a ∈ R and b ∈ l1 if and only if (ND1) or (ND2) is satisfied. If (ND2) holds then a in the representation (3.3.2) is unique while if (ND1) then for each a there exists b such that (3.3.2) holds. Proof Let m ≤ d denote the dimension of Gd and assume for simplicity that the first m-vectors a1 , a2 , . . . , am are linearly independent. Then for any sequence γ1 , γ2 , . . . , γm there exists b ∈ l1 such that b, ai = γi , i = 1, 2, . . . , m, which means that the image of the operator A given by (3.3.1) is of dimension m. Writing (3.3.2) in the form ⎛ ⎞ ⎛ ⎞ γ1 1 ⎜ γ2 ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ (3.3.3) ⎜ . ⎟ = Ab + a ⎜ . ⎟ , b ∈ l1 , ⎝ .. ⎠ ⎝ .. ⎠ γd

1

we see that the representation of an arbitrary sequence γ1 , γ2 , . . . , γd is possible if and only if the image of the right side of (3.3.3) equals Rd . This may happen in two situations only. Either m = d or m = d−1 and (1, 1, . . . , 1) ∈ / ImA, which correspond to (ND1) and (ND2), respectively. If (ND1) holds then clearly for any a ∈ R we can find b ∈ l1 such that (3.3.3) is satisfied. If (ND2) holds then γ = (γ1 , . . . , γd ) has a unique decomposition of the form γ = γ  + γ  , where γ  ∈ ImA and γ  ∈ (ImA)⊥ . Since (ImA)⊥ = span{(1, 1, . . . , 1)}, the constant a in (3.3.3) is unique. In general, there are many vectors b ∈ l1 solving Ab = γ  . Proposition 3.3.4 Let a1 , . . . , ad be a finite set of vectors in m. Then there exists n ∈ N such that for any ϕ ∈ l1 there exists ϕ˜ ∈ Rn such that ϕ, ai = ϕ, ˜ a(n) i Rn ,

i = 1, 2, . . . , d. (n)

Above ·, · Rn stands for the scalar product in Rn and ai is the truncation of ai = (ai (1), ai (2), . . .) restricted to the first n coordinates, i.e. (n)

ai

:= (ai (1), ai (2), . . . , ai (n)), i = 1, 2, . . . , d.

Proof Let dim Gd = m ≤ d, where Gd := span{a1 , a2 , . . . , ad } and assume for simplicity that a1 , a2 , . . . , am are linearly independent. We prove first that for some (n) (n) n ∈ N also a(n) 1 , a2 , . . . , am are linearly independent. Otherwise, for each n ≥ 1, n such that we can find non-vanishing α1n , α2n , . . . , αm n (n) α1n a(n) 1 + · · · αm am = 0,

n and without loss of generality we can assume that m i=1 | αi |= 1. Then we can find a subsequence {nk } such that n

nk ) −→ (α˜ 1 , . . . , α˜ m ), (α1 k , . . . , αm

3.3 Sufficient Conditions for Completeness 73

where (α˜ 1 , . . . , α˜ m ) is some vector in Rn such that m ˜ i |= 1. Since for each i=1 | α l > 1 we have n

(l)

(l)

nk (l) k α˜ 1 a1 + · · · + α˜ m a(l) m = lim α1 a1 + · · · + αm am = 0, k→+∞

it follows that α˜ 1 a1 + · · · + α˜ m am = 0, which contradicts the linear independence of a1 , a2 , . . . , am . (n) (n) 1 It follows from the linear independence of a(n) 1 , a2 , . . . , am , that for any ϕ ∈ l , we can find ϕ˜ ∈ Rn such that (n)

Since ak =

m

ϕ, ak =

ϕ, ai = ϕ, ˜ ai Rn ,

j=1 βj aj , k

m 

i = 1, 2, . . . , m.

= d + 1, . . . , m for some {βj }, we have

βj ϕ, aj =

j=1

m 

(n)

(n)

βj ϕ, ˜ aj Rn = ϕ, ˜ ak Rn ,

k = d + 1, . . . , m,

j=1

and the assertion follows. ˆ 0 , η1 , η2 , . . . , ηt ), t > 0, we are looking for Proof of Theorem 3.3.2 For Xˆ = X(η a ∈ R and a process ϕs = ϕs (η0 , η1 , . . . , ηs ) such that Xˆ = a +

t−1  ϕs , ηs+1 .

(3.3.4)

s=0

(Sufficiency) For fixed t > 0 let us consider the path of the discounted price process η0 = x0 , . . . , ηt−1 = xt−1 , which appears with positive probability. Given this path, the random variable ˆ 0 , x1 , . . . , xt−1 , ηt ) is a sequence of numbers, i.e. X(x ˆ 0 , x1 , . . . , xt−1 , ati ), ˆ 0 , x1 , . . . , ηt ) = X(x X(x

i = 1, 2, . . . , dt ,

and in view of Proposition 3.3.3 there exist a real number at−1 = at−1 (x0 , x1 , . . . , xt−1) and a vector ϕt−1 = ϕt−1 (x0 , x1 , . . . , xt−1 ) in l1 such that ˆ 0 , x1 , . . . , xt−1 , ati ) = at−1 (x0 , x1 , . . . , xt−1 ) X(x + ϕt−1 (x0 , x1 , . . . , xt−1 ), ati ,

i = 1, 2, . . . , dt .

Since the preceding relation holds for every path of positive probability, we obtain ˆ 0 , η1 , . . . , ηt ) = at−1 (η0 , η1 , . . . , ηt−1 ) + ϕt−1 (η0 , η1 , . . . , ηt−1 ), ηt . (3.3.5) X(η

74

Completeness

By induction, we obtain analogous representations for at−1 (η0 , η1 , . . . , ηt−1 ), i.e. at−1 (η0 , η1 , . . . , ηt−1 ) = at−2 (η0 , η1 , . . . , ηt−2 ) + ϕt−2 (η0 , η1 , . . . , ηt−2 ), ηt−1 , (3.3.6) and continue the procedure till the final formula for a1 (η0 , η1 ): a1 (η0 , η1 ) = a0 (η0 ) + ϕ0 (η0 ), η1 .

(3.3.7)

Combining (3.3.5), (3.3.6) and (3.3.7) yields ˆ 0 , η1 , . . . , ηt ) = a0 (η0 ) + X(η

t−1  ϕs (η0 , η1 , . . . , ηs ), ηs+1 , s=0

and thus (3.3.4) holds. (Necessity) We show that completeness implies that for any t > 0 the set {at1 , at2 , . . . , atdt } satisfies (ND1) or (ND2). Since the market is complete, it follows ˆ 0 , η1 , . . . , ηt ) and a fixed trajectory that occurs from (3.3.4) that for any function X(η with positive probability η0 = x0 , . . . , ηt−1 = xt−1 , we have the representation ˆ 0 , x1 , . . . , xt−1 , ati ) = at−1 (x0 , x1 , . . . , xt−1 ) + ϕ(x0 , x1 , . . . , xt−1 ), ati , X(x i = 1, 2, . . . , dt . In view of Proposition 3.3.3 the set {at1 , at2 , . . . , atdt } satisfies (ND1) or (ND2).

3.4 Approximate Completeness In this section we characterize bond markets that are Lp -approximately complete, with p ≥ 1, under the minimal filtration {Ft }. Recall that for any discounted claim ˆ 0 , η1 , . . . , ηt ) ∈ Lp (, Ft , P) and any ε > 0 we are looking for an initial Xˆ = X(η capital x ∈ R and l1 -valued strategy (ϕs ) such that  t−1  E |x+ ϕs , ηs+1 − Xˆ |p < ε. (3.4.1) s=0

The strategy (ϕs ) is assumed to be adapted to {Fs } such that the corresponding discounted wealth is p-integrable, i.e.  t−1  p p ˆ E(| X(t) | ) = E | x + ϕs , ηs+1 | < +∞. s=0

For the case p = 2 we will assume that E(| ϕs |2l1 ) < +∞,

s = 0, 1, . . . ,

(3.4.2)

3.4 Approximate Completeness

75

that is, ϕs ∈ L2 (, Fs , P; l1 ). Since, for each s > 0, ηs takes values in m and its coordinates are in the interval [0, 1], (3.4.2) implies that t−1      ˆ |2 ≤ 2t+1 x2 + E | X(t) E(| ϕs |2l1 ) < +∞. s=0

Hence, the requirement (3.4.2) makes the problem of L2 -approximate completeness well posed.

3.4.1 General Characterization Here we characterize L2 -approximate completeness under the minimal filtration in the case when (ηt ) is a general process taking values in m. Theorem 3.4.1 The bond market is L2 -approximately complete in the class of claims Xˆ ∈ L2 (, Ft , P) and strategies satisfying (3.4.2) if and only if for any t = 0, 1, 2, . . . and Yt+1 ∈ L2 (, Ft+1 , P) the following implication holds: E[Yt+1 | Ft ] = 0,

E[Yt+1 ηt+1 | Ft ] = 0

⇒

Yt+1 = 0.

(3.4.3)

Remark 3.4.2 If P is a martingale measure then the condition (3.4.3) is equivalent to the implication Xt Pˆ t is a martingale

⇒

Xt = 0, t = 1, 2, . . .

for a square integrable martingale Xt with X0 = 0. To see this note that the left side of (3.4.3) can be written with the use of the square integrable martingale Xt defined by X0 := 0,

Xt :=

t 

Ys ,

t = 1, 2, . . .

s=1

in the following way: E[(Xt+1 − Xt )(Pˆ t+1 − Pˆ t ) | Ft ] = 0,

t = 0, 1, . . . .

(3.4.4)

Since Pˆ t is a martingale, (3.4.4) is equivalent to the condition E[Xt+1 Pˆ t+1 | Ft ] = Xt Pˆ t ,

t = 0, 1, . . . ,

which clearly means that the product Xt Pˆ t is a martingale. Proof of Theorem 3.4.1 We simplify notation in the proof by writing L2 (Ft ) := L2 (, Ft , P) and L2 (Ft ; l1 ) := L2 (, Ft , P; l1 ), L2 (Ft ; m) := L2 (, Ft , P; m). First we show that the market is L2 -approximately complete if and only if for each t = 0, 1, . . . the operator Kt : L2 (Ft ) × L2 (Ft ; l1 ) −→ L2 (Ft+1 ) defined by Kt (Z, ϕ) = Z + ϕ, ηt+1

76

Completeness

has a dense image in L2 (Ft+1 ). If the market is L2 -approximately complete then for any Xˆ ∈ L2 (Ft ) and ε > 0 there exist x ∈ R and {ϕs } such that (3.4.1) holds with

p = 2. Consequently, for Z := x + t−2 s=0 ϕs , ηs+1 and ϕ := ϕt−1 we obtain   E | Xˆ − Kt−1 (Z, ϕ) |2 < ε, which means that the image of Kt−1 is dense in L2 (Ft ). Conversely, if the image of Kt is dense in L2 (Ft+1 ) for any t, then for Xˆ ∈ L2 (Ft ) and ε > 0 we can find Zt−1 ∈ L2 (Ft−1 ) and ϕt−1 ∈ L2 (Ft−1 ; l1 ) such that   E | Xˆ − (Zt−1 + ϕt−1 , ηt ) |2 < ε. Since the image of Kt−1 is also dense, for Zt−1 we can find Zt−2 ∈ L2 (Ft−2 ) and ϕt−2 ∈ L2 (Ft−2 , l1 ) such that   E | Zt−1 − (Zt−2 + ϕt−2 , ηt−1 ) |2 < ε, which yields

  E | Xˆ − (Zt−2 + ϕt−2 , ηt−1 + ϕt−1 , ηt ) |2 < 4ε.

Repetition of the procedure provides x ∈ R and {ϕs } such that t−1    ϕs , ηs+1 ) |2 < k(t)ε, E | Xˆ − (x + s=0

where k(t) is some constant. Hence, the market is L2 -approximately complete. Now we show that (3.4.3) is equivalent to the fact that Kt has a dense image in 2 L (Ft+1 ). For Yt+1 ∈ L2 (Ft+1 ) we have E[Kt (Z, ϕ) · Yt+1 ] = E[ZYt+1 ] + E[ ϕ, ηt+1 Yt+1 ]     = E Z · E[Yt+1 | Ft ] + E ϕ, E[ηt+1 Yt+1 | Ft ] .

(3.4.5)

Let us notice that E[Yt+1 | Ft ] ∈ L2 (Ft ) and E[ηt+1 Yt+1 | Ft ] ∈ L2 (Ft , m), so the operator   (3.4.6) Ht (Yt+1 ) := E[Yt+1 | Ft ], E[ηt+1 Yt+1 | Ft ] acts from L2 (Ft+1 ) into L2 (Ft ) × L2 (Ft , m). But L2 (Ft ) × L2 (Ft , m) ⊆ (L2 (Ft ) × L2 (Ft , l1 ))∗ , which follows from Remark 3.4.3. It follows from (3.4.5) and (3.4.6) that   , Kt (Z, ϕ), Yt+1 L2 (Ft+1 );L2 (Ft+1 ) = (Z, ϕ), Ht (Yt+1 ) 2 2 1 2 2 1 ∗ L (Ft )×L (Ft ,l );(L (Ft )×L (Ft ,l ))

Kt∗

= Ht . The image of Kt is dense in so leads to (3.4.3).

L2 (F

t+1 )

if and only if Ker Kt∗ = {0} that

3.4 Approximate Completeness

77

Remark 3.4.3 Let E be a Banach space with dual E∗ and V be a measurable space with a measure μ. Then  ∗ L2 (μ, E∗ ) ⊆ L2 (μ, E) .    For g ∈ L2 (μ, E∗ ), let us define Gg (f ) := V g(v), f (v) E∗ ;E μ(dv) for f ∈ L2 (μ, E). Then    | Gg | = sup | g(v), f (v) E∗ ;E μ(dv) | |f |≤1

≤ sup

V



|f |≤1 V

| g(v) |E∗ | f (v) |E μ(dv)

≤| g |,  ∗ so Gg belongs to L2 (μ, E) . Remark 3.4.4 The general bond market is not L2 -approximately complete if trading strategies are restricted to finite portfolios only. To see this, let us consider the operator K0 : R × l1 −→ L2 (, F1 , P) given by K0 (x, ϕ0 ) = x + ϕ0 , η1 ,

(3.4.7)

which we used in the proof of Theorem 3.4.1. Portfolios with a number of bonds bounded by n restrict the domain of K0 to R × Rn and then the image of K0 cannot be dense in L2 (, F1 , P) because it is a finite dimensional space.

3.4.2 Bond Curves in a Finite Dimensional Space Let us now consider the Lp -approximate completeness problem in the particular case when the process (ηs ) takes values is a finite dimensional subspace of m. The underlying filtration {Ft } is still assumed to be the minimal one. Theorem 3.4.5 Assume that ηt , for each t > 0, takes values in a finite dimensional subspace of m. The following conditions are equivalent. (a) The market is Lp -approximately complete in the class Xˆ ∈ Lp (, Ft , P) for some p ∈ [1, +∞). (b) The market is Lp -approximately complete in the class Xˆ ∈ Lp (, Ft , P) for each p ∈ [1, +∞). (c) For each t > 0, ηt takes a finite number of values and conditional values of ηt given η0 , η1 , . . . ηt−1 satisfy (ND1) or (ND2). Proof Let ηt have a support Kt in some finite dimensional subspace Ht of m. We show that if the market is Lp -approximately complete for some p ≥ 1 then Kt is

78

Completeness

finite for each t > 0 and the market is complete. The rest of the assertion follows from Theorem 3.3.2. First we show the assertion for t = 1. Let μ be the distribution of η1 and H1 ⊆ Lp (m, μ) be the linear space of functions of the form H1 := {h(y) : h(y) := x + ϕ0 , y ,

x ∈ R, ϕ0 ∈ l1 }.

Since H1 is finite dimensional, so is H1 . Since an arbitrary claim Xˆ ∈ Lp (m, μ) can be approximated by elements from H1 it follows that H1 = Lp (m, μ). In particular Lp (m, μ) is finite dimensional, say dim Lp (m, μ) = d. We claim that the cardinality of K1 must be no greater than d. Assume to the contrary that #K1 ≥ d + 1. Then there exist disjoint sets A1 , A2 , . . . , Ad+1 such that μ(Aj ) > 0, j = 1, 2, . . . , d + 1. The functions 1A1 , 1A2 , . . . , 1Ad+1 are linearly independent and all belong to Lp (m, μ). It follows that the dimension of Lp (m, μ) is greater than or equal to d + 1, which is a contradiction. Consequently K1 is finite and the measure μ is a sum of a finite number of atoms. By fixing η1 = x1 we can use the preceding arguments to show that the conditional distribution of η2 is finite, which implies that K2 is finite as well. By induction Kt is finite for any t > 0.

3.4.3 Bond Curves in Hilbert Spaces In this section we characterize L2 -approximately complete markets under the minimal filtration {Ft } in the case when the process (ηt ) is a square integrable process taking values in some Hilbert space H. Since H should be a subspace of m, a good choice of H can be, for instance, the space of square summable sequences l2 or its weighted version lρ2 . Let {ek } be an orthonormal complete basis in H. Then ηt =

+∞ 

t = 1, 2, . . .

αn (t)en ,

(3.4.8)

n=1

for some adapted square integrable functions {αn (t)} such that E(| ηt |2H ) = E

+∞ 

| αn (t) |2



< +∞,

t = 1, 2, . . . .

(3.4.9)

n=1

This model setting allows us to construct some useful examples presented in the second part of this section. They are based on the following result. Theorem 3.4.6 Let (ηt ) be an H-valued process with the representation (3.4.8) satisfying (3.4.9).

3.4 Approximate Completeness

79

(a) Then the market is L2 -approximately complete in the class Xˆ ∈ L2 (, Ft , P) if and only if, for each t ≥ 0, the set & ' A¯ 1 (t + 1) := 1A0 , α1 (t + 1)1A1 , α2 (t + 1)1A2 , . . . ; A0 , A1 , . . . ∈ Ft is linearly dense in L2 (, Ft+1 , P). In particular, if, for any t = 1, 2, . . ., the functions A1 (t) := {1, α1 (t), α2 (t), . . .} are linearly dense in L2 (, Ft , P), then the market is L2 -approximately complete. (b) If there exists a martingale measure Q with square integrable density then, for each t ≥ 0, the set & ¯ + 1) := α1 (t + 1)1A , α2 (t + 1)1A , . . . ; A(t 1 2

A 0 , A1 , . . . ∈ F t

'

is not linearly dense in L2 (, Ft+1 , P). Proof (a) We characterize condition (3.4.3) form Theorem 3.4.1 in terms of the sequence {αn (t)}. For any Yt+1 ∈ L2 (, Ft+1 , P), we have E(ηt+1 Yt+1 | Ft ) = E

+∞  

  αn (t + 1)en Yt+1 | Ft

n=1

=E

+∞  

  αn (t + 1)Yt+1 en | Ft ,

n=1

which, in view of (3.4.9), yields E(ηt+1 Yt+1 | Ft ) =

+∞    E αn (t + 1)Yt+1 | Ft en . n=1

Hence condition (3.4.3) can be reformulated to the form E[Yt+1 | Ft ] = 0,

E(Yt+1 αn (t + 1) | Ft ) = 0,

n = 1, 2, . . .

⇒

Yt+1 = 0,

which means that if for all sets A0 , A1 , . . . ∈ Ft E[Yt+1 1A0 ] = 0,

E(Yt+1 1An αn (t + 1)) = 0,

then Yt+1 = 0. This, however, clearly means that A¯ 1 (t + 1) is linearly dense in L2 (, Ft+1 , P). Since A1 (t) ⊆ A¯ 1 (t), it follows that the linear density of A1 (t) in L2 (, Ft , P) implies L2 -completeness.

80

Completeness

(b) If Q is a martingale measure then EQ [ηt+1 | Ft ] = 0,

t = 0, 1, . . . .

(3.4.10)

By the Bayes rule, EQ [ηt+1 | Ft ] =

E[ρt+1 ηt+1 | Ft ] , E[ρt+1 | Ft ]

t = 0, 1, . . . ,

(3.4.11)

where (ρt ) stands for the density process. By (3.4.10) and (3.4.11) we obtain E[ρt+1 ηt+1 | Ft ] = 0,

t = 0, 1, . . . .

(3.4.12)

If (ρt ) is square integrable, then, arguing as in (a), we see that (3.4.12) is equivalent to the condition   E ρt+1 αn (t + 1) | Ft = 0, n = 1, 2, . . . t = 0, 2, . . . . Hence, for any A1 , A2 , . . . ∈ Ft , E[ρt+1 αn (t + 1)1An ] = 0,

n = 1, 2, . . . ,

¯ + 1). Thus A(t ¯ + 1) which means that ρt+1 is orthogonal to all elements of A(t 2 is not linearly dense in L (, Ft+1 , P). It follows from Theorem 3.4.6 that each model of the form ηt =

+∞ 

αn (t)en ,

n=1

where, for each t > 0, {1, α1 (t), α2 (t), . . .} are linearly dense in L2 (, Ft , P), is L2 -approximately complete. Let us recall that the necessary condition for the market with finite dimensional noise to be complete was that ηt , for each t ≥ 0, takes a finite number of values only (see Theorem 3.4.5). This may suggest that L2 -approximate completeness can take place only if ηt is discretely distributed. In Example 3.4.7 we show that this conjecture is not true. We construct a one-period model that is L2 -approximately complete and η1 has a non-atomic distribution. On the other hand, discrete distribution of η1 does not imply L2 -approximate completeness. This we show in Example 3.4.8. Example 3.4.7 The distribution of η1 has an atom a ∈ H if and only if P(η1 = a) = c > 0. If a =

+∞

n=1 an en

then   P(η1 = a) = P αn (1) = an , n = 1, 2, . . . .

3.4 Approximate Completeness

81

Now, let {1, f1 , f2 , . . .} be an orthonormal complete basis in L2 ([0, 1], dx) such that the distribution of at least one fn has no atoms. Then for any sequence {γn } such that

γn = 0, n = 1, 2, . . . and γn2 < +∞ the random variable η1 :=

+∞ 

γn fn en

n=1

has a non-atomic distribution. Since the set {1, γ1 f1 , γ2 f2 , . . .} is linearly dense in L2 ([0, 1], dx), by Theorem 3.4.6, the market is L2 -approximately complete. Example 3.4.8 Let {ei } be an orthonormal basis in H and the distribution of η1 be given by the following P(η1 = ei ) = pi > 0,

+∞ 

pi < 1,

P(η1 = a) = pa := 1 −

i=1

where H  a :=



+∞ 

pi ,

i=1

ai ei and the sequence {ai } satisfies the conditions

+∞ 

a2i < +∞,

i=1

+∞ 2  a i

i=1

pi

< +∞,

+∞ 

ai = 1.

i=1

For the random variable Y defined by Y := −pa

+∞  ai i=1

pi

1{η=ei } + 1{η=a} ,

we have   ai 2

 a2

i + pa < +∞, pi pi  ai  pi + pa = pa (1 − ai ) = 0, E(Y) = −pa pi  ai  ai ei + pa a = 0. ei pi + pa a = −pa E(η1 Y) = −pa pi

E(Y 2 ) = p2a

pi + pa = p2a

In view of Theorem 3.4.1 the market is not L2 -approximately complete. We close this section with an example showing that the uniqueness of the martingale measure with square integrable density does not imply L2 -approximate completeness. Example 3.4.9 Let η1 be given by (3.4.8) satisfying (3.4.9) such that E[αn (1)] = 0,

n = 1, 2, . . . .

82

Completeness

It follows from Theorem 3.4.6 that such a market is L2 -approximately complete if ¯ and only if the orthogonal complement of A(1) = {α1 (1), α2 (1), . . .} satisfies  ⊥ ¯ = span{1}. (3.4.13) span{A(1)} ⊥ is at least two ¯ To see this, assume that (3.4.13) is not satisfied. Then (span{A(1)}) ⊥ ¯ dimensional and hence there exists Y ∈ (span{A(1)}) , which is orthogonal to the vector 1, i.e. E[Y] = 0. This implies, however, that the set {1, α1 (1), α2 (1), . . .} is not linearly dense in L2 (, F1 , P). Now let us assume that  ⊥ span{α1 (1), α2 (1), . . .} = span{1, Y},

where Y ∈ L2 (, F1 , P) is a random variable which is unbounded from below and from above. This market is not L2 -approximately complete. Since each element in span{1, Y} is of the form Z(a, b) = a + bY,

a, b ∈ R,

it follows that it takes negative values with positive probability providing that b = 0. Thus all positive elements are of the form Z(a, 0), a > 0. It follows that there exists a unique martingale measure with density in L2 (, F1 , P) and its density is given by ρ = Z(1, 0) = 1.

3.5 Models with Martingale Prices Contingent claims do not necessarily have to be functions of bond curves. One can consider claims depending on some economical factors describing randomness in the bond market. Replicating strategies for such claims are constructed also with the use of information about evolution of the factors. Therefore we introduce here a new filtration {Gt }, t = 0, 1, . . ., which can be greater than the minimal one {Ft } generated by the bond curves, i.e. F t ⊆ Gt ,

t = 0, 1, . . . ,

and consider the completeness problem under {Gt }. So, for a Gt -measurable discounted claim Xˆ paid at time t > 0 we are looking for the representation Xˆ = x +

t−1 

ϕs , ηs+1 ,

P − a.s.,

(3.5.1)

s=0

where x ∈ R and (ϕs ) is an l1 -valued process adapted to {Gt }. Recall that ηt is a sequence {ηt (T); T = t + 1, t + 2, . . .}, belonging to the space m, defined by ηt (T) = ˆ T) − P(t ˆ − 1, T). P(t,

3.5 Models with Martingale Prices

83

In the particular case, when {Gt } is the natural filtration of the martingale Z(t) = ξ1 + ξ2 + · · · + ξt ,

t = 1, 2, . . . ,

where {ξi } are zero-mean independent and identically distributed random variables in U = Rd , (3.5.1) reads as ˆ 0 , ξ1 , . . . , ξ t ) = x + X(ξ

t−1  ϕs (ξ0 , ξ1 , . . . , ξs ), ηs+1 ,

P − a.s.

(3.5.2)

s=0

By definition ξ0 = 0. Recall that by the generalized martingale representation property of Z, any martingale N adapted to {Gt } can be written in the form t−1   N(t) = N0 + ψ(s, y)π˜ ({s + 1}, dy) (3.5.3) s=0

U

(see Section 2.3.2 for details), where ψ(·, y) is an adapted process, that is, ψ(s, y) = h(s, ξ1 , . . . , ξs , y) for some function h and   ψ(s, y)π˜ ({s + 1}, dy) := h(s, ξ1 , . . . , ξs , ξs+1 ) − h(s, ξ1 , . . . , ξs , y)μ(dy). U

U

Here μ stands for the distribution of ξ1 . If Xˆ ∈ L1 (, Gt , P) we can apply (3.5.3) with Ns = E[Xˆ | Fs ] to obtain t−1   ˆ + ψXˆ (s, y)π˜ ({s + 1}, dy) (3.5.4) Xˆ = E[Xˆ | Gt ] = E[X] s=0

U

for some adapted process ψXˆ . With the use of (3.5.4) one can formulate sufficient conditions for determining (3.5.2) but one needs to know that E[ηs+1 | Gs ] = 0,

s = 0, 1, . . . , t − 1,

ˆ T) is a martingale for each T = 1, 2, . . .. i.e. the discounted bond price process P(s, Therefore we assume in the present section that the condition (MP) is satisfied. The method of determining replicating strategy for Xˆ from (3.5.4) leads to the so-called hedging equation, which is commonly used in the continuous time setting. Therefore we discuss it in the sequel for concrete models in a detailed way.

3.5.1 HJM Models Recall that in the HJM model f (t + 1, T) = f (t, T) + α(t, T) + σ (t, T), ξt+1 ,

0≤t 0.

(3.5.13)

Then the model is L2 -approximately complete under minimal filtration if and only if for each t ≥ 0, almost surely +∞ 

1 = +∞.

T−1 v=t σ (t, v) T=t+2

(3.5.14)

3.5 Models with Martingale Prices

87

Proof Condition (3.5.13) implies that {Gt } is identical with the minimal filtration. Taking into account this and also the fact that

T−1

ˆ T)[e−ϕξ ( ηt+1 (T) = P(t,

v=t

σ (t,v))− T−1 v=t σ (t,v)ξt+1

− 1],

the conditions in (3.4.3) in Theorem 3.4.1 take the form E[Yt+1 | Gt ] = 0 and

T−1

ˆ T)[e−ϕξ ( E[Yt+1 P(t,

v=t

σ (t,v))− T−1 v=t σ (t,v)ξt+1

−1] | Gt ] = 0,

T = t + 2, t + 3, . . . .

Recall that ηt+1 (T) = 0 for T ≤ t + 1, so these values could be omitted in the preceding equation. Hence L2 -approximate completeness takes place if and only if for Yt+1 ∈ L2 (, Gt+1 , P) the following conditions E[Yt+1 | Gt ] = 0,

E[Yt+1 e−

T−1 v=t

σ (t,v)ξt+1

| Gt ] = 0,

T = t + 2, t + 3, . . . (3.5.15)

imply that Yt+1 = 0. Since Yt+1 can be written in the form Yt+1 = h(ξ1 , ξ2 , . . . , ξt , e−ξt+1 ), where h is a deterministic function, we can write (3.5.15) in the form  

T−1 h(ξ1 , . . . , ξt , y)μ(dy) = 0, h(ξ1 , . . . , ξt , y)y v=t σ (t,v) μ(dy) = 0, I

I

(3.5.16)

T = t + 2, t + 3, . . . , where μ(dy) stands for the distribution of e−ξ1 and I for its bounded support in R+ . So, the market is L2 -approximately complete if and only if (3.5.16) implies that h ≡ 0. This means, however, that the functions 1,

T−1

y

v=t

σ (t,v)

,

T = t + 2, t + 3, . . .

are linearly dense in L2 (I, μ). In view of Theorem 3.5.4 this is possible if and only if (3.5.14) is satisfied. Remark 3.5.6 satisfies

Let us assume that in the preceding HJM model the volatility +∞ 

| σ (t, s) |< +∞,

t = 0, 1, . . . .

(3.5.17)

s=t

Then, since (MP) holds, it follows from Theorem 2.2.3 that P is the unique martingale measure. Since (3.5.17) implies (3.5.14), the market is L2 -approximately complete in this case.

88

Completeness

3.5.2 Multiplicative Factor Model We focus now on the factor model from Proposition 2.4.15 where bond prices are given by P(t, T) = F(T − t, Xt )

(3.5.18)

with F(1, x) = xγ , x > 0 and Xt+1 = aXt + bXt ξt+1 ,

X0 = x > 0,

t = 1, 2, . . . .

(3.5.19)

Above a, b ∈ R, b = 0 and ξ1 , ξ2 , . . . are independent and identically distributed. By Proposition 2.4.15 the model satisfies (MP) if and only if F(k, x) =

 k−1 $

 cjγ xkγ ,

(3.5.20)

j=1

where cγ := E[(a + bξ1 )γ ] and forward rates are positive if γ > 0,

x ∈ (0, 1),

0 < a + bξ1 < 1

(3.5.21)

a + bξ1 > 1.

(3.5.22)

or γ < 0,

x > 1,

In what follows, we consider only models satisfying (3.5.20) and (3.5.21) or (3.5.22). Then (T−t)γ

P(t, T) = cγ c2γ . . . c(T−t−1)γ Xt

(3.5.23)

and B(t) = e

t−1

s=0 R(s)

t−1

=e

s=0 G(0,Xs )

=

t−1 $

Xs−γ .

(3.5.24)

s=0

The completeness problem is considered under the filtration Gt := σ {ξ1 , ξ2 , . . . , ξt },

t = 1, 2, . . . ,

which, clearly satisfies Gt = GtX := σ {X1 , X2 , . . . , Xt },

t = 1, 2, . . . .

Theorem 3.5.7 A claim Xˆ ∈ L1 (, Gt , P) with the representation (3.5.4) is attainable if the hedging equation  ψXˆ (s, y) = ϕs (T)s (T, y), s = 0, 1, . . . , t − 1, y ∈ supp{ξ1 } (3.5.25) T≥s+2

3.5 Models with Martingale Prices

89

has a solution ϕ taking values in l1 . Where   1 ˆ T) s (T, y) := P(s, (a + by)(T−s−1)γ − 1 . c(T−s−1)γ If the hedging equation has a solution for any {Gt }-adapted process ψ(s, y), then the market is complete in the class of claims Xˆ ∈ L1 (, Gt , P). Proof that

First let us determine the process (ηt ). It follows from (3.5.23) and (3.5.24)

ˆ T) = P(t, T)/B(t) = P(t,

 T−t−1 $

cjγ

t−1  $

j=1

 (T−t)γ Xsγ Xt ,

s=0

and consequently ˆ + 1, T)/P(t, ˆ T) = P(t =

(T−t−1)γ

1 c(T−t−1)γ 1 c(T−t−1)γ

γ Xt

Xt+1

(T−t)γ

Xt

=

1



c(T−t−1)γ

Xt+1 Xt

(T−t−1)γ

(a + bξt+1 )(T−t−1)γ .

Hence ˆ + 1, T) − P(t, ˆ T) = P(t, ˆ T)[P(t ˆ + 1, T)/P(t, ˆ T) − 1] ηt+1 (T) = P(t   1 (T−t−1)γ ˆ = P(t, T) −1 (a + bξt+1 ) c(T−t−1)γ = t (T, ξt+1 ). The discounted wealth of a pair (x, ϕ) is given thus by ˆ =x+ X(t)

t−1 t−1    ϕs , ηs+1 = x + ϕs (T)ηs+1 (T) s=0

=x+

s=0 T≥s+2

t−1  

ϕs (T)s (T, ξs+1 )

s=0 T≥s+2

and can be written in the form ˆ =x+ X(t)

t−1   s=0 T≥s+2

 s (T, y)π˜ ({s + 1}, dy).

ϕs (T)

(3.5.26)

U

ˆ = Xˆ and If (ϕs ) is such that (3.5.25) holds, then, by (3.5.4) and (3.5.26), X(t) ˆ x = E[X].

90

Completeness

Corollary 3.5.8 If ξ1 takes a finite number of values {y1 , y2 , . . . , yK } and if, for any s = 0, 1, . . . , t − 1, there exist maturities T1 < T2 < · · · < TK such that det([s (Tk , yk )]k=1,2,...,K ) = 0, then the hedging equation has a solution for any ψXˆ and the market is complete in the class Xˆ ∈ L1 (, Gt , P). Corollary 3.5.9

It is clear that b = 0

⇒

s (T, y) = s (T, z)

for y = z.

Consequently, if ξ1 takes an infinite number of values then so does ηs+1 (T) = s (T, ξs+1 ). If b = 0 then {Gt } is identical with the minimal filtration and, consequently, by Theorem 3.2.1, the market is not complete in the class Xˆ ∈ L∞ (, Gt , P).

Approximate Completeness Now we pass to the L2 -approximate completeness of the multiplicative factor model in the class Xˆ ∈ L2 (, Gt , P). Theorem 3.5.10 If the distribution of ξ1 is absolutely continuous with respect to the Lebesgue measure, then the model is L2 -approximately complete in the class of claims Xˆ ∈ L2 (, Gt , P). In the proof we use again the M¨untz theorem 3.5.4. Proof

By (3.5.23) and (3.5.24) we obtain ˆ T) = P(t, T)/B(t) = P(t,

t−1 $

(T−t)γ

Xsγ cγ c2γ . . . c(T−t−1)γ Xt

s=0

and consequently ˆ + 1, T) − P(t, ˆ T) ηt+1 (T) = P(t

  γ γ γ (T−t−1)γ (T−t)γ = cγ c2γ . . . c(T−t−2)γ xγ X1 . . . Xt−1 Xt Xt+1 − c(T−t−1)γ Xt ,

T = t + 2, t + 3, . . . for t = 0, 1, . . .. Since, for b = 0, the filtration {Gt } is identical with the minimal one, for the characterization of L2 -approximate completeness we can apply Theorem 3.4.1. One can check that condition (3.4.3) has the form E(Yt+1 | Ft ) = 0,

(T−t−1)γ

E(Yt+1 Xt+1

| Ft ) = 0,

T = t + 2, t + 3, . . . ⇒ Yt+1 = 0. Since Yt+1 = h(X1 , X2 , . . . , Xt+1 ), for a deterministic function h , the Markovianity of X implies that the left side of the preceding implication can be written as

3.5 Models with Martingale Prices

 

R+ R+

91

h(X1 , X2 , . . . , Xt , y) P(Xt , dy) = 0, h(X1 , X2 , . . . , Xt , y)ynγ P(Xt , dy) = 0,

n = 1, 2, . . . .

(3.5.27)

In (3.5.27) P(x, ·) stands for the transitions function of the process X. Denoting, μ(dy) := P(Xt , dy) and ψ(y) := h(X1 , X2 , . . . , Xt , y) we see that (3.5.27) has the form  ψ(y)yγ n μ(dy) = 0, n = 0, 1, 2, . . . . (3.5.28) R+

It follows that the model is L2 -approximately complete if and only if the functions 1, yγ , y2γ , . . .

(3.5.29)

are linearly dense in L2 (R, μ(dy)). Let us consider two cases when γ > 0 and γ < 0. If γ > 0 then the measure μ is concentrated on (0, 1). Since +∞  1 = +∞, nγ

γ > 0,

n=1

it follows from Lemma 3.5.4 that (3.5.29) are linearly dense L2 ((0, 1), μ). If γ < 0 then μ is concentrated on (1, +∞) and (3.5.28) can be written in the form  +∞ 1 ψ(y) |γ |n μ(dy) = 0, n = 0, 1, 2, . . . . (3.5.30) y 1 Let us consider the transformation T(y) = 1y , y ∈ (1, +∞) and the measure ν := T ◦ μ on (0, 1) given by ν(A) = (T ◦ μ)(A) := μ{z ∈ (1, +∞) : T(z) ∈ A}, Then

 1

+∞



1

ψ(T(y)) μ(dy) =

A ⊆ (0, 1).

ψ(z) (T ◦ μ)(dz).

0

Since μ = T −1 ◦ ν , (3.5.30) can be reformulated to  +∞  +∞ 1 1 ψ(y) |γ |n μ(dy) = ψ(y) |γ |n (T −1 ◦ ν)(dy) y y 1 1  1  1 |γ |n = ψ [T ◦ (T −1 ◦ ν)](dz) z z 0  1  1 |γ |n ψ ν(dz) = 0. z = z 0 To show that (3.5.29) is linearly dense we can use the same arguments as in the case γ > 0.

92

Completeness

3.5.3 Affine Models Let us now consider the affine model P(t, T) = e−C(T−t)−D(T−t)R(t) ,

t ≤ T < +∞,

(3.5.31)

where the short-rate R is a Markov chain with transition function P(x, dy) = (μx ∗ ν)(dy),

x, y ≥ 0.

(3.5.32)

Here μx is a convolution semigroup and ν a probability measure, both on R+ . In Section 2.4.2 we described functions C(·), D(·) for which the model satisfies (MP). Recall that functions ϕ, ψ given by   e−λy μx (dy) = e−ψ(λ)x , e−λy ν(dy) = e−ϕ(λ) , R+

R+

provide the following formulas C(T) − C(T − 1) = ϕ(D(T − 1)),

D(T) = ψ(D(T − 1)) + 1,

and C(0) = D(0) = 0. Moreover, ψ admits the representation  ψ(λ) = βλ + (1 − e−λy )m(dy), λ ≥ 0, R+

T = 1, 2, . . . . (3.5.33)

(3.5.34)

where β ≥ 0 and the measure m satisfies  (1 ∧ y)m(dy) < +∞. R+

The completeness problem is considered under the filtration Gt := σ {R(0), R(1), . . . , R(t)},

t ≥ 0,

which, however, in view of (3.5.31) is identical with the minimal filtration, i.e. Gt = Ft ,

t = 0, 1, . . . .

(3.5.35)

First we argue that non-trivial affine models are not complete. Let us notice that R(t) takes a finite number of values if and only if the convolution (μx ∗ ν)(dy) has finite support. This is possible only when the support of ν is finite and μx is concentrated on the point βx, i.e. μx = 1{βx} . In the opposite case R(t) takes infinite many values and, by Theorem 3.2.1 and (3.5.35), we obtain the following corollary. Corollary 3.5.11 If in the transition function (3.5.32) of the short-rate process the support of ν is infinite or μx = 1{βx} then the affine model is not complete in the class Xˆ ∈ L∞ (, Gt , P).

3.5 Models with Martingale Prices

93

Approximate Completeness The following result characterizes the L2 -approximate completeness of the affine models. Theorem 3.5.12 Let the affine model (3.5.31) driven by a Markovian short rate with the transition function (3.5.32) satisfy (MP) and β be given by (3.5.34). Let, in addition, the transition function be absolutely continuous with respect to Lebesgue measure. Then the following statements are true. (a) If β < 1 then the market is L2 -approximately complete. (b) If β > 1 then the market is not L2 -approximately complete. (c) If β = 1 and the measure in (3.5.34) is γ -stable, i.e. m(dy) =

1 y1+γ

1[0,+∞) (y)dy,

γ ∈ (0, 1),

then the market is not L2 -approximately complete. Proof

By (3.5.35) we can use Theorem 3.4.1. Since

ˆ + 1, T) − P(t, ˆ T) ηt+1 (T) = P(t = e−

t

s=0 R(s)

e−C(T−t−1)−D(T−t−1)R(t+1) − e−

t−1

s=0 R(s)

e−C(T−t)−D(T−t)R(t) ,

we can write (3.4.3) in the form E(Yt+1 | Ft ) = 0,

E(Yt+1 e−D(T−t−1)R(t+1) | Ft ) = 0,

T = t + 2, t + 3, . . . ⇒ Yt+1 = 0.

(3.5.36)

Since Yt+1 = h(R(1), . . . , R(t), e−R(t+1) ), for some function h, we can write the left side of the preceding implication in the form  1 h(R(1), . . . , R(t), y) μ(dy) = 0, 

0 1

h(R(1), . . . , R(t), y)yD(T−t−1) μ(dy) = 0,

T = t + 2, t + 3, . . . ,

0

where μ(dy) stands for the conditional distribution of e−R(t+1) given R(t). Denoting g(y) := h(R(1), . . . , R(t), y) we can thus write (3.5.3) in the form  1  1 g(y) μ(dy) = 0, g(y)yD(n) μ(dz) = 0, n = 1, 2, . . . ⇒ h = 0. 0

0

(3.5.37) So, it follows that the market is L2 -approximately complete if and only if the functions zD(n) , n = 1, 2, 3, . . . are linearly dense in L2 ((0, 1), μ). By Theorem 3.5.4 this is the case if and only if

94

Completeness +∞  n=1

1 = +∞. D(n)

(3.5.38)

In the proof of Theorem 2.4.11 we have shown that D(n) is increasing and lim

n−→+∞

D(n) = +∞

⇐⇒

β ≥ 1.

(3.5.39)

It follows from (3.5.39) that (3.5.38) is satisfied for β < 1. For the case β > 1, we obtain from (3.5.34) that ψ(λ) ≥ βλ, λ ≥ 0. This together with (3.5.33) yields D(n) ≥ 1 + β + β 2 + · · · + β n−1 ,

n ≥ 1.

(3.5.40)

Consequently, D(n) ≥ β n−1 and (3.5.38) clearly fails for β > 1. So, the model is L2 -approximately complete for β < 1 and is not for β > 1. Hence (a) and (b) follow. Now we prove (c). One can check (see Example 5.3.7 for details) that  +∞ 1 (1 − e−λy ) 1+γ dy = cλγ , y 0 where c =

(1−γ ) γ

and  stands for the gamma function. Consequently, ψ(λ) = βλ + cλγ .

By (2.4.20) we have that D(n + 1) = D(n) + c(D(n))γ + 1,

D(0) = 0.

(3.5.41)

It is clear that D(n) ≥ n, n = 0, 1, . . ., so (3.5.41) yields D(n + 1) = (n + 1) + c

n n   (D(k))γ ≥ (n + 1) + c kγ . k=1

Now we show that n 

kγ ≥

(3.5.42)

k=1

 n 1+γ 2

k=1

.

(3.5.43)

If n is even, i.e. n = 2l, l = 1, 2, . . ., then n 

2l 

kγ ≥

k=1

kγ ≥ l · lγ = l1+γ =

 n 1+γ

k=l+1

2

.

If n = 2l + 1, l = 1, 2, . . . then n 

kγ ≥

k=1

2l+1 

kγ ≥ (l + 1)(l + 1)γ = (l + 1)γ +1

k=l+1

 ≥



1+γ

n−1 +1 2

=

n+1 2

1+γ ≥

 n 1+γ 2

.

From (3.5.42) and (3.5.43) we obtain that (3.5.38) fails, so the market is not L2 -approximately complete.

3.6 Replication with Finite Portfolios

95

Remark 3.5.13 The transition functions P(x, ·), x > 0 are absolutely continuous with respect to Lebesgue measue if, for instance, ν or μx , x > 0 are. Specific condition for the latter can be found in Grzywacz, Le˙zaj and Trajan [65]. We apply Theorem 3.5.12 to the affine model for which the short-rate process R(t), t ≥ 0 solves the stochastic equation R(0) = R0 .

R(t + 1) = F(R(t)) + G(R(t))ξt+1 ,

(3.5.44)

We know from Section 2.4.3 that (3.5.44) defines a model satisfying (MP) if it is of the form ˜ α ξt+1 , R(t + 1) = (aR(t) + a˜ ) + (bR(t) + b) 1

t = 0, 1, . . . ,

(3.5.45)

where ξt is a sequence of independent standard α-stable distributions and a, a˜ , b, b˜ are nonnegative numbers, or R(t + 1) = aR(t) + ξt+1 ,

t = 0, 1, . . . ,

(3.5.46)

where (ξt ) is an arbitrary sequence of independent, identically distributed nonnegative random variables and a ≥ 0. Theorem 3.5.14 The affine model with short rate given by (3.5.45) or (3.5.46) is L2 -approximately complete if and only if a < 1. Proof

By Proposition 2.4.8 and Proposition 2.4.9 we know that ψ(λ) = aλ + bλα , or

Thus

 ψ(λ) = aλ + c

R+

ψ(λ) = aλ,

(1 − e−λy )

λ ≥ 0.

1 dy y1+α

with some c ≥ 0. The result follows from Theorem 3.5.12.

3.6 Replication with Finite Portfolios In this section we show how martingale measures and martingale representation property can be used to characterize replicating portfolios. Our analysis is concerned with contingent claims at time t0 of the form   (3.6.1) X = h P(t0 , T1 ), P(t0 , T2 ), . . . , P(t0 , Tn ) , which are some function of prices of bonds with future maturities T1 < T2 < · · · < Tn , where t0 < T1 . Many important contingent claims can be represented in this way. In particular, the so called LIBOR spot rate (London Interbank Offered Rate) for [t0 , T1 ] defined by

96

Completeness L(t0 , T1 ) :=

1 − P(t0 , T1 ) , (T1 − t0 )P(t0 , T1 )

and the related caplet, respectively floorlet, given by (L(t0 , T1 ) − K)+ , respectively (K − L(t0 , T1 ))+ , with some K > 0 (see Filipovi´c [52]). Our aim is to formulate conditions allowing to replicate claims of the form (3.6.1) only with the use of bonds with maturities t0 , T1 , . . . , Tn and the bank account, of course. The problem may seem to be embraced by the classical model setting with a finite number of trading assets, where the uniqueness of martingale measure implies completeness. There is, however, a subtle difference concerning the bank account process that makes the problem more involved. Recall that by (1.1.5) the bank account process is given by B(t) =

1 , P(0, 1)P(1, 2) · . . . · P(t − 1, t)

t = 1, 2, . . . .

This means that for the determination of the bank account process on the interval [0, t0 ] we need all bonds with maturities lying in [0, t0 ]. In other words, the bank account process is not predictable with respect to the σ -field generated merely by the bonds with maturities t0 , T1 , . . . , Tn , in general. The predictability of B(t) with respect to the σ -field generated by risky assets is, however, necessary for the implication of completeness by the uniqueness of martingale measure (see Shiryaev [116, p. 482 and p. 495]). Hence, to solve our problem it is not sufficient to find conditions for the uniqueness of a martingale measure for the bonds with maturities t 0 , T1 , . . . , T n . Our solution of the problem formulated above provides conditions for the increments of discounted prices of bonds with maturities t0 , T1 , . . . , Tn . The analysis is based on the condition (MM), i.e. we assume here that there exists a measure Q ˆ T), t ∈ [0, T] are Q-martingales for any such that the discounted bond prices P(t, T = 1, 2, . . .. Theorem 3.6.1 Any contingent claim of the form (3.6.1) can be replicated by a strategy involving bank account and bonds with maturities t0 , T1 , . . . , Tn if and only if the Q-martingale ˆ t0 ), P(t, ˆ T1 ), . . . , P(t, ˆ Tn )), S(t) := (P(t,

t = 0, 1, . . . , t0

is such that, for each t = 1, 2, . . . , t0 , its increment S(t) = S(t) − S(t − 1) for a given path S(0), S(1), . . . , S(t − 1), is concentrated on a finite set At in Rn+1 such that dim(span At ) = m(t) − 1,

(3.6.2)

where m(t) := At stands for the number of elements of At . If this is the case then the price of X equals EQ [X/B(t0 )] and the replicating strategy is unique providing that m(t) = n + 2 for each t = 1, 2, . . . , t0 .

3.6 Replication with Finite Portfolios

97

Proof In view of Proposition 1.4.1, the contingent claim X can be replicated by a strategy involving bonds with maturities t0 , T1 , . . . , Tn if and only if the discounted claim Xˆ = X/B(t0 ) admits the representation Xˆ = x +

t 0 −1

ϕs , S(s + 1)

(3.6.3)

s=0

F S -adapted

for some x ∈ R and process ϕ. First we show that the discounted value of the claim (3.6.1) admits the representation   ˆ 0 , T1 ), . . . , P(t ˆ 0 , Tn ) ˆ 0 − 1, t0 ), P(t (3.6.4) Xˆ = g P(t for some function g. Due to the specific form of the discount factor 1/B(t) = P(0, 1)P(1, 2) · . . . · P(t − 1, t),

t = 1, 2, . . .

(3.6.5)

(see (1.1.5)), we obtain the following formula ˆ T) = P(0, 1)P(1, 2) · . . . · P(t − 1, t)P(t, T), P(t,

t ≤ T.

(3.6.6)

ˆ T) = P(0, T) yields Recursive application of (3.6.6) with the initial condition P(0, ˆ T)/P(t ˆ − 1, t), P(t, T) = P(t,

t ≤ T.

(3.6.7)

As a consequence of (3.6.5) and (3.6.7) we obtain ˆ 1) · P(1, ˆ 2)/P(0, ˆ 1) · . . . · P(t ˆ − 1, t)/P(t ˆ − 2, t − 1) = P(t ˆ − 1, t) 1/B(t) = P(0, (3.6.8) for t = 1, 2, . . .. By (3.6.1), (3.6.7) and (3.6.8) we obtain   ˆ 0 − 1, t0 ) · h P(t ˆ 0 , T1 )/P(t ˆ 0 − 1, t0 ), . . . , P(t ˆ 0 , Tn )/P(t ˆ 0 − 1, t0 ) , Xˆ = P(t so (3.6.4) holds. By Theorem 2.3.1, condition (3.6.2) is necessary and sufficient for S to have the martingale representation property. Since the process M(t) = EQ [Xˆ | F S (t)],

t = 0, 1, . . . , t0

is a Q-martingale, it can be represented in the form M(t) = M(0) +

t−1  ϕ(s), S(s + 1) ,

t = 0, 1, . . . , t0

s=0

for some F S -adapted process ϕ. Since Xˆ is FtS0 -measurable, we have that M(t0 ) = Xˆ and consequently (3.6.3) is satisfied with x = M(0). Taking Q-expectations in (3.6.3) ˆ The uniqueness of ϕ in the case when yields that the price of X equals EQ [X]. m(t) = n + 2 follows from Theorem 2.3.1.

98

Completeness

Proposition 3.6.2 Let us consider the HJM model f (t + 1, T) = f (t, T) + α(t, T) + σ (t, T)ξt+1 ,

t, T = 0, 1, . . . , T ∗ − 1,

where {ξt } is a sequence of i.i.d. real valued random variables taking k values, where k ≥ 1. Let us assume that   α(t, T) = G t, T, σ (t, t + 1), σ (t, t + 2), . . . , σ (t, T) (3.6.9) for some function G, and, for given constants t0 < T1 < · · · < Tn ≤ T ∗ ,   ˆ t0 ), P(t, ˆ T1 ), . . . , P(t, ˆ Tn ) σ (t, T) = g t, T, P(t,

(3.6.10)

for some function g. Let the model admit a martingale measure Q. Then the property of replicating claims of the form   X = h P(t0 , T1 ), P(t0 , T2 ), . . . , P(t0 , Tn ) (3.6.11) with the use of strategies involving bonds with maturities t0 , T1 , . . . , Tn is preserved for k ≤ n + 1 and fails for k ≥ n + 3. Remark 3.6.3 Recall that in Proposition 2.2.2 we constructed a market satisfying (3.6.9) that admits a martingale measure. Condition (3.6.10) can be viewed as an additional requirement specifying volatility of the model. Proof of Proposition 3.6.2 We examine condition (3.6.2). By (2.2.2) we obtain the ˆ + 1, T) = P(t ˆ + 1, T) − P(t, ˆ T), with T = 1, 2, . . . , T ∗ , following formula for P(t ˆ + 1, T) = e− P(t = e−

T−1 s=0

T−1 s=0

f (t+1,s) f (t,s)

− e−

e−

T−1

T−1 s=0

s=0

f (t,s)

{α(t,s)+σ (t,s)ξt+1 }

!

−1

! ˆ T) eA(t,T)+(t,T)ξt+1 − 1 , = P(t,

(3.6.12)

T−1

T−1 with A(t, T) := − s=t α(t, s), (t, T) := − s=t σ (t, s). Consequently, the increments of the Q-martingale   ˆ t0 ), P(t, ˆ T1 ), . . . , P(t, ˆ Tn ) S(t) := P(t, have the form

⎛

! ⎞

ˆ t0 ) eA(t,t0 )+(t,t0 )ξt+1 − 1 P(t,

⎜ ! ⎜ ˆ ⎜ P(t, T1 ) eA(t,T1 )+(t,T1 )ξt+1 − 1 ⎜ S(t + 1) := ⎜ .. ⎜ . ⎜ ⎝ ! ˆ Tn ) eA(t,Tn )+(t,Tn )ξt+1 − 1 P(t,

⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

3.6 Replication with Finite Portfolios

99

and by (3.6.9) and (3.6.10) we obtain that for given values S(0), S(1), . . . , S(t), the number of values of S(t + 1) equals the number of values of ξt+1 , providing that (t, t0 ), (t, T1 ), . . . , (t, Tn ) do not vanish. Let a1 , a2 , . . . , ak stand for the values of ξ1 . Then the values of S(t + 1) are v1 , v2 , . . . , vk with ! ⎞ ⎛ ˆ t0 ) eA(t,t0 )+(t,t0 )ai − 1 P(t, ⎜ ! ⎟ ⎟ ⎜ ˆ ⎜ P(t, T1 ) eA(t,T1 )+(t,T1 )ai − 1 ⎟ ⎟ ⎜ vi := ⎜ ⎟ , i = 1, 2, . . . , k. .. ⎟ ⎜ . ⎟ ⎜ ⎝ ! ⎠ ˆ Tn ) eA(t,Tn )+(t,Tn )ai − 1 P(t, It follows from the existence of a martingale measure that the vectors v1 , v2 , . . . , vk are linearly dependent. We argue now that in the set v1 , . . . , vk there are k − 1 linearly independent vectors providing that k ≤ n + 1. By Corollary 2.2.6 we see that the vectors ⎞ ⎛ eA(t,t0 )+(t,t0 )ai ⎟ ⎜ ⎜ eA(t,T1 )+(t,T1 )ai ⎟ ⎟ ⎜ ⎟ , i = 1, 2, . . . , k ⎜ ⎟ ⎜ .. ⎟ ⎜ . ⎠ ⎝ A(t,T )+(t,T )a n n i e are linearly independent. It follows from Lemma 3.6.4 that the set ⎞ ⎛ eA(t,t0 )+(t,t0 )ai − 1 ⎟ ⎜ ⎜ eA(t,T1 )+(t,T1 )ai − 1 ⎟ ⎟ ⎜ ⎟ , i = 1, 2, . . . , k ⎜ ⎟ ⎜ .. ⎟ ⎜ . ⎠ ⎝ eA(t,Tn )+(t,Tn )ai − 1 contains k − 1 linearly independent vectors. The same result is clearly true for the original set {v1 , . . . , vk }. Consequently, (3.6.2) is satisfied if k ≤ n + 1 and fails if k ≥ n + 3. Lemma 3.6.4 the set

Let the vectors w1 , w2 , . . . , wn be linearly independent. Then in {w1 − z, w2 − z, . . . , wn − z},

(3.6.13)

where z is some vector, there exist n − 1 linearly independent vectors. Proof Let us assume to the contrary that each arbitrarily chosen n − 1 vectors from (3.6.13) are linearly dependent. Then there exist at least constants c1 , . . . , cn−1 , not all equal zero, such that c1 (w1 − z) + c2 (w2 − z) + · · · + cn−1 (wn−1 − z) = 0.

100

Completeness

Equivalently, c1 w1 + c2 w2 + · · · + cn−1 wn−1 − (c1 + · · · + cn−1 )z = 0. Since w1 , . . . ., wn−1 are linearly independent, we have that c1 + · · · + cn−1 = 0 and consequently z = c˜ 1 w1 + · · · + c˜ n−1 wn−1

(3.6.14)

for some constants c˜ 1 , . . . , c˜ n−1 . We may assume that c˜ 1 = 0. Repeating the same procedure with the set {w2 , . . . , wn }, we obtain z = c˜˜ 2 w2 + · · · + c˜˜ n wn .

(3.6.15)

Subtracting (3.6.15) from (3.6.14) yields 0 = c˜ 1 w1 + (˜c2 − c˜˜ 2 )w2 + · · · + (˜cn−1 − c˜˜ n−1 )wn−1 − c˜˜ n wn with the non-vanishing sequence c˜ 1 , c˜ 2 − c˜˜ 2 , . . . , c˜ n−1 − c˜˜ n−1 , c˜˜ n . This contradicts the linear independence of w1 , . . . , wn .

3.7 Completeness and Martingale Measures In the previous section we showed that completeness of the market may hold only if increments of the discounted bond prices η1 , η2 , . . . take a finite number of values. Such a market, with a finite time horizon t > 0, can be treated as a classical stock market with a finite number of shares, which additionally take a finite number of values. To see this, let us notice that all possible values of the random variables η 0 , η1 , . . . , η t form also a finite set in m and, by Proposition 3.3.4, one can find n > 0 such that for any ϕ ∈ l1 , (n)

ϕ, ηi = ϕ (n) , ηi Rn ,

i = 1, 2, . . . , t,

P − a.s.,

where b(n) stands for the truncation of b ∈ l1 to the first n coordinates. This means that, without losing generality, one can trade with bonds with maturities from the set 1, 2, . . . , n only. In addition to that, for each T ∈ {0, 1, . . . , n} ˆ T) takes a finite number of values for each s = the discounted bond price P(s, 1, 2, . . . , t. This simple model setting allows us to use the classical Fundamental Theorems of Asset Pricing describing properties of the model in terms of martingale measures. In the sequel {Ft } stands for the minimal filtration. Theorem 3.7.1 Assume that, for each t > 0, the random variable ηt takes a finite number of values.

3.7 Completeness and Martingale Measures

101

(a) Then the market is arbitrage free if and only if there exists a martingale measure. (b) Assume that the market admits a martingale measure. Then the market is complete under the minimal filtration if and only if the martingale measure is unique. In view of Theorem 3.3.2 and Theorem 3.7.1 we obtain the following result, which, in particular, shows that completeness does not imply existence of a martingale measure. Proposition 3.7.2 Let the bond market be complete, i.e. for any t the conditional distribution of ηt with respect to η0 , η1 , . . . , ηt−1 is concentrated on the set {at1 , at2 , . . . , atdt }, with dt < +∞, of vectors in m which satisfies (ND1) or (ND2). Then the following statements are true. (a) If {at1 , at2 , . . . , atdt } satisfy (ND1) for some t > 0 then the market does not admit a martingale measure. (b) Let, for each t > 0, the vectors {at1 , at2 , . . . , atdt } satisfy (ND2). Assume for simplicity that at1 , at2 , . . . , atdt −1 are linearly independent. Define the positive cone of the set at1 , at2 , . . . , atdt −1 by Con+ t := {u ∈ m : u =

d t −1

αi ati ; αi > 0}.

i=1

Then i) If, for each t > 0, −adt ∈ Con+ t then the market admits a unique martingale measure. / Con+ ii) If, for some t > 0, −adt ∈ t then the market does not allow a martingale measure. Proof

By definition, a measure Q is a martingale measure if ˆ T) | Ft−1 ] = P(t ˆ − 1, T), EQ [P(t,

T > t,

or, equivalently, EQ [ηt | Ft−1 ] =

dt 

ati qi = 0,

(3.7.1)

i=1

where qi := Q(ηt = ati | η0 = x0 , η1 = x1 , . . . , ηt−1 = xt−1 ) > 0 for i = 1, 2, . . . , dt . (a) If a martingale measure exists then condition (3.7.1) clearly means that vectors at1 , . . . , atdt are linearly dependent which contradicts (ND1). (b) If −adt ∈ Con+ t then from the unique representation of −adt , −adt =

d t −1 i=1

αi ati

102

Completeness we can construct a martingale measure in a unique way via αi qi := d −1 , i = 1, 2, . . . , dt − 1, t j=1 αj + 1

1 . qdt := d −1 t j=1 αj + 1

If −adt ∈ / Con+ t then at least one qi defined above is nonpositive. A Counterexample It turns out that when ηt , t = 1, 2, . . . take an infinite number of values, uniqueness of the martingale measure does not imply completeness. Now we construct an example, in the spirit of Proposition 1.5.3, of a one period regular bond market for which there exists a unique martingale measure but the model is not complete. Proposition 3.7.3 There exists a one period regular bond market model that admits a unique martingale measure and is not complete in the class Xˆ ∈ L∞ (, F1 , P), where F1 is the minimal σ -field at time 1. Proof We construct the required market on the probability space  = {ω1 , ω2 , . . .} consisting of natural numbers, i.e. ωk = k, k = 1, 2, . . ., with the measure P({ωk }) = 1 , k = 1, . . .. Let the initial prices be given by 2k P(0, T) =

1 , 2T

T = 0, 1, . . . .

If the market is regular then the prices at time 1 satisfy P(1, 1) = 1,

0 < P(1, T) ≤ 1,

P(1, T) ≥ P(1, T + 1),

T = 1, 2, . . . . (3.7.2)

Using ˆ T) − P(0, ˆ T) = P(1, T)P(0, 1) − P(0, T), η(T) = η1 (T) := P(1,

T = 1, 2, . . . ,

one can check, as we did in the proof of Proposition 1.5.3, that (3.7.2) is equivalent to η(1) = 0,

η(T) + P(0, T) > 0,

η(T + 1) − η(T) ≤ P(0, T) − P(0, T + 1), Let us define η(T) by

η(1) ≡ 0,

η(T)(ωk ) :=

T = 1, 2, . . . .

⎧ ⎪ ⎪ ⎨

for k = T − 1,

⎪ ⎩ 0

elsewhere.

1 2T+3 − 1 ⎪ 2T+2

for k = T,

(3.7.3)

T = 2, 3, . . . ,

One can check directly that (3.7.3) is satisfied, so the market is regular. Let Q be a measure equivalent to P and set qi := Q(ωi ), i = 1, 2, . . . . The martingale condition EQ [η] = 0

3.7 Completeness and Martingale Measures

103

reads as 0 = EQ [η(T)] = qT−1

1 2T+3

− qT

1 2T+2

,

T = 2, 3, . . . .

This yields qT = 12 qT−1 for T = 2, 3, . . . and consequently +∞ 

+∞  1 = 2q1 = 1 2i

qi = q1

i=1

i=0

if and only if q1 = 12 . This means that there exists exactly one martingale measure Q and it is equal to P. Since η takes an infinite number of values, by Theorem 3.2.1, we see that the market is not complete. Remark 3.7.4 The bond market from Proposition 3.7.3 is L2 -approximately complete. To see this, we use Theorem 3.4.1. For Y ∈ L2 (, F1 , P), we have E[Y] =

+∞ 

Y(ωk )

k=1

1 2k

and for T = 2, 3, . . ., E[Yη(T)] =

+∞ 

Y(ωk )η(T)(ωk )

k=1

1 2k

= Y(ωT−1 )η(T)(ωT−1 ) = Y(ωT−1 ) =

1

1

2T+3

2T−1

1 2T−1

+ Y(ωT )η(T)(ωT )

− Y(ωT )

1 2T+2

1 2T

1 2T

1 (Y(ωT−1 ) − Y(ωT )). 22T+2

Hence E[Y] = 0,

E[Yη(T)] = 0,

T = 2, 3, . . .

if and only if Y ≡ 0. So, condition (3.4.3) is satisfied and the assertion follows.

Part II Fundamentals of Stochastic Analysis

4 Stochastic Preliminaries

This chapter is devoted to general stochastic analysis and covers such topics as square integrable martingales, Doob–Meyer decomposition, semimartingales and their characteristics, random measures, compensators and compensating measures. Stochastic integration with respect to semimartingales and random measures is presented as well. The material is compiled with almost no proofs and is based on Protter [102], Jacod and Shiryaev [75], where more detailed presentations are available.

4.1 Generalities Let us recall that a measurable space (E, E) consists of a set E and a σ -field E of its subsets. If E is a metric space then B(E) stands for the Borel σ -field, the smallest σ -field containing all open subset of E. If (E1 , E1 ), (E2 , E2 ) are two measurable spaces then the product of E1 and E2 is denoted by E1 × E2 and it is the smallest σ -field containing all sets of the form A × B, where A ∈ E1 , B ∈ E2 . If μ1 , μ2 are measures on (E1 , E1 ), (E2 , E2 ), respectively, then μ1 × μ2 denotes the product measure on (E1 × E2 , E1 × E2 ). Sometimes we simplify the notation and write also μ1 (de1 )μ(de2 ) for the product measure. Let (, F, P) be a probability space with filtration Ft , t ≥ 0, satisfying F = F∞ . ( The filtration is assumed to be right continuous, i.e. Ft = s>t Fs for t ≥ 0, and to satisfy the usual conditions, which means that F is complete, i.e. B ⊆ A, A ∈ F, P(A) = 0

⇒

B ∈ F,

and Ft contains all P-null sets of F for each t ≥ 0. A transformation X :  × R+ → U, where U is a separable Hilbert space with the Borel σ -field B(U), is called a stochastic process if it is a measurable function from ( × R+ , F × B(R+ ) to (U, B(U)). It will be denoted by Xt , t ≥ 0, (Xt ), X(t) or just X for short. For a fixed time point t ≥ 0, the function ω → Xt (ω)

108

Stochastic Preliminaries

is a U-valued random variable and for any ω ∈ , t → Xt (ω) is a U-valued function. It is called a path or trajectory of X. The paths of X are c`adl`ag if they are right continuous on [0, +∞) and have left limits on (0, +∞) in U. If this is the case then the process of left limit X− : Xt− = lim Xs , s↑t

t > 0,

(4.1.1)

and the process of jumps X: Xt := Xt − Xt− ,

t > 0,

(4.1.2)

are well defined in U. If Xt = 0, t ≥ 0, then X is called continuous. Two processes X and Y are indistinguishable if their paths are identical almost surely, i.e. P(X(t) = Y(t), t ≥ 0) = 1.

(4.1.3)

If the laws of Xt and Yt are equal for each t ≥ 0, i.e. P(Xt = Yt ) = 1,

t ≥ 0,

(4.1.4)

then Y is a modification or version of X. It is clear that (4.1.3) implies (4.1.4) and it is easy to construct an example showing that the converse implication is not true. However, under the assumption that both X and Y have c`adl`ag paths, (4.1.3) and (4.1.4) are equivalent. The following boundedness properties of c`adl`ag processes can be proven by application of elementary arguments to their paths. Proposition 4.1.1 Let X be a process with c`adl`ag paths in U. Then for any t > 0 (a) and any ε > 0, we have   P {s ∈ [0, t] :| X(s) |> ε} < +∞ = 1, (b) we have



P

 sup | X(s) |< +∞ = 1,

t ≥ 0.

s∈[0,t]

Above | · | stands for the norm in U. The result tells us that, on each finite time interval, the number of jumps of a c`adl`ag process exceeding a positive constant is finite and that the process is bounded. The second property is the same as for continuous processes, but a c`adl`ag process does not have to take its extreme values on compacts. Recall that if the process X is such that Xt is Ft -measurable for any t ≥ 0 then it is called adapted. Adapted processes generate two σ -fields on the space  × R+ , which we introduce now.

4.2 Doob–Meyer Decomposition

109

˜ =  × R+ generated by all adapted processes Definition 4.1.2 The σ -field on  with c`adl`ag (resp. continuous) paths is called optional (predictable) and denoted by O, (P). A stochastic processes is called optional (predictable) if it is O (P)-measurable. It is clear that P ⊆ O, so each predictable process is also optional. It can be proved that any optional process is adapted and that the σ -field P is equal to that generated by all adapted left continuous processes. Since every process X with continuous paths can be pointwise approximated by the sequence X n (t) := X0 1{0} (t) +

+∞ 

X k−1 1( k−1 , k ] (t),

k=1

n

n

n

t ≥ 0,

n = 1, 2, . . . ,

we obtain the following alternative characterization of P. Proposition 4.1.3 The σ -field P is generated by the following family of sets A × {0}, A ∈ F0

and

A × (s, t], A ∈ Fs , s < t.

4.2 Doob–Meyer Decomposition In this section we present some basic facts concerning martingales and submartingales taking values in U = Rd including the Doob–Meyer decomposition, which is of prime importance for the sequel. Definition 4.2.1 An adapted process X taking values in U satisfying E[| Xt |] < +∞, t ≥ 0 is a martingale if E[X(t) | Fs ] = X(s),

0 ≤ s ≤ t.

If U = R and E(Xt | Fs ) ≥ Xs , (E(Xt | Fs ) ≤ Xs ) then X is called a submartingale (supermartingale). A martingale M is square integrable if sup E(| M(t) |2 ) < +∞. t≥0

The space of all square integrable U-valued martingales with c`adl`ag paths will be denoted by M2 (U) while M2,c (U) stands for all elements of M2 (U) with continuous paths. For U = R we abbreviate the notation by setting M2 := M2 (R) and M2,c := M2,c (R). Recall that a [0, +∞]-valued random variable τ is a stopping time with respect to the filtration Ft if {τ ≤ t} ∈ Ft for each t ≥ 0.

110

Stochastic Preliminaries

Definition 4.2.2 An adapted process M is called a local martingale if there exists a sequence of stopping times {τn } such that τn ≤ τn+1 ,

lim τn = +∞,

P − a.s.

n→+∞

(4.2.1)

and the stopped process M τn (t) := M(τn ∧ t) is a martingale for each n. The sequence {τn } satisfying (4.2.1) will be called a localizing sequence for M. In the definition of local martingale one can, additionally, require that M τn is a uniformly integrable martingale. In fact both definitions are equivalent. If M is a martingale then τn := n, n = 1, 2, . . . defines its localizing sequence and hence M is also a local martingale. There are local martingales that fail to be martingales. By M2loc (U) we denote the class of local martingales M such that there exist localizing sequences {τn } such that M τn ∈ M2 (U). Likewise we define the classes M2,c loc (U), M2loc and M2,c . loc For the path regularity of martingales and submartingales we need the concept of stochastic continuity. Recall that a U-valued process is stochastically continuous if lim P(| Xt − Xs |> ε) = 0,

s→t

ε > 0, t ≥ 0.

Clearly, any process with continuous paths is stochastically continuous but the opposite implication is not true. For stochastically continuous submartingales we have, however, the following regularity result. Theorem 4.2.3 Any stochastically continuous submartingale has a modification with c`adl`ag paths. It follows from Theorem 4.2.3 that any stochastically continuous submartingale X the processes X− , X, (see (4.1.1), (4.1.2)) are well defined. This holds, in particular, if X is a real-valued martingale. Theorem 4.2.4 (Optional Sampling) Let X be a c`adl`ag submartingale and τ a bounded stopping time. Then X(τ ) is integrable and E[X(τ ) | Fσ ] ≥ X(τ ∧ σ ) for any stopping time σ . It follows immediately that E[X(t ∧ τ ) | Fs ] ≥ X(s ∧ τ ),

0≤s≤t

for any stopping time τ and a submartingale X from Theorem 4.2.4, which means that the stopped process X τ (t) := X(τ ∧ t) is also a c`adl`ag submartingale. The following result, which is due to Doob and Meyer, describes decomposition of submartingales and plays a central role in further study of the properties of square

4.2 Doob–Meyer Decomposition

111

integrable martingales and compensators of increasing processes in the sequel. We need an auxiliary definition. Definition 4.2.5

A process X is of class (D) if the set of random variables {Xτ : τ − finite valued stopping time}

is uniformly integrable. Recall, that a family of random variables (Yα )α∈A is uniformly integrable if lim

sup E[| Yα | 1{|Yα |≥n} ] = 0.

n−→+∞ α∈A

Theorem 4.2.6 (Doob–Meyer) For any c`adl`ag submartingale X of class (D) there p exists a unique predictable, increasing, integrable process X p with X0 = 0 such that the process p

Xt − Xt ,

t≥0

is a uniformly integrable martingale.

4.2.1 Predictable Quadratic Variation of Square Integrable Martingales The following result concerned with a U = Rd -valued square integrable martingale is a consequence of Theorem 4.2.6. Theorem 4.2.7 For any M ∈ M2loc (U) there exists a unique R-valued process M, M t ; t ≥ 0 that is predictable, increasing, M, M 0 = 0 and such that | Mt |2 − M, M t = Mt , Mt − M, M t ;

t ≥ 0,

(4.2.2)

is a local martingale. If M ∈ M2 (U) then M, M t ; t ≥ 0 is integrable and (4.2.2) is a martingale. The process M, M is called a predictable quadratic variation or an angle bracket of M. Notice a hardly noticeable difference in notation between the angle bracket ·, · and the scalar product ·, · . The definition of quadratic variation can be extended. For any M, N ∈ M2loc (U) we can write ! 1 Mt , Nt = | Mt + Nt |2 − | Mt − Nt |2 , t ≥ 0, 4 which immediately yields, that Mt , Nt −

! 1 Mt + Nt , Mt + Nt − Mt − Nt , Mt − Nt , 4

t≥0

(4.2.3)

112

Stochastic Preliminaries

is a local martingale. The process

! 1 M + N, M + N t − M − N, M − N t , t≥0 (4.2.4) 4 is called a predictable quadratic covariation or an angle bracket of M and N. Clearly, M, N inherits the properties of quadratic variations of M + N and M − N, i.e. it is unique, predictable and starts from zero. If, additionally, M, N ∈ M2 (U), then M, N is integrable and (4.2.3) is a martingale. If M i are coordinates of the martingale M then a predictable operator quadratic variation is defined by M, N t :=

M, M

t := ( M i , M j t )i,j . Moreover, there exists a predictable process Qt taking values in the space of positive symmetric d × d matrices such that  t M, M

t = Qs d M, M s (4.2.5) 0

(see Theorem 8.2 in Peszat and Zabczyk [100]).

4.2.2 Compensators of Finite Variation Processes The aim of this section is to introduce the notion of compensator of an increasing process. Since increasing and finite variation processes are closely related we extend our discussion on finite variation processes. The class of all R-valued adapted increasing process with c`adl`ag paths starting from 0 will be denoted V + . For any A ∈ V + there exists an R ∪ {+∞}-valued random variable A∞ := lim At . t→+∞

If E[A∞ ] < +∞ then the process A is called integrable and the corresponding class is denoted by A+ , that is, & ' A+ := A ∈ V + , E[A∞ ] < +∞ . If there exists an increasing sequence of stopping times {τn }n=1,2,... that diverges to +∞ and such that the stopped process Aτn belongs to A+ for each n, then A will be called locally integrable. A+ loc stands for the class of all locally integrable increasing processes. An R-valued adapted process X is called of finite variation if its (total) variation defined by TV(X)t := lim

n→+∞

n−1  k=1

| X (k+1)t − X kt | n

n

is finite for any t ≥ 0 and any path. By V we denote the family of R-valued adapted processes starting from 0 with c`adl`ag paths of finite variation. Clearly, any process

4.2 Doob–Meyer Decomposition

113

from V + is of finite variation. On the other hand, any R-valued process of finite variation X is a difference of two increasing processes because it can be decomposed to the form Xt = Xt1 − Xt2 ,

t ≥ 0,

(4.2.6)

with X 1 := 12 (TV(X)+X) and X 2 := 12 (TV(X)−X). If A ∈ V and E[TV(A)∞ ] < +∞ then A is said to be of integrable variation and & ' A := A ∈ V, E[TV(A)∞ ] < +∞ stands for the corresponding class. By Aloc we denote the localized class. It is clear that a process A from V does not have to belong to Aloc nor to A. The situation changes, however, if A is predictable. Proposition 4.2.8 Let A ∈ V be a predictable process. Then A ∈ Aloc . The variation of a local martingale turns out to be locally integrable providing that it is finite. Let M be an R-valued local martingale that belongs to V. Then

Proposition 4.2.9 M ∈ Aloc .

The variation of local martingales is, however, typically infinite. This happens always if the martingale is continuous or, more generally, predictable. Proposition 4.2.10 Let M be an R-valued predictable local martingale that belongs to V. Then M(t) = M(0), t ≥ 0. Further stochastic properties of increasing and finite variation processes arise from Theorem 4.2.6. Theorem 4.2.11 such that

+ p For A ∈ A+ loc there exists a unique predictable process A ∈ Aloc

A(t) − Ap (t),

t≥0

is a local martingale. The process Ap in Theorem 4.2.11 is called the compensator of A. It can be alternatively characterized as follows. Proposition 4.2.12 only if

+ p Let A ∈ A+ loc . Then A ∈ Aloc is the compensator of A if and

E[A(τ )] = E[Ap (τ )] for any stopping time τ .

(4.2.7)

114

Stochastic Preliminaries

Since every real valued process of finite variation can be represented as a difference of two increasing processes (see (4.2.6)), as a consequence of Theorem 4.2.11 we obtain the following. Theorem 4.2.13 For A ∈ Aloc there exists a unique predictable process Ap ∈ Aloc such that A − Ap is a local martingale. Also in this case Ap will be called the compensator of A.

4.3 Semimartingales A real valued process of the form Xt = X0 + Mt + At ,

t ≥ 0,

(4.3.1)

where X0 ∈ R, M is a local martingale with M0 = 0 and c`adl`ag paths, A adapted process with A0 = 0 and c`adl`ag paths of finite variation, is called a semimartingale. The sum in (4.3.1) will be referred to as a decomposition of the semimartingale X. It is, in general, not unique unless the process A is predictable. If A is predictable then Z is called a special semimartingale. If X is a special semimartingale then its decomposition with predictable A is called canonical and its uniqueness one can easily prove with the use of Proposition 4.2.10. The class of special semimartingales can be characterized as follows. Theorem 4.3.1 Let X be a semimartingale. Then X is a special semimartingale if and only if the process Xt∗ := sup | Xs |,

t≥0

s∈[0,t]

belongs to A+ loc . Application of Theorem 4.3.1 yields a useful criteria for identifying special semimartingales. Corollary 4.3.2 Every semimartingale X with bounded jumps, i.e. such that | X |≤ a with a ≥ 0, is special. In particular, every continuous semimartingale is special. It turns out that the property of bounded jumps is inherited by the canonical decomposition of a semimartingale. More precisely, the following result can be proven. Theorem 4.3.3 Let X be a semimartingale such that | X |≤ a, where a ≥ 0, and X = X0 + M + A be its canonical decomposition. Then | A |≤ a and | M |≤ 2a. Specifically, both components of the canonical decomposition of a continuous semimartingale are continuous.

4.3 Semimartingales

115

As already mentioned, a general semimartingale admits many decompositions with a non-predictable finite variation part. A wide and useful class of decompositions arises from the following result dealing with decompositions of local martingales. Theorem 4.3.4 Let M be a local martingale. For any constant α > 0 there exist two local martingales M 1 and M 2 such that | M1 |≤ α, M 2 is of finite variation and M = M 1 + M 2 . Corollary 4.3.5 It follows from Theorem 4.3.4 that any decomposition of a semimartingale can be modified in such a way that the local martingale part has bounded jumps, hence is locally square integrable. So, any semimartingale X admits the representation X(t) = X(0) + M(t) + A(t),

t > 0,

where M ∈ M2loc and A ∈ V. Theorem 4.3.4 can, in a sense, be extended by proving that any local martingale M can be written as a sum of two unique local martingales M = M 1 +M 2 such that M 1 is continuous and M 2 is such that the product M 2 · N is again a local martingale for any continuous local martingale N. In contrast to the decomposition in Theorem 4.3.4, M 2 doesn’t have to be of finite variation, in general. What may seem surprising is that M 1 is robust for the martingale part of a given semimartingale. Let us formulate this fact precisely. Theorem 4.3.6 For any semimartingale X there exists a unique continuous local martingale X c such that any decomposition X = X0 + M + A satisfies M c = X c . X c defined in Theorem 4.3.6 is called a continuous martingale part of X. Recall that for any semimartingale X there exists a quadratic variation process [X, X] defined by [X, X]t := lim

n→+∞

n−1  k=1

| X (k+1)t − X kt |2 , n

n

t ≥ 0,

where the convergence holds in probability uniformly on every compact interval. Clearly, [X, X] is an adapted, increasing process with c`adl`ag paths, [X, X]0 = 0 and [X, X]t = (Xt )2 . If X is a local martingale then its quadratic variation has the following property. Theorem 4.3.7

If M is an R-valued local martingale then the process Mt2 − [M, M]t ,

is a local martingale.

t≥0

116

Stochastic Preliminaries

Let us notice that Theorem 4.3.7 is, in a sense, similar to Theorem 4.2.7 if considered in the class M2loc . In this case both of the processes M 2 − [M, M],

M 2 − M, M ,

are local martingales, but the quadratic variation [M, M] and predictable quadratic variation M, M differ in general. This fact does not contradict the uniqueness of M, M formulated in Theorem 4.2.7 because [M, M] is, usually, not predictable. Since [M, M] is increasing one may suppose that it meets the requirements allowing to construct its compensator. This is the case that shows the next result. Theorem 4.3.8 If M ∈ M2loc then [M, M] is locally integrable, i.e. [M, M] ∈ A+ loc and M, M is the compensator of [M, M]. If M ∈ M2,c loc then [M, M] has continuous paths, hence is predictable, and thus both quadratic variations are equal, i.e. [M, M]t = M, M t ,

t ≥ 0,

M ∈ M2,c loc .

(4.3.2)

If the paths of M ∈ M2loc are not continuous then the relation between quadratic variations is  [M, M]t = M c , M c t + | Ms |2 , t ≥ 0. s∈[0,t]

Notice, that M c , M c is well defined because M c has continuous paths and hence is a local square integrable martingale. The preceding formula can be extended also for semimartingales. Theorem 4.3.9 Then

Let X be a semimartingale and X c its continuous martingale part. [X, X]t = X c , X c t +



| Xs |2 ,

t ≥ 0.

(4.3.3)

s∈[0,t]

The process [X, X] is increasing with [X, X] =| X |2 , so its jumps are summable. Moreover, one defines [X, X]c by   [X, X]ct = [X, X]t − [X, X]s = [X, X]t − | Xs |2 , t ≥ 0, s∈[0,t]

s∈[0,t]

which is the pathwise continuous part of [X, X]. The quadratic variation is a useful tool for estimating the supremum of a local martingale (see Kallenberg [79], Theorem 26.12). Theorem 4.3.10 (Burkholder–Davis–Gundy) For each p ≥ 1 there exists a constant 0 < cp < +∞ such that for any local martingale M with M(0) = 0 and any t ≥ 0,   p p 1 E[M, M]t2 ≤ E sup | Ms |p ≤ cp E[M, M]t2 . (4.3.4) cp s∈[0,t]

4.4 Stochastic Integration

117

If the local martingale M in the preceding result has continuous paths, then (4.3.2) takes the form   p p 1 E M, M t2 ≤ E sup | Ms |p ≤ cp E M, M t2 . (4.3.5) cp s∈[0,t]

4.4 Stochastic Integration Here we recall construction of the stochastic integrals  t  t g(s)dX(s), h(s, y)π(ds, dy), 0

0

U

with respect to a semimartingale X and its jump measure π . We consider first stochastic integrals  t g(s)dX(s), t > 0,

(4.4.1)

0

where the integrator X is a semimartingale with values in U = Rd and the integrand g is a predictable process taking values in the set of matrices Mk×d . The definitions and results extend easily to the case when U, H are infinite dimensional separable Hilbert spaces and g is an operator valued process.

4.4.1 Bounded Variation Integrators In view of Corollary 4.3.5, any semimartingale X can be represented in the form X(t) = X(0) + M(t) + A(t),

t > 0,

(4.4.2)

where M ∈ M2loc (U) and the coordinates of A belong to V. Then (4.4.1) can be defined by  t  t  t g(s)dX(s) := g(s)dM(s) + g(s)dA(s), t > 0. (4.4.3) 0

0

0

The second integral in (4.4.3) can be viewed as the classical Lebesgue integral. In fact, A(t) admits the representation     A(t) = A11 (t), . . . , A1d (t) − A21 (t), . . . , A2d (t) , t > 0, where A1l , A2l , l = 1, . . . , d, are increasing, right continuous functions. The functions define measures μ1l , μ2l , l = 1, . . . , d on [0, +∞) and if g(s) = (gj,l (s))j,l , then one sets,  d    t  t 1 2 g(s)dA(s) = gj,l (s)(μl − μl )(ds) , j = 1, 2, . . . , k. (4.4.4) 0

l=1

0

j

If g is locally bounded then the integral (4.4.4) is well defined.

118

Stochastic Preliminaries

4.4.2 Square Integrable Martingales as Integrators Let M be a square integrable martingale taking values in U = Rd and the integrand g be a process taking values in the set of matrices Mk×d , i.e. g(s) = gj,l (s),

j = 1, 2, . . . , k,

l = 1, 2, . . . , d.

For the columns of g(s) we use a single index, i.e. gl (s) = [g1,l (s), g2,l (s), . . . , gk,l (s)],

l = 1, 2, . . . , d.

Similarly, the rows of g(s) are denoted by gl (s) = [gl,1 (s), gl,2 (s), . . . , gl,d (s)], Then the stochastic integral



l = 1, 2, . . . , k.

t

I(g)t :=

g(s)dM(s),

t>0

(4.4.5)

0

will take values in H := Rk . The scalar valued integral (4.4.5), i.e. when k = 1, will be denoted by  t g(s), dM(s) , t > 0. 0

its coordinates M j , j = 1, 2, . . . , d belong to M2 . Consequently, Since M ∈ quadratic covariations are well defined and M2 (U),

M j (t)M l (t) − M j , M l t ,

t > 0,

j, l = 1, 2, . . . , d

are martingales. For a compact formulation of the so-called isometric formula we need some more notation. If A is in Mk×d then its Hilbert–Schmidt | A |2 is given by   1/2 | A |2 := a2i,j . i,j

More generally, a linear operator A acting from a Hilbert space U into a Hilbert space H is called Hilbert–Schmidt if, for an orthonormal basis (ek ) in U: | A |2 =

+∞ 

|Aek |2H

1/2

< +∞.

k=1

Recall that Qt is a d × d positive symmetric matrix given by (4.2.5). Theorem 4.4.1

Let g(s) be a predictable process satisfying  ∗  T 1 2 2 E | g(s)Qs |2 d M, M s < +∞ 0

(4.4.6)

4.4 Stochastic Integration

119

for some T ∗ > 0. Then the integral (4.4.5) is a well-defined H-valued square integrable martingale on [0, T ∗ ] satisfying the following isometric formula  t 1 2 2 2 E(| I(g)t |H ) = E | g(s)Qs |2 M, M s , t ∈ [0, T ∗ ]. (4.4.7) 0

Remark 4.4.2  E 0

t

|

One can check by direct calculation that  ∗ 1 2

g(s)Qs |22

M, M s

T

=E

Trace(g(s)Qs g (s))d M, M s

0

⎛



T∗

= E⎝

0

⎛ = E⎝

 ∗

d 

⎞ ⎠ Qj,l s gj (s), gl (s) d M, M s

j,l=1

d   j,l=1 0

T∗

⎞ gj (s), gl (s) d M j , M l s ⎠ ,

which enables us to write the isometric identity (4.4.7) in several equivalent forms. Proof of Theorem 4.4.1 form

We show the assertion only for simple processes of the

g(s) = a(0)1{s=0} +

n−1 

a(ti )1(ti ,ti+1 ] (s),

i=0

where 0 = t0 < t1 < · · · < tn = T ∗ and a(ti ) is Fti -measurable, bounded and takes values in Mk×d . Then I(g)t =

n−1 

a(ti )(M(ti+1 ∧ t) − M(ti ∧ t)),

t ∈ [0, T ∗ ].

i=0

To avoid technical difficulties, we check the martingale condition and the isometric formula for the partition points only. For 0 < m < n we have ) m *  a(ti )(M(ti+1 ) − M(ti )) | Ftm E[I(g)tm+1 | Ftm ] = E i=0

=

m−1 

a(ti )(M(ti+1 ) − M(ti )) + a(tm ) E[M(tm+1 ) − M(tm ) | Ftm ]

i=0

=

m−1  i=0

a(ti )(M(ti+1 ) − M(ti )) = I(g)tm .

120

Stochastic Preliminaries

Since  t   t   t  E | g(s)dM(s) |2 = E | g1 (s), dM(s) |2 + · · · + E | gk (s), dM(s) |2 , 0

0

0

(4.4.8) we determine first

   t l 2 g (s), dM(s) | , E |

l = 1, 2, . . . , k.

0

We have   E |

tm

)

 g (s), dM(s) | l

2

=E |

0

*

m−1 

a(ti ) , M(ti+1 ) − M(ti ) | l

2

i=0

=E

)m−1  ⎡

* | a(ti )l , M(ti+1 ) − M(ti ) |2

i=0

m−1 

⎢ = E⎣

⎛ ⎝

i=0

⎡

⎞2 ⎤ ⎥ a(ti )l,j (M j (ti+1 ) − M j (ti ))⎠ ⎦

j=1

m−1 

= E⎣

d 

d 

⎤ a(ti )l,j a(ti )l,r (M j (ti+1 ) − M j (ti ))(M r (ti+1 ) − M r (ti ))⎦ .

i=0 j,r=1

Changing the order of the sums and using the quadratic covariation of M j and M r we obtain finally ⎤ ⎡    tm d m−1   gl (s), dM(s) |2 = E ⎣ a(ti )l,j a(ti )l,r ( M j , M r t − M j , M r t )⎦ E | i

i+1

0

j,r=1 i=1

⎡ = E⎣

d  

j,r=1 0

⎤ tm

g(s)l,j g(s)l,r d M j , M r s ⎦ .

In view of (4.4.8) and (4.4.9) we obtain ⎡    tm d  k   2 ⎣ g(s)dM(s) | = E E | 0

l=1 j,r=1 0

⎡ = E⎣

d  

j,r=1 0

(4.4.9)

⎤ tm

g(s)l,j g(s)l,r d M j , M r s ⎦ ⎤

tm

g(s)j g(s)r d M j , M r s ⎦ .



4.4 Stochastic Integration

121

The definition of the integral can be extended to integrands g satisfying the condition  ∗  T 1 2 2 P | g(s)Qs |2 d M, M s < +∞ = 1, (4.4.10) 0

by the usual technique of localization. As a localizing sequence {τn } one may take  s 1 | g(s)Qs2 |22 d M, M s ≥ n, n = 1, 2, . . . . inf ∗ s∈[0,T ] 0

Then, for each n, the integrand g(t ∧ τn ) satisfies the condition (4.4.6) from the theorem and one sets:  t  t g(s)dM(s) = lim g(s ∧ τn )dM(s). n→+∞ 0

0

In particular, if g is locally bounded then the integral is well defined.

4.4.3 Integration over Random Measures Here we discuss the construction of the integral  t h(s, y)π(ds, dy), 0

U

where π(dt, dy) is the so-called jump measure of an U = Rd -valued semimartingale X and h(·, ·) takes values in some Hilbert space H. Since X has c`adl`ag paths, one may count its jumps along a set I ∈ B(R+ ) which belong to A ∈ B(U). This number is denoted by π(I × A), i.e. π(I × A) := {s ∈ I : X(s) ∈ A},

I ∈ B(R+ ),

A ∈ B(U).

(4.4.11)

¯ where A¯ stands for the closure of A, and I If A is separated from zero, that is, 0 ∈ / A, is bounded, then it follows from Proposition 4.1.1 that π(I × A) is finite. If A is not separated from zero or I is unbounded then π(I × A) may be infinite. For a sequence {Ii × Ai } of disjoint sets in R+ × U π

 +∞ %

+∞   Ii × Ai = π(Ii × Ai ),

i=1

i=1

hence π(·) can be viewed as a measure on B(R+ ) × B(U). In fact,  π= 1X(s) δs,X(s) , s≥0

and it can be regarded as a measure valued random variable. It will be called the jump measure of X. It is clear that the jump measure takes values in the set of natural numbers, is σ -finite and random, as it depends on paths of X.

122

Stochastic Preliminaries

˜ × U −→ H one defines an integral over π in the For h :  × R+ × U =  natural way  t  (h ∗ π )t := h(s, y)π(ds, dy) := h(s, X(s)). 0

U

s∈[0,t]

If the function h is measurable with respect to the product σ -field P × B(U) then it is called predictable process or predictable random field. With π(·) we can associate another random measure, the so-called (predictable) compensating measure π p , which we characterize in the following proposition. For the sake of brevity we write π(t, A) := π([0, t] × A) in the sequel. Proposition 4.4.3 Let π be a jump measure of an adapted, U-valued process with c`adl`ag paths. There exists a unique random measure π p (·, ·) on B(R) × B(U \ {0}), such that π(t, A) − π p (t, A), is a local martingale for any set A separated from zero. Moreover, if h :  × R+ × U −→ R is a predictable process such that (| h | ∗π )· ∈ A+ loc , then (| h | ∗π )· ∈ A+ loc ,

(4.4.12)

and (h ∗ π p ) is the compensator of (h ∗ π ). Proof

If A ∈ B(U) is separated from zero then the process t −→ π(t, A),

t≥0

is adapted, increasing and c`adl`ag with jump sizes equal to 1, hence locally integrable. It follows from Theorem 4.2.11 that there exists its compensator, which one denotes by Y(A), i.e. π(t, A) − Yt (A), t ≥ 0, is a local martingale. Since for any disjoint, sets A, B separated from zero ∈ B(U) we have π(t, A ∪ B) = π(t, A) + π(t, B),

t ≥ 0,

it follows from the uniqueness of the compensator that Yt (A ∪ B) = Yt (A) + Yt (B),

t ≥ 0, Yt (∅) = 0,

so Yt (·) is a finitely additive measure defined on the ring of separated from zero Borel sets of U. By the Carath´eodory theorem there exists an extension of Yt (·) to the measure defined on the σ -field generated by the ring that is identical with B(U \ {0}). Moreover, since U \ {0} is a union of separated from zero sets, i.e. ∞ & ' % u ∈ U :| u |≥ 1/n , U \ {0} = i=1

4.4 Stochastic Integration

123

it follows that Yt (·) is σ -finite and thus the extension is unique. We set π p (t, A) := Yt (A),

t ≥ 0,

A ∈ B(U \ {0}),

p

and extend πX (·, ·) to the measure π p (I × A),

I ∈ B(R),

A ∈ B(U \ {0})

in a standard way. Let us consider the function h(ω, w, x) := 1A×(s,v]×B (ω, w, x),

¯ s < v. (4.4.13) A ∈ Fs , B ∈ B(U), 0 ∈ / B,

Since A × (s, t], where A ∈ Fs , s < t belongs to P, h is predictable. Integration of (4.4.13) over π and π p yields the following expressions (h ∗ π )t = 1A×(s,+∞) (ω, t)π((s, t ∧ v] × B) = 1A×(s,+∞) (ω, t)(π(t ∧ v, B) − π(s, B)),

(4.4.14)

(h ∗ π p )t = 1A×(s,+∞) (ω, t)π p ((s, t ∧ v] × B) = 1A×(s,+∞) (π p (t ∧ v, B) − π p (s, B)).

(4.4.15)

It is clear that (4.4.14) is optional, that (4.4.15) is predictable and also that (h ∗ π )t − (h ∗ π p )t ,

t≥0

is a local martingale. Hence, it follows that (4.4.15) is the compensator of (4.4.14). If h is a general predictable function, then, by Proposition 4.1.3, it can be approximated by finite combinations of functions of the form (4.4.13). Hence, if only (h ∗ π ) is of locally integrable variation then (h ∗ π p ) is its compensator. Remark 4.4.4

The measure π˜ defined by π˜ (t, A) := π(t, A) − π p (t, A)

is called a compensated jump measure of X. The integral (h ∗ π˜ ) := (h ∗ π ) − (h ∗ π p ) is a local martingale for predictable h such that (| h | ∗π )· ∈ A+ loc .

4.4.4 Itˆo’s Formula Let us start with the classical Itˆo formula for a real valued semimartingale X (see for instance Protter [102]).

124

Stochastic Preliminaries

Theorem 4.4.5 Let X be a semimartingale and f : R −→ R, f ∈ C 2 (R). Then f (X) admits the representation 

t

f (Xt ) = f (X0 ) +

f  (Xs− )dXs +

0

+

 &

1 2



t

f  (Xs− )d[X, X]cs

0

' f (Xs ) − f (Xs− ) − f  (Xs− )Xs ,

t ≥ 0.

(4.4.16)

s∈[0,t]

In particular, f (X) is a semimartingale. Let us notice that the integral over X in formula (4.4.16) is well defined because the process f  (X(s−)) is predictable and locally bounded. In (4.4.16) [X, X]c stands for the continuous part of the quadratic variation process. One can show that [X, X]ct = [X c , X c ]t ,

t ≥ 0,

where X c stands for the continuous martingale part of X (see Theorem 4.3.6). Using the fact that [X c , X c ] = X c , X c (see (4.3.2)) or the relation [X, X] = X c , X c +

| Xs |2 (see (4.3.3)) we can obtain several equivalent formulations for (4.4.16). To see that f (X) is a semimartingale, we can use (4.4.2) and write (4.4.16) in the form f (Xt ) = f (X0 ) + Nt + Bt , where, for t ≥ 0, 

t

Nt := 

f  (Xs− )dMs ,

0 t

Bt := 0

1 f (Xs− )dAs + 2 



t

0

f  (Xs− )d[X, X]cs +

&

' f (Xs ) − f (Xs− ) − f  (Xs− )Xs .

s∈[0,t]

Moreover, N is a local martingale and B is of finite variation. Now, let X be an Rn -valued semimartingale, that is, each of its coordinate is a real valued semimartingale. Then for a twice differentiable function f : Rn −→ R the following multidimensional version of Itˆo’s formula can be proven f (Xt ) = f (X0 ) +

n   i=1

+

t

0

fxi (Xs− )dXsi +

n  1  t  fxi xj (Xs− )d[X i , X j ]cs 2 0

 &

i,j=1

f (Xs ) − f (Xs− ) −

s∈[0,t]

n 

' fxi (Xs− )Xsi ,

t ≥ 0.

(4.4.17)

i=1

A direct application of (4.4.17) for two real-valued semimartingales X, Y and the function f (x, y) = xy provides the Itˆo product formula, also called the integration by parts formula

 Xt Yt = X0 Y0 + 0

 = X0 Y0 +

4.4 Stochastic Integration  t t  Xs− dYs + Ys− dXs + [X, Y]ct + Xs Ys 0

t



s∈[0,t] t

Xs− dYs +

0

125

Ys− dXs + [X, Y]t ,

t ≥ 0.

(4.4.18)

0

As a direct application of the Itˆo formula we can prove the following. Theorem 4.4.6 Let X be a semimartingale. There exists a unique semimartingale Y solving the equation  t Ys− dXs , t ≥ 0, Yt = Y0 + 0

and it has the form 1

Yt = Y(0)eXt −X0 − 2 [X,X]t

c

$

(1 + Xs )e−Xs ,

t ≥ 0.

(4.4.19)

s∈[0,t]

The solution Y is called the Dol´eans-Dade exponential or stochastic exponential of the semimartingale X. Again, similar to the Itˆo formula, using the relations between [X, X], [X, X]c , [X c , X c ] and X c , X c we can write Y in several equivalent forms.

5 L´evy Processes

This chapter is devoted to L´evy processes, a basic tool used in this book. We start from general definitions and describe specific classes of L´evy processes. For the missing proofs, the readers are referred to Applebaum [2], Sato [114] and Peszat and Zabczyk [100].

5.1 Basics on L´evy Processes L´evy processes are of basic importance for the present book. We will be mainly, but not solely, concerned with L´evy processes taking values in U = Rd , but the definition and several properties are the same if the processes take values in separable Hilbert spaces U. Definition 5.1.1 A U-valued L´evy process {Z(t), t ≥ 0} is an adapted process satisfying the following conditions: (a) Z0 = 0, (b) its increments are independent, i.e. the random variables Z(t0 ), Z(t1 ) − Z(t0 ), Z(t2 ) − Z(t1 ), . . . , Z(tn ) − Z(tn−1 ) are independent for any sequence of times 0 ≤ t0 ≤ t1 ≤ · · · ≤ tn , (c) the law of Z(t + h) − Z(t), t ≥ 0, h > 0, does not depend on t, that is, the increments of Z are stationary, (d) Z is stochastically continuous, i.e. for each t ≥ 0: ∀ε > 0,

P(| Zt+h − Zt |> ε) −→ 0. h→0

L´evy processes have c`adl`ag modifications. We will work solely with such versions. Let μt , t ≥ 0 be the laws of the random variables Zt , t ≥ 0 of a L´evy process Z. Then the family (μt ) is infinitely divisible, that is, μ0 = δ{0} ,

μt ∗ μs = μt+s , t, s ≥ 0,

μt ({u : |u| ≤ r}) → 1 as t ↓ 0, ∀r > 0.

5.1 Basics on L´evy Processes

127

If (μt ) is an arbitrary infinitely divisible family of measures on U then there exists a L´evy process Z such that μt is the law of Zt . This is a direct consequence of the Kolmogorov’s existence theorem. In fact the family pt1 ,...,tn , 0 < t1 < · · · < tn ,

n = 1, 2, . . . ,

pt1 ,t2 ,...,tn (1 , 2 , . . . , n )  = 11 (x1 )12 (x1 + x2 ) . . . 1n (x1 + · · · + xn )μt1 (dx1 )μt2 −t1 (dx2 ) . . . μtn −tn−1 (dxn ), where 1 , . . . , n are arbitrary Borel subsets of U, satisfies the consistency conditions and therefore there exists a stochastic process Z on the probability space (, F, P) such that  E(ϕ(Zt1 , . . . , Ztn )) = ϕ(x1 , x1 + x2 , . . . , x1 + · · · + xn )μt1 (dx1 ) . . . μtn −tn−1 (dxn ), for arbitrary bounded Borel function ϕ. Setting ϕ(x1 , . . . , xn ) = 11 (x1 )12 (x2 − x1 ) . . . 1n (xn − xn−1 ) we get P(Zt1 ∈ 1 , Zt2 − Zt1 ∈ 2 , . . . , Ztn − Ztn−1 ∈ n ) = =

n $ k=1 n $

μtk −tk−1 (k ) =

n $

P(Ztk −tk−1 ∈ k )

k=1

P(Ztk − Ztk−1 ∈ k ),

k=1

where t0 := 0, and the independence of increments follows. Basic examples of L´evy processes are the deterministic function Zt := at, with a ∈ U, the Wiener process, the Poisson process, the compound Poisson process and the finite activity process. Wiener process: If Q = [qij ]i,j=1,2,...,d is a symmetric and positive definite matrix in Rd×d then the U-valued Q-Wiener process is characterized by the following properties: W0 = 0, W has independent and stationary increments, Wt has Gaussian distribution with mean 0 and covariance tQ, W has continuous trajectories. The matrix Q is called the covariance operator of W = (W 1 , . . . , W d ) as it describes covariance of its coordinates, i.e. j

Cov(Wti , Wt ) = tqi,j ,

i, j = 1, 2, . . . , d,

t > 0.

A real valued Wiener process is called standard if Q = q = 1. Compound Poisson process: Let τn , n = 1, 2, . . . be a sequence of independent identically distributed (i.i.d.) random variables with exponential distribution with parameter λ > 0, that is, P(τn > t) = e−λt ,

t ≥ 0.

128

L´evy Processes

A Poisson process {Nt , t ≥ 0} with intensity λ is defined by Nt = 0 if t < τ1 ,

Nt = n

n 

if

τi ≤ t
0 and Y1 , Y2 , . . . a sequence of i.i.d. U-valued random variables that are also independent on N. The compound Poisson process is defined by Z0 = 0

if Nt = 0,

Zt =

Nt 

Yi ,

if Nt > 0,

t ≥ 0.

(5.1.1)

i=1

Its paths are piecewise constant c`adl`ag functions with jump moments governed by the Poisson process and jump magnitudes determined by the sequence {Yi }. Finite activity processes: By a finite activity process we mean a process that admits only a finite number of jumps on any bounded time interval. If Z is a sum of independent components: linear function, Wiener process and compound Poisson process, i.e. Zt := at + Wt +

Nt 

Yi = at + Wt +

i=1



Zs ,

(5.1.2)

s∈[0,t]

then it is clearly of finite activity. It turns out that each L´evy process of finite activity is of the form (5.1.2). Other examples of L´evy processes will be discussed in Section 5.3.

5.2 L´evy–Itˆo Decomposition To describe the general form of a L´evy process let us start from the description of its jump measure. Since Z has c`adl`ag paths we can define, as in (4.4.11), the jump measure of Z by π(I × A) := {s ∈ I : Z(s) ∈ A},

I ∈ B(R+ ), A ∈ B(U),

which counts all jumps of Z, over the time set I, which lie in A. If I is bounded and A separated from zero, then π(I × A) < +∞. We briefly write π(t, A) := π([0, t] × A). The jump measure of a L´evy process is called a Poisson random measure because for any set A separated from zero the function t → π(t, A),

t ≥ 0,

5.2 L´evy–Itˆo Decomposition

129

is a Poisson process. Its intensity is given by the formula ν(A) := E[π(1, A)].

(5.2.1)

Since the processes π(t, A) and π(t, B) are independent if A ∩ B = ∅, it follows that ν(·) is a finitely additive measure defined on the ring of Borel subsets of U separated from zero. Hence, by the Carath´eodory theorem, it can be extended in a unique way to the measure on B(U \ {0}). This measure is called the L´evy measure or also the intensity measure of the process Z. It can be shown that each L´evy measure ν(dy) satisfies  (| y |2 ∧ 1) ν(dy) < +∞. (5.2.2) U

Conversely, if ν(dy) is a measure satisfying (5.2.2), then it is a L´evy measure of some L´evy process. Condition (5.2.2) has a direct interpretation for jumps of Z, that is, for any ε > 0 and t > 0  1{|Z(s)|>ε} < +∞, P − a.s., s∈[0,t]

and



| Z(s) |2 1{|Z(s)|≤ε} < +∞,

P − a.s.

s∈[0,t]

This means that the number of large jumps of Z on a finite interval is finite while small jumps of Z are square summable. Example 5.2.1 Let Z be a compound Poisson process (5.1.1) with intensity λ > 0 of the underlying Poisson process and g(dy) be the distribution of Yi . Then the L´evy measure of Z is given by ν(dy) = λg(dy). Indeed, for a set A separated from zero we have π(t, A) =

Nt 

Yˆ i ,

where

Yˆ i := 1A (Yi ),

t ≥ 0.

i=1

Since P(Yˆ i = 1) = g(A) = 1 − P(Yˆ i = 0), we see that π(t, A), t ≥ 0 is a Poisson process with intensity E[π(1, A)] = λg(A). Here ν is a finite measure. Since for any separated set A from zero the process π(t, A) − tν(A),

t ≥ 0, A ∈ B(U)

is a compensated Poisson process, hence a martingale, it follows that dtν(dy) is the compensating measure of π(dt, dy). Consequently, the compensated jump measure of Z is of the form

130

L´evy Processes π˜ (dt, dy) := π(dt, dy) − dt ν(dy).

Let us consider the integrals  t  t yπ˜ (ds, dy) := I ε (t) := 0

{ε1

(see Kallenberg [79, p. 536]).

5.3 Special Classes

131

It turns out that the quadratic variation of a L´evy process has the form  t   i j [Z, Z]t = t Q + (Z (s)Z (s) = t Q + y ⊗ y π(ds, dy), i,j

s≤t

0

(5.2.4)

U

where y ⊗ y := (yi yj )i,j=1,...,d for y = (y1 , . . . , yd ) ∈ Rd . Another consequence of (5.2.3) is the form of the characteristic function of Z(t). Using the independence of all components in (5.2.3) we obtain E[ei u,Zt ] = etψ(u) , with 1 ψ(u) := i a, u − Qu, u + 2



t ≥ 0, u ∈ U,



 ei u,y − 1 − i u, y 1B (y) ν(dy),

(5.2.5)

(5.2.6)

U

where B := {y :| y |≤ 1}. (5.2.6) is called the characteristic exponent of Z and (5.2.5) the L´evy–Khinchin formula. In much the same way we can obtain the direct formula for the Laplace transform of Z defined by E[e− u,Zt ],

u ∈ U, t ≥ 0.

(5.2.7)

It can be shown that (5.2.7) is well defined for u ∈ U for any t ≥ 0 if and only if  e− u,y ν(dy) < +∞ (5.2.8) {|y|>1}

(see Theorem 25.17 in Sato [114] and Theorem 4.30 in Peszat and Zabczyk [100]). If (5.2.8) holds then E[e− u,Zt ] = etJ(u) ,

t ≥ 0,

where the Laplace exponent J(u) of Z is given by    1 e− u,y − 1 + 1B (y) u, y ν(dy). J(u) = − a, u + Qu, u + 2 U

(5.2.9)

5.3 Special Classes 5.3.1 Finite Variation Processes Here we characterize L´evy processes of finite variation with the use of their characteristic triplets. Proposition 5.3.1 A real-valued L´evy process Z with characteristic triplet (a, q, ν) is a process of finite variation if and only if  q = 0, and | y | ν(dy) < +∞. (5.3.1) {|y|≤1}

132

L´evy Processes

Proof form

If (5.3.1) is satisfied then, in view of (5.2.3), Z can be represented in the

Z(t) = at + Z0 (t) + Z1 (t)   t  t   = a − yν(dy) t + yπ(ds, dy) + yπ(ds, dy), B

0

B

0

Bc

t ≥ 0, (5.3.2)

with B := {y :| y |≤ 1}. It is clear that the first and the third component in (5.3.2) are processes of finite variation. The variation of the second satisfies  !  t  yπ(ds, dy) = t | y | ν(dy) < +∞, t ≥ 0. E TV 0

B

B

Now assume that Z is of finite variation. For any ε > 0  t  | Z(s) | 1{ε 0. y 0 1

Putting z = uy we obtain



+∞ 

 1 dz = uα J(1). z1+α

e−z − 1

J(u) = uα 0

Integration by part yields  J(1) =

+∞ 



e−y − 1

0

1 y1+α

dy

+∞ 1  +∞ 1  − y−α e−y dy = − (e−y − 1)y−α  0 α α 0 1 = − (1 − α), α

 +∞ where (z) stands for the Gamma function, i.e. (z) := 0 xz−1 e−x dx, z > 0. So, finally we obtain 1 J(u) = − (1 − α)uα , u > 0. α

5.3.3 L´evy Martingales Here we characterize L´evy processes that are martingales and square integrable martingales. The following equivalence is valid for α > 0    α E | Zt | < +∞, t ≥ 0 ⇐⇒ | y |α ν(dy) < +∞ (5.3.8) {|y|>1}

(see Sato [114], Theorem 25.3 and Proposition 25.4). Using the L´evy–Itˆo decomposition and (5.2.9) one can deduce the following. Proposition 5.3.4 Let Z be a U-valued L´evy process with characteristic triplet (a, Q, ν). Then the following conditions are equivalent: (a) Z is a martingale, (b)  | y | ν(dy) < +∞, {|y|>1}

 and

a=−

{|y|>1}

yν(dy),

(5.3.9)

5.3 Special Classes

135

(c) Z admits the representation Z(t) = W(t) +

 t 0

y π˜ (ds, dy),

t ≥ 0,

(5.3.10)

U

(d) the Laplace exponent of Z equals    1 J(u) = Qu, u + e− u,y − 1 + u, y ν(dy). 2 U

(5.3.11)

Moreover, if a L´evy process Z is a martingale, then it is a square integrable martingale if and only if  | y |2 ν(dy) < +∞. (5.3.12) {|y|>1}

If Z is a square integrable L´evy martingale, then its predictable quadratic variation equals   2 Z, Z t = t TraceQ + | y | ν(dy) ; t≥0 U

and predictable operator quadratic variation  y ⊗ y ν(dy), Z, Z

t = tQ + t U

compare to (4.2.5). Above a ⊗ b := (ai bj )i,j where a, b ∈ Rd . It is rather surprising that L´evy processes are local or local square integrable martingales if and only if they are martingales resp. square integrable martingales (see Kallenberg [79, p. 518, 536]). We prove this fact under the square integrability condition. Proposition 5.3.5 If Z is a real valued local square integrable L´evy martingale then it is a square integrable martingale. Proof The following direct proof was communicated to us by Słomi´nski [118]. In view of the L´evy–Itˆo decomposition (5.2.3) of Z we can omit the Wiener part and compensated small jumps, as they are both square integrable martingales, and consider Z of the form  t yπ(ds, dy). Z(t) = at + Z1 (t) = at + 0

|y|>1

Then there exists an increasing sequence of stopping times {τn } such that EZ 2 (t ∧ τn ) < +∞, n = 1, 2, . . .. However, for each n, EZ 2 (t ∧ τn ) = E[Z, Z]t∧τn < +∞

136

L´evy Processes

(see Section 4.3 or Protter [102, p. 66]). But   t∧τn  2 E[Z, Z]t∧τn = E y π(ds, dy) |y|>1

0

t∧τn 

 =E

y2 dsν(dy) =

|y|>1

0

so

 |y|>1

 y2 ν(dy) E(t ∧ τn ),

 |y|>1

y2 ν(dy) < +∞.

Therefore

 E[Z, Z]t ≤ t

|y|>1

y2 ν(dy) < +∞,

and by Theorem 4.3.10 Z is a square integrable martingale. The following α-stable martingale will frequently appear in the sequel. Example 5.3.6 (Stable martingale with index α ∈ (1, 2)) A real valued α-stable martingale, with α ∈ (1, 2), is a process  t  +∞ yπ˜ (ds, dy), t ≥ 0, Z(t) := 0

0

1 where the related L´evy measure equals ν(dy) = y1+α dy on (0, +∞). As in Example 5.3.3 we can determine the Laplace exponent of Z. By direct calculation we obtain  +∞  −uy  1 J(u) = e − 1 + uy 1+α dy = uα J(1), y 0

and J(1) =

1 (2 − α). α(α − 1)

This yields J(u) =

1 (2 − α)uα , α(α − 1)

u > 0.

5.4 Stochastic Integration In this section we are concerned with stochastic integrals when the integrator Z is a U-valued L´evy martingale. We allow the space H, where the integrands take values, to be infinite dimensional Hilbert spaces. More specific properties and formulae will be derived for



5.4 Stochastic Integration

 t

t

g(s, y)π˜ (ds, dy),

g(s)dZ(s), 0

137

t ∈ [0, T ∗ ], T ∗ < +∞,

(5.4.1)

U

0

where the integrands are predictable processes taking values in the space of linear operators from U into H and H -valued predictable random fields, respectively. The second integral is with respect to the compensated jump measure π˜ determined by Z. If Z is a martingale then both integrals are local martingales.

5.4.1 Square Integrable Integrators Recall that by Proposition 5.3.4, a L´evy process Z with characteristic triplet (a, QW , ν) belongs to M2 (U) if and only if   | y |2 ν(dy) < +∞, a=− yν(dy). {|y|>1}

{|y|>1}

If this is the case then Z admits the representation  t yπ˜ (ds, dy) = W(t) + Z0 (t), Z(t) = W(t) +

t ≥ 0.

U

0

Moreover,





Z, Z

t = t QW +

 y ⊗ y ν(dy) = t Q,

U

where Q is the covariance operator of Z(1), equals to the sum of the covariance operators of W and Z0 . In addition, the angle bracket of Z is equal to Z, Z t = t TrQ = t E|Z(1)|2 and Qt = Q/TrQ (see the definition (4.2.5)). Therefore the general isometric formula (4.4.7) takes the following transparent form:    t  E | g(s)dZ(s) |2H = E 0

0

t

1 | g(s)Q 2 |22 ds ,

t ∈ [0, T ∗ ].

(5.4.2)

In view of Theorem 4.4.1 we obtain the following characterization. Theorem 5.4.1

If g(s) is a predictable process satisfying  ∗  T

E 0

1

| g(s)Q 2 |22 ds < +∞,

then the integral 

t

g(s)dZ(s), 0

t ∈ [0, T ∗ ],

(5.4.3)

138

L´evy Processes

is a well-defined H-valued square integrable martingale on [0, T ∗ ] and the following isometric formula holds  t   t  1 E | g(s)dZ(s) |2H = E | g(s)Q 2 |22 ds , t ∈ [0, T ∗ ]. (5.4.4) 0

0

Moreover, if



T∗

P

 | g(s)Q

0

1 2

|22

ds < +∞ = 1,

then the stochastic integral is a well-defined local martingale. Remark 5.4.2 For arbitrary linear operators A, B, | AB |2 ≤| A | · | B |2 , therefore the sufficient condition for the integral to be local martingale can be formulated in a simpler way as   ∗ T 2 | g(s) | ds < +∞ = 1. P 0

5.4.2 Integration over Compensated Jump Measures In the case when the jump measure π is related to a L´evy process, then the integral  t g(s, y)π˜ (ds, dy), t ∈ [0, T ∗ ] 0

U

can be defined for more general integrands than those discussed in Section 4.4.3. The process g = g(t, y) is simple if it has the form ⎛ ⎞ mi n−1   ⎝ s ∈ [0, T ∗ ], y ∈ U, gij 1(ti ,ti+1 ] (s)1Aij ⎠ , g(s, y) = g(0, y)1{s=0} + i=0

j=1

(5.4.5) where 0 = t0 < t1 < · · · < tn = T ∗ is a partition of [0, T ∗ ] and Aij is a family of sets in U that are separated from zero, i.e. 0 ∈ / A¯ ij . For a given subinterval (ti , ti+1 ] the process g is a linear combination of terms gij 1(ti ,ti+1 ] (s)1Aij , where gij are H-valued bounded Fti - measurable random variables and Aij , j = 1, 2, . . . , mi are disjoint. Obviously, it is a predictable process (random field). Denote the class of simple processes by S. It is clear that  t mi n   g(s, y)π˜ (ds, dy) := gij π((t ˜ i ∧ t, ti+1 ∧ t] × Aij ), t ∈ [0, T ∗ ]. (g ∗ π˜ )t = 0

U

i=0 j=1

It is a simple consequence of the following lemma that the introduced stochastic integral is a square integrable martingale.

5.4 Stochastic Integration

139

Lemma 5.4.3 For sets A, B ∈ U separated from zero and s < t, s, t ∈ [0, T ∗ ] hold E[π˜ 2 ((s, t] × A) | Fs ] = (t − s)ν(A), E[π˜ ((s, t] × A) · π˜ ((s, t] × B) | Fs ] = 0, E[π˜ ((s, t] × A) · π˜ ((u, v] × B) | Fu ] = 0,

if A ∩ B = ∅, for t ≤ u < v ≤ T ∗ .

We have the following result. Proposition 5.4.4 The stochastic integral  t (g ∗ π˜ )t = g(s, y)π˜ (ds, dy), 0

t ∈ [0, T ∗ ]

U

can be extended from S to all predictable random fields g, as a local martingale, such that ) ∗  * T P | g(s, y) |2H dsν(dy)) < +∞ = 1. 0

U

Moreover, if )

T∗

E

*

 | g(s, y)

0

U

|2H

dsν(dy)) < +∞,

then (g ∗ π˜ )t , t ∈ [0, T ∗ ], is an H-valued square integrable martingale and the following isometric formula holds   t  ! 2 2 E | (g ∗ π˜ )t |H = E | g(s, y) |H dsν(dy)) , t ∈ [0, T ∗ ]. (5.4.6) U

0

The proof of the result is rather standard. First, with the help of the lemma, one shows the isometric formula for simple processes and using it one establishes the second part of the proposition. The first part is established by a localization technique. Denote by 2 (H) the class of all predictable processes satisfying  T∗  2 (H) : | g(s, y) |2H dsν(dy) < +∞, P − a.s., 0

U

and by 1 (H) the class all predictable processes satisfying  T∗  1 (H) : | g(s, y) |H dsν(dy) < +∞, 0

P − a.s.

U

Using localizing arguments one can show that for g ∈ 2 (H) the integral  t g(s, y)π˜ (ds, dy), t ∈ [0, T ∗ ] 0

U

140

L´evy Processes

is a well-defined H-valued locally square integrable martingale and for g ∈ 1 (H) a local martingale. For H = R the corresponding classes will be denoted briefly by 1 , 2 .

5.4.3 Stochastic Fubini’s Theorem In some situations the integrands depend on a parameter a from a measurable space (E, E) with a finite positive measure λ(dx) and the problem of changing the order of integration arises. Results of this type are usually called Fubini’s theorems. It turns out that, under rather weak assumptions, the following formulae hold for t ∈ [0, T ∗ ]:   t  t  g(s, a)dZ(s) λ(da) = g(s, a)λ(da) dZ(s), (5.4.7) E

0

 t   g(s, y, a)π˜ (ds, dy) λ(da) = g(s, y, a)λ(da) π˜ (ds, dy),

  t  E

0

E

0

U

0

U

E

(5.4.8)  t   g(s, y, a)π(ds, dy) λ(da) = g(s, y, a)λ(da) π(ds, dy).

  t  E

0

U

0

U

E

(5.4.9) The following theorem formulates sufficient conditions for the validity of (5.4.7), (5.4.8) and (5.4.9). The identity (5.4.7) is a special case of Fubini’s theorem from Protter’s monograph [102, p. 160]. One can arrive at (5.4.8) using similar methods as in [102, p. 157–162]. Formula (5.4.9) is a corollary from the classical Fubini theorem applied pathwise. Theorem 5.4.5 (Stochastic Fubini) (a) Let Z be a L´evy process and g(t, ω, a), (t, ω, a) ∈ [0, T ∗ ] ×  × E be measurable with respect to σ -field P × E and  ∗   T

P

| g(s, a) |2 ds λ(da) < +∞ = 1.

0

E

t If X(t, a) = 0 g(s, a)dZ(s), (t, a) ∈ [0, T ∗ ]×E, is B([0, T ∗ ])×E ×F-measurable and X(·, a) is c`adl`ag for each a, then (5.4.7) holds. (b) Let π˜ be the compensated random measure corresponding to a L´evy process Z and g(t, ω, y, a), (t, ω, y, a) ∈ [0, T ∗ ] ×  × U × E be measurable with respect to σ -field P × U × E and  ∗    T

P 0

| g(s, y, a) |2 ds ν(dy)λ(da) < +∞ = 1. U

E

5.4 Stochastic Integration

141

t If X(t, a) = 0 U g(s, y, a)π˜ (ds, dy), (t, a) ∈ [0, T ∗ ] × E, is B([0, T ∗ ]) × E × F measurable and X(·, a) is c`adl`ag for each a,then (5.4.8) holds. The result is true if g is a Hilbert space valued process as well. (c) Let π be the random measure corresponding to a L´evy process Z and g(t, ω, y, a), (t, ω, y, a) ∈ [0, T ∗ ]××U×E be measurable with respect to σ -field P ×U ×E and  ∗    T

P 0

| g(s, y, a) | ds ν(dy)λ(da) < +∞ = 1. U

E

t If X(t, a) = 0 U g(s, y, a)π˜ (ds, dy), (t, a) ∈ [0, T ∗ ] × E is B([0, T ∗ ]) × E × Fmeasurable and X(·, a) is c`adl`ag for each a, then (5.4.9) holds. The result is true if g is a Hilbert space valued process as well.

5.4.4 Ito’s Formula for L´evy Processes From Theorem 4.4.5 one derives easily the stochastic representation of the process f (X(t)) for a regular function f and semimartingale X of the form  t  t  t g1 (s)ds + g2 (s), dWs + g3 (s, y)π˜ (ds, dy) Xt = 0 0 0 U  t g4 (s, y)π(ds, dy), t ≥ 0. + 0

U

Here W is a Wiener process with covariance operator Q and π a jump measure – both related to the L´evy process Z. The integrals are assumed to exist. Moreover,  t 1 · 2  t 1 g2 (s), dWs = g2 (s)Qg∗2 (s)ds = | Q 2 g2 (s) |2 ds, t ∈ [0, T ∗ ], [X, X]ct = 0

and

t

0

0

  Xs = g3 (s, Zs ) + g4 (s, Zs ) 1{Zs =0} .

Thus (4.4.16), for a twice continuously differentiable function f from U into R yields  t  1 1 f  (Xs )g1 (s) + f  (Xs ) | Q 2 g2 (s) |2 ds f (Xt ) = f (X0 ) + 2 0  t f  (Xs− ) g2 (s), dWs + f  (Xs− )g3 (s, y)π˜ (ds, dy) 0 0 U  t  f (Xs− )g4 (s, y)π(ds, dy) + 

+

0

+

t

U

 t &

' f (Xs ) − f (Xs− ) − f  (Xs− )(g3 (s, y) + g4 (s, y)) π(ds, dy),

0

t ∈ [0, T ∗ ].

U

(5.4.10)

6 Martingale Representation and Girsanov’s Theorems

We present the theorem on representation of local martingales adapted to the filtration generated by a L´evy process as a sum of stochastic integrals over the Wiener process and compensated random measure. The representation theorem is applied in the proof of the Girsanov formula for densities of equivalent probability measures.

6.1 Martingale Representation Theorem Let Z be a L´evy process with characteristic triplet (a, Q, ν) and let Ft = σ {Z(s), s ≤ t} be its natural filtration. Let W denote its Wiener part and π˜ its compensated random measure. The class of predictable U-valued processes φ such that  T∗ | φ(s) |2 ds < +∞ 0

is denoted by (U). For such processes the stochastic integral  t φ(s), dW(s) , t ∈ [0, T ∗ ] 0

is well defined. The class of predictable processes g(s, y), s ∈ [0, T ∗ ], y ∈ U, satisfying  T∗  (| g(s, y) |2 ∧ | g(s, y) |)dsν(dy) < +∞, P − a.s. 0

U

will be denoted by 1,2 . For g ∈ 1,2 one defines the integral  t  t  t g(s, y)π˜ (ds, dy) := g1{|g|≤1} π˜ (ds, dy) + g1{|g|>1} π˜ (ds, dy), 0

U

which is a local martingale on

0

U

[0, T ∗ ].

0

U

6.2 Girsanov’s Theorem and Equivalent Measures

143

Now we formulate the martingale representation theorem. Theorem 6.1.1 Let M be a real valued P-local martingale on [0, T ∗ ] adapted to (Ft ). Then there exist predictable processes φ and ψ such that, P − a.s., φ ∈ (U), for which



t

Mt = M0 +

φ(s), dW(s) +

0

 t 0

ψ ∈ 1,2 ,

(6.1.1)

ψ(s, y)π˜ (ds, dy),

t ∈ [0, T ∗ ]. (6.1.2)

U

Moreover, the pair (φ, ψ) is unique i.e. if (φ  , ψ  ), φ  ∈ (U), ψ  ∈ 1,2 , satisfy (6.1.2) then Qφ = Qφ  ,

dP × dt − a.s. and

ψ = ψ ,

dP × dt × dν − a.s.

Moreover, if M is a square integrable martingale then  t  t 2 2 1/2 2 2 | Q φ(s) | ds + | ψ(s, y) | dsν(dy) < +∞. EMt = EM0 + E 0

0

U

The problem of representing martingales as stochastic integrals is of fundamental importance in financial modelling and has a very long history. Its importance was noticed in the 1960s in connection with stochastic filtering (Fujisaki–Kallianpur– Kunita filtering equation) and the general theory was developed in the book of Jacod [74]. For early papers dealing with jump processes see Dellacherie [38] and [39], and Davis [34]. The general version of the theorem in a semimartingale framework, can be found in Jacod and Shiryaev [75], but it is not directly applicable to specific cases. The presented version is due to Kunita [84] who used some results of Itˆo [73]. The result is well known if Z is a Wiener process, and we present in Appendix A a detailed proof of Theorem 6.1.1 in the case when Z is a jump process without the Wiener part.

6.2 Girsanov’s Theorem and Equivalent Measures For a U-valued L´evy process Z, with U = Rd , defined on (, F, P) let us consider an equivalent to P measure Q on (, FT ∗ ). Then there exists a density process dQ ⏐ ⏐ t ∈ [0, T ∗ ], ρt := ⏐ dP Ft which is adapted to the filtration (Ft ) generated by Z. The aim of this section is to prove the Girsanov theorem that provides an explicit form of ρ and characterizes the process Z under the measure Q. Theorem 6.2.1 (Girsanov) Let Q ∼ P and Z be a U-valued L´evy process with characteristic triplet (a, Q, ν) under P. The following statements are true.

144

Martingale Representation and Girsanov’s Theorems

(a) There exists a pair of processes (φ, ψ) such that φ ∈ (U) and eψ − 1 ∈ 1,2 such that the density process ρ has the form    ψ(t,y) dρ(t) = ρ(t−) φ(t), dW(t) + (e − 1)π˜ (dt, dy) , U

ρ(0) = 1, with E[ρt ] = 1, t ∈ [0, T ∗ ]. (b) Under the measure Q the process ˜ W(t) := W(t) −



t

φ(s)ds,

t ∈ [0, T ∗ ],

(6.2.1)

t ∈ [0, T ∗ ]

0

is a Q-Wiener process in U and the measure νQ (dt, dy) := eψ(t,y) dtν(dy),

t ∈ [0, T ∗ ], y ∈ U

is the compensating measure for the jump measure π(dt, dy) of Z. Moreover, it satisfies  T∗  (| y |2 ∧ 1)νQ (ds, dy) < +∞. (6.2.2) 0

U

(c) Under Q the process Z admits the representation  t  t ˜ Z(t) = a˜ t + W(t) + y π˜ Q (ds, dy) + 0

with

 a˜ t := at + 0

t

{|y|≤1}

φ(s)ds +

0

 t 0

{|y|≤1}

{|y|>1}

y π(ds, dy), (6.2.3)

y(eψ(s,y) − 1)dsν(dy),

(6.2.4)

and π˜ Q (ds, dy) := π(ds, dy) − νQ (ds, dy). (d) Under Q the process Z is still a L´evy process if and only if φ is a deterministic constant and ψ is a deterministic function independent of time, i.e. φ(ω, t) = φ,

ψ(ω, t, y) = ψ(y),

t ∈ [0, T ∗ ], y ∈ U.

(6.2.5)

The pair (φ, ψ) appearing in the theorem will be called a generating pair of the measure Q. First let us comment on Theorem 6.2.1. The Dol´eans-Dade equation (6.2.1) can be solved explicitly to see that ρ has the form ρt = eYt with  t   t 1 1 t φ(s), dW(s) − | Q 2 φ(s) |2 ds + (eψ(s,y) − 1)π˜ (ds, dy) Y(t) = 2 0 0 0 U −

 t 0

(eψ(s,y) − 1 − ψ(s, y))π(ds, dy), U

t ∈ [0, T ∗ ].

(6.2.6)

6.2 Girsanov’s Theorem and Equivalent Measures

145

It follows from Theorem 6.2.1 that if Q is a measure equivalent to P with the generating pair (φ, ψ) then Z is not a L´evy process under Q, in general. In particular, the measure Q changes stochastic properties of the jumps of Z because the new compensating measure νQ is random and time dependent. Hence π is no longer a Poisson random measure. It is clear that under Q, small jumps are square summable on compacts and since Q ∼ P, there are only a finite number of big jumps. However, (6.2.2) does not imply that their expectations are finite just as it was under P. By (6.2.2) there exists an increasing sequence of stopping times {τn , n = 1, 2, . . .} such that  τn  ! (| y |2 ∧ 1)νQ (ds, dy) < +∞, n = 1, 2, . . . , EQ 0

U

which implies that !  | Zs |2 1{|Zs |≤1} < +∞, EQ EQ s∈[0,τn ]



! 1{|Zs |>1} < +∞,

n = 1, 2, . . . .

s∈[0,τn ]

Moreover, in view of (6.2.9), for any set A ∈ U with 0 ∈ / A¯ holds  t eψ(s,y) dsν(dy) < +∞, νQ ([0, t], A) = 0

(6.2.7)

A

and using similar arguments as in the preceding text one can argue that the process  t νQ (ds, dy), t ≥ 0, π˜ Q (t, A) := π(t, A) − 0

A

is a Q-local martingale and not a Q-martingale, in general. For the proof of Theorem 6.2.1 we need two auxiliary results, which we recall now. Lemma 6.2.2 Let Q be equivalent to P and have the density process ρt , t ∈ [0, T ∗ ]. Then the process M(t) is a Q-local martingale if and only if M(t)ρ(t) is a P-local martingale. Proof Assume that M(t) is a Q-local martingale and let {τn }, n = 1, 2, . . . be a localizing sequence. Then EQ [1 M(t ∧ τn )] = EQ [1 M(s ∧ τn )],

 ∈ Fs ,

s < t,

and thus EP [ρ(t)1 M(t ∧ τn )] = EP [ρ(s)1 M(s ∧ τn )],

 ∈ Fs ,

s < t,

which implies that for each n the process ρ(t)M(t ∧ τn ) is a P-martingale. For a bounded stopping time τˆn := τn ∧ n the stopped process (ρt M(t ∧ τn ))τˆn = ρ(t ∧ τˆn )M(t ∧ τˆn )

146

Martingale Representation and Girsanov’s Theorems

is also a P-martingale and thus {τˆn }, n = 1, 2, . . . forms a localizing sequence for the process ρ(t)M(t). Moreover, since τˆn is bounded, we have lim ρ(t ∧ τˆn )M(t ∧ τˆn ) = ρ(τn )M(τn ),

t→+∞

and uniform integrability of ρ(t ∧ τˆn )M(t ∧ τˆn ) for each n follows. Hence ρ(t)M(t) is a P-local martingale. 1 . The opposite implication can be shown analogously using the density process ρ(t) Proposition 6.2.3 Let X be a semimartingale under P and Q be a measure equivalent to P. Then X is a semimartingale under Q and the quadratic variation of X under Q is the same as under P. Proof The proof that X is a Q-semimartingale can be found in Protter [102, Theorem 2, p. 53]. By definition of quadratic variation we have P( sup | Zsn − [X, X]s |> ε) −→ 0, n

s∈[0,t]

t > 0, ε > 0,

with Ztn := n−1 − X kt |2 . Let ρ = dQ k=1 | X (k+1)t dP be the corresponding density of Q. n n Since ρ is P-integrable it follows that  Q( sup | Zsn − [X, X]s |> ε) = 1{sups∈[0,t] |Zsn −[X,X]s |>ε} ρdP −→ 0, n



s∈[0,t]

and consequently that [X, X] is the quadratic variation of X under Q. Proof of Theorem 6.2.1 (a) To prove (6.2.1) one can use elements of the proof of the martingale representation theorem presented in Appendix A. The positivity of the martingale ρ allows representing it in the exponential form that leads to the Dol´eansDade equation (6.2.1). Details are presented in Lemma A.1.8 and Remark A.1.10 in the pure jump case, but the analysis containing the Wiener part is similar. ˜ is a Q-Wiener process in U under Q. We show that, for (b) First we prove that W any u ∈ U, the process ˜ t = u, W(t) − ˜ tu := u, W W



t

u, φ(s) ds,

t ∈ [0, T ∗ ],

(6.2.8)

0

is, under Q, a real valued Wiener process with variance Qu, u . Application of the ˜ tu yields product Itˆo formula (4.4.18) to the product ρt · W ˜ tu = ρt · W

 0

t

˜ su + ρs− dW



t 0

˜ u ]t . ˜ su dρs + [ρ, W W

6.2 Girsanov’s Theorem and Equivalent Measures

147

Since stochastic integration  t of ua locally bounded process over a local martingale ˜ s dρs is a P-local martingale. Moreover, yields local martingale, so 0 W 

! ρs− φs , dWs , Wtu 0  t  t !  t ρs− φs , dWs , u, dWs = ρs− φs , u ds, =

˜ u ]t = [ρ, W

t

0

0

0

where Wtu := u, Wt , so  t  t  t  t u u u ˜ ˜ ρs− dWs + [ρ, W ]t = ρs− dWs − ρs− u, φs ds + ρs− u, φs ds 0

0

 =

0

0

t

0

ρs− dWsu ,

˜ u is a P-local martingale. Hence W ˜ u is a continuous Qand, consequently, ρ · W u ˜ and W u are the same. By local martingale. By (6.2.8) the quadratic variations of W Proposition 6.2.3 the quadratic variation of W u under Q and P remains the same and equals t Qu, u . So, we obtain ˜ u ]t = [W u , W u ]t = t Qu, u . ˜ u, W [W Since the quadratic variation is continuous, it is equal to the predictable quadratic ˜ u under Q, i.e. variation of W ˜ u t = t Qu, u . ˜ u, W W ˜ u is a Wiener process under Q. With the use of (4.2.4) one can It follows that W ˜ v , which is ˜ u and W determine the predictable quadratic covariation of W ˜ v t = t Qu, v U , ˜ u, W W

t > 0,

u, v ∈ U.

˜ is a Q-Wiener process under Q. This proves that W Now we will show that νQ is a compensating measure of π under Q. It follows from the condition eψ − 1 ∈ 1,2 that for any set A separated from zero 

T∗ 0



 | eψ(s,y) − 1 | dsν(dy) = A

0

 + 0

T∗

T∗

 A

| eψ(s,y) − 1 | 1{|eψ −1|≤1} dsν(dy)

 A

| eψ(s,y) − 1 | 1{|eψ −1|>1} dsν(dy)

≤ T ∗ ν(A) +

 0

T∗

 A

| eψ(s,y) − 1 | 1{|eψ −1|>1} dsν(dy) < +∞.

(6.2.9)

148

Martingale Representation and Girsanov’s Theorems t Hence the integral 0 A eψ(s,y) dsν(dy), t ∈ [0, T ∗ ] is well defined and  t  t A ψ(s,y) e dsν(dy) = π˜ (t, A)+ (1 − eψ(s,y) )dsν(dy), Xt := π(t, A)− 0 A

It follows from the Itˆo product formula that  t  t A Xs− dρs + ρs− dXsA + [ρ, X A ]t , XtA ρt = 0

Since  0

t ∈ [0, T ∗].

0 A

t ∈ [0, T ∗ ].

0

t

ρs− dXsA + [ρ, X A ]t  t  t = ρs− π(ds, ˜ dy) + ρs− (1 − eψ(s,y) )dsν(dy) A

0

+

 t

ρs− π(ds, ˜ dy) + A

0

A

ρs− (eψ(s,y) − 1)π(ds, dy) A

0

=

0

 t

 t 0

ρs− (eψ(s,y) − 1)π˜ (ds, dy),

t ∈ [0, T ∗ ],

A

t we conclude that XtA = π(t, A) − 0 A eψ(s,y) dsν(dy) is a Q-local martingale. So, the measure eψ(s,y) dsν(dy) is the compensating measure of π under Q. To prove (6.2.2) let us notice that  T∗  (| y |2 ∧ 1)eψ(s,y) dsν(dy) < +∞ 0

U



T∗

⇐⇒



0

Using the fact that 

eψ

T∗ 0

(| y |2 ∧ 1) | eψ(s,y) − 1 | dsν(dy) < +∞. U

− 1 ∈ 1,2 we obtain

 (| y |2 ∧ 1) | eψ(s,y) − 1 | dsν(dy) U

 ≤

T∗



0



T∗

+ 0

U

(| y |2 ∧ 1)1{|eψ −1|≤1} dsν(dy)

 U

| eψ(s,y) − 1 | 1{|eψ −1|>1} dsν(dy) < +∞.

(c) The decomposition (6.2.3) follows immediately decompo from the L´  evy–Itˆo ψ ye dsν(dy). sition (5.2.3) by adding and subtracting the terms φds and {|y|≤1} We need, however, to show that all terms in (6.2.3) are well defined. It follows from the estimation

6.2 Girsanov’s Theorem and Equivalent Measures 

T∗

0

=

149

 | y(eψ(s,y) − 1) | dsν(dy)

{|y|≤1}  T∗ 

{|y|≤1}

0





T∗

+

{|y|≤1}

0



| y(eψ(s,y) − 1) | 1{|eψ −1|≤1} dsν(dy)

 1 

T ∗

≤

T∗

+

T ∗

2

{|y|≤1}

0



| y(eψ(s,y) − 1) | 1{|eψ −1|>1} dsν(dy)

| y |2 dsν(dy) 0

{|y|≤1}

1 2

| eψ(s,y) −1 |2 1{|eψ −1|≤1} dsν(dy)



0

{|y|≤1}

| eψ(s,y) − 1 | 1{|eψ −1|>1} dsν(dy) < +∞

t that a˜ t is well defined. In view of (6.2.2) the integral 0 {|y|≤1} yπ˜ (ds, dy) is also well defined. (d) Let us assume that (6.2.5) is satisfied. Then the measure νQ (dt, dy) splits into the product of eψ(y) ν(dy) and dt. It follows from (6.2.2) that  (| y |2 ∧1)eψ(y) ν(dy) < +∞, U

so eψ(y) ν(dy) is a L´evy measure and consequently π(dt, dy) becomes a Poisson random measure under Q. Since  t  t at + φ(s)ds + y(eψ(y) − 1)dsν(dy) = ct, 0

0

{|y|≤1}



with c := a + φ + {|y|≤1} y(eψ(y) − 1)ν(dy), formula (6.2.3) provides the L´evy–Itˆo decomposition of Z under Q. This shows that Z is a L´evy process under Q. Let us now assume that Z is a L´evy process under Q. Then νQ (dt, dy) splits into the product of dt and some L´evy measure. Consequently, for any set A separated from zero there exists a constant cA such that νQ ([0, t] × A) = cA · t, Since νQ ([0, t] × A) = we obtain that

 t 0

eψ(s,y) dsν(dy),

t ∈ [0, T ∗ ],

A

 eψ(t,y) ν(dy) = cA , A

t ∈ [0, T ∗ ].

t ∈ [0, T ∗ ],

150

Martingale Representation and Girsanov’s Theorems

and, consequently, that ψ(ω, t, y) = ψ(y). It follows from (6.2.3) that  t  t ˜ a˜ t := Z(t) − W(t) − y π˜ Q (ds, dy) − y π(ds, dy), 0

{|y|≤1}

0

{|y|>1}

is a L´evy process. However, by (6.2.4), a˜ t is a finite variation process, so it must be a linear function. Hence there exists c such that  t  φ(s)ds + t y(eψ(y) − 1)ν(dy) = ct, t ∈ [0, T ∗ ]. a˜ t = at + 0

{|y|≤1}

This implies that  t   φ(s)ds = c − a − 0

{|y|≤1}

 y(eψ(y) − 1)ν(dy) t,

which yields that φ(ω, t) = φ for t ∈ [0, T ∗ ].

t ∈ [0, T ∗ ],

Part III Bond Market in Continuous Time

7 Fundamentals

The chapter deals with the mathematical description of bond market models and their elementary properties. We introduce three basic models that are at the centre of attention in further parts of the book: the Heath–Jarrow–Morton forward rate model, the Markovian factor forward curves model and the affine term structure model. Relation to modelling of forward rates in function spaces is presented as well.

7.1 Prices and Rates A bond market can be regarded as a collection of two random fields P(t, T),

T ∈ R+ ,

f (t, T),

0 ≤ t ≤ T,

(7.1.1)

and two stochastic processes B(t),

R(t),

t ∈ R+ ,

(7.1.2)

called respectively, bond prices, forward rates, bank account and short rate. They are defined on a probability space (, F, (Ft ), P) and adapted. The value P(t, T) is interpreted as the price, at time t, of the bond that matures at time T. That is, the owner of this T-bond will receive at time T the so-called nominal value, specified on the bond, which we assume to be 1. Hence it is natural to assume that P(T, T) = 1. The relation between bond prices and forward rates is given by the formula P(t, T) = e−

T t

f (t,s)ds

,

0 ≤ t ≤ T.

(7.1.3)

154

Fundamentals

If one spends 1 at fixed time t on buying bonds that mature at time T then the number of bonds one receives is equal to e

T t

f (t,s)ds

= (P(t, T))−1 ,

which is identical with the amount of money one gets at T. So, f (t, T) as a function of T plays a role of rates for bond prices with respect to maturity times T. Empirical observations show that for fixed T > 0, the function t → P(t, T),

t ≥ 0,

is chaotic,

(7.1.4)

T ≥ t,

is regular,

(7.1.5)

and for any fixed t ≥ 0, the function T → P(t, T),

so they may be assumed to be differentiable. Moreover, in typical situations bond prices satisfy 0 < P(t, T) ≤ 1,

0 ≤ t ≤ T,

(7.1.6)

0 ≤ t ≤ S ≤ T,

(7.1.7)

and are monotone functions of T, i.e. P(t, S) ≥ P(t, T),

which means that higher gains require more time. Models satisfying (7.1.6) and (7.1.7) will be called regular. In view of (7.1.3) it is clear that models with positive forward rates are regular. By (7.1.3) we have also the following obvious formula for the forward rates in term of the bond prices ∂ ln P(t, T), t ≤ T. (7.1.8) ∂T More specific assumptions concerned with forward rates and bond prices are motivated by further empirical observations. These are discussed in Section 7.1.2. The value B(t) defined by T → f (t, T) := −

B(t) := e

t 0

R(s)ds

,

B(0) = 1,

(7.1.9)

is interpreted as the amount of money on the bank account at time t from depositing 1 at time 0. It satisfies the equation dB(t) = R(t)B(t)dt,

t > 0.

One requires that the short rate is directly related to the forward rates by R(t) := f (t, t),

t ≥ 0.

(7.1.10)

The relation (7.1.10) might look surprising and therefore is discussed in detail in the next section. Since R given by (7.1.10) is adapted, it follows from (7.1.9) that B is a predictable process.

7.1 Prices and Rates

155

With the use of the bank account and (7.1.10) one can extend the definition of bond prices also for t ≥ T by “cashing” the bond at its maturity T. We assume thus that 1 is put at T on the bank account. This yields P(t, T) = e

t T

R(s)ds

t

=e

T

f (s,s)ds

,

t ≥ T.

(7.1.11)

If, additionally, forward rates satisfy f (t, T) = f (T, T) = R(T);

t > T,

(7.1.12)

which will often be the case, then (7.1.3) and (7.1.11) yield P(t, T) = e−

T t

f (t,u)du

t, T ≥ 0,

,

(7.1.13)

which extends the definition of the bond prices on R+ × R+ .

7.1.1 Bank Account and Discounted Bond Prices Here we give a financial justification of the relation R(t) := f (t, t),

t ≥ 0,

(7.1.14)

by showing that the bank can offer such rate by constructing a “risk-free strategy”. Let us recall that having capital a at time t one can buy T  −1 = a · e t f (t,s)ds , (7.1.15) a · P(t, T) T-bonds with the risk free gain at time T equal exactly to (7.1.15). A roll-over strategy consists of frequently buying bonds maturing in short times, selling them at their maturity times and investing the gain again into bonds. Let us fix time t and a natural number n. Starting with the capital Bn (0) = 1 at time 0, one invests it into bonds maturing at time t/n and reinvests the gain  t/n

Bn (t/n) = e

0

f (0,s)ds

,

at time t/n, in bonds maturing at time 2t/n and so on. After n operations, at time t, the gain will be equal to  −1 (7.1.16) Bn (nt/n) = P(0, t/n)P(t/n, 2t/n) . . . P((n − 1)t/n, tn/n) , or, more explicitly, Bn (t) = e

n−1  (k+1)t/n k=0 kt/n

f (kt/n,s)ds

.

(7.1.17)

Under very mild conditions on the forward rates one has that lim Bn (t) = e

n→+∞

t 0

f (u,u)du

= B(t),

(7.1.18)

for t > 0 with B(t) given by (7.1.9), so (7.1.14) holds. In particular, we have the following result.

156

Fundamentals

Proposition 7.1.1 Assume that the forward rate f (s, T), s, T ∈ [0, T ∗ ], with T ∗ < +∞ , is bounded and one of the following two conditions holds: i) for almost all T ∈ [0, T ∗ ], lim[f (s, T) − f (T, T)] = 0, s↑T

ii) for almost all T ∈

[0, T ∗ ], lim[f (s, T) − f (s, s)] = 0, s↑T

and the points of discontinuity of f (s, s), s ∈ [0, T ∗ ] are of measure 0. Then the convergence (7.1.18) holds for t ∈ [0, T ∗ ]. Proof

It is enough to show that  t n−1  (k+1)t/n  lim f (kt/n, T)dT = f (T, T)dT,

n→+∞

k=0 kt/n

t ∈ [0, T ∗ ].

(7.1.19)

0

Let us define the sequence of functions Fn (T), T ∈ [0, t] by Fn (T) = f (kt/n, T) − f (T, T), if kt/n ≤ T < (k + 1)t/n. Then n−1  

(k+1)t/n

 f (kt/n, T)dT −

k=0 kt/n

t

 f (T, T)dT =

0

t

Fn (T)dT, 0

and by i) and the Lebesgue dominated convergence theorem (7.1.19) holds. Similarly, define Gn (T) = f (kt/n, T) − f (kt/n, kt/n), if kt/n ≤ T < (k + 1)t/n. Since f (s, s), s ∈ [0, T ∗ ] is Riemann integrable, by the assumption on points of discontinuity, one has:  t n−1  f (kt/n, kt/n)t/n = f (T, T)dT. lim n→+∞

0

k=0

By the first part of ii),

 lim

n→+∞ 0

t

Gn (T)dT = 0,

so the result follows under ii) as well. The regularity required in the latter part of ii) in Proposition 7.1.1, is satisfied, for instance, if the short rate is right continuous with finite left limits. This condition will be satisfied in models considered in the sequel.

7.1 Prices and Rates

157

ˆ T) defined by the An important concept is the so-called discounted bond price P(t, formula ˆ T) = B−1 (t)P(t, T) = P(t, T)e− P(t,

t 0

R(s)ds

t, T ≥ 0.

,

(7.1.20)

If t > T then, by (7.1.20) and (7.1.11), ˆ T) = e− P(t,

t 0

R(s)ds

t

e

T

R(s)ds

= e−

T 0

R(s)ds

,

so the discounted price of the T-bond is constant after T. If, additionally, (7.1.12) holds, then ˆ T) = e− P(t,

t 0

f (u,u)du −

e

T t

f (t,u)du

= e−

T 0

f (t,u)du

,

t, T ≥ 0.

(7.1.21)

If P(t, T) is a semimartingale for each T > 0, then the Itˆo product formula applied in (7.1.20) yields,   t t ˆ T) = P(t−, T)d e− 0 f (s,s)ds + e− 0 f (s,s)ds dP(t, T) (7.1.22) dP(t, and thus we obtain a simple but important relation ˆ T) dP(t, dP(t, T) , = −f (t, t)dt + ˆ P(t−, T) P(t−, T)

t, T ≥ 0.

(7.1.23)

7.1.2 Prices and Rates in Function Spaces Bond prices and forward rates can be regarded as function-valued processes. For fixed running time t ∈ [0, T ∗ ], T ∗ < +∞, they are functions of maturities on [0, +∞), i.e. T → f (t, T),

T → P(t, T),

T ∈ [0, +∞),

and they can be elements of some Hilbert spaces. We alternatively use a modified notation to stress that forward rates, bond prices and discounted bond prices are treated as function valued processes. We denote them by ft := ft (T) = f (t, T),

Pt := Pt (T) = P(t, T),

ˆ T), Pˆ t := Pˆ t (T) = P(t,

where t ∈ [0, T ∗ ], T ∈ [0, +∞) and call them forward curves, bond curves and discounted bond curves. Under the assumption that ft (u) = fu (u) for t > u, we clearly obtain by (7.1.13) and (7.1.21) that Pt (T) = e−

T t

ft (u)du

,

Pˆ t (T) = e−

T 0

ft (u)du

,

t ∈ [0, T ∗ ], T ∈ [0, +∞). (7.1.24)

Working with function valued forward rates is suggested by financial practice. Central banks of many countries determine forward curves in the class of linear

158

Fundamentals

combinations of exponential-polynomial functions of time to maturity u := T − t, i.e. ft (u) = G(u), where: G(u) = p1 (u)e−α1 u + · · · + pn (u)e−αn u ,

u ≥ 0.

(7.1.25)

Here pi (·) are some polynomials and α = (α1 , . . . , αn ) are parameters in Rn+ . In particular, the Nelson–Siegel family of functions ! (7.1.26) β1 + β2 + β3 u e−α1 u , and the Svensson family of functions ! β1 + β2 + β3 u e−α1 u + β4 ue−α2 u ,

(7.1.27)

are widely used in Europe. Since working with forward curves taking values in Hilbert spaces is mathematically preferable, we will use the following state space    +∞ H = L2,γ := h : [0, +∞) → R : |h|2L2,γ := h2 (x)eγ x dx < +∞ , (7.1.28) 0

indexed with positive γ . This choice preserves exponential decay of forward curves at infinity suggested by (7.1.26) and (7.1.27). To obtain positive limits at infinity we can enlarge H to the space & ' ˆ = Lˆ 2,γ := g : [0, +∞) → R : g = c + h, c a constant, h ∈ L2,γ . (7.1.29) H Both spaces are separable Hilbert spaces with norms  |h|L2,γ =

+∞

1/2 2

γx

h (x)e dx 0

,

|h + c|Lˆ 2,γ = (|c|2 + |h|2L2,γ )1/2 .

Bond prices, as functions of maturities, are usually differentiable, so the following Sobolev space is often taken as a state space for them:    +∞ 1,γ 2 2  2 γx G = W := h : [0, +∞) → R : |h|H 1,γ :=| h(0) | + | h (x) | e dx < +∞ . 0

(7.1.30) The following proposition is concerned with forward rates taking values in L2,γ . Its extension to the space Lˆ 2,γ , containing constant functions, is discussed in the corollary following the proof. Proposition 7.1.2 If ft , t ∈ [0, T ∗ ] is a c`adl`ag process in H = L2,γ then the processes Pt , Pˆ t , t ∈ [0, T ∗ ], are c`adl`ag processes in G = W 1,γ .

7.1 Prices and Rates

159

The proof of Proposition 7.1.2 for Pˆ t follows from the local Lipschitz property of the transformation F(h)(T) := e−

T 0

h(s)ds

,

T ≥ 0,

h ∈ H,

(7.1.31)

which we prove in Proposition 7.1.3. Then Pt is also c`adl`ag because t

Pt (T) = Pˆ t (T)e t

and t → e

0 ft (u)du

0

R(u)du

t

= Pˆ t (T)e

0 ft (u)du

,

is c`adl`ag (see Proposition 7.1.5 and Lemma 7.1.6).

Proposition 7.1.3 For arbitrary M > 0 there exists C > 0 such that if | h |H ≤ M,

| g |H ≤ M,

then F given (7.1.31) satisfies | F(h) − F(g) |G ≤ C | h − g |H . Since F(h)(0) = F(g)(0) = 1, so

Proof

I := | F(h) − F(g) |2G  +∞ !2 T T = eγ T e− 0 h(s)ds h(T) − e− 0 g(s)ds g(T) dT 

0 +∞

=

eγ T e−

T 0

h(s)ds

(h(T) − g(T)) + g(T)(e−

T 0

h(s)ds

− e−

T 0

g(s)ds

!2 ) dT.

0

Consequently, I ≤ 2(I1 + I2 ), with



+∞

I1 :=

eγ T (h(T) − g(T))2 e−2

T 0

h(s)ds

dT,

0



+∞

I2 :=

 T 2 T eγ T g2 (T) e− 0 h(s)ds − e− 0 g(s)ds dT.

0

We estimate I2 . For some constants c1 , c2  T T T | e− 0 h(s)ds − e− 0 g(s)ds |2 ≤ c1 | (h(s) − g(s))ds |2 ≤ c2 | h − g |2H , 0

and thus

 I2 ≤ 0

+∞

eγ T g2 (T)dT · c2 | h − g |2H

≤ c2 | g |2H · | h − g |2H ≤ c2 M | h − g |2H .

160

Fundamentals

In the estimate we used the fact that if | g |H ≤ M then 

T

|

g(s)ds |≤



T

−γ s

e

0

1   ds

T

2

0

eγ s | g(s) |2 ds

1 2

0

M ≤ √ | g |H . γ

To estimate I1 notice that | e−2

T 0

h(s)ds

|≤ e2

 +∞ 0

2 √Mγ

|h(s)|ds

≤e

.

Therefore  I1 ≤

+∞

eγ T (h(T) − g(T))2 e

0

2M √ γ

dT ≤| h − g |2H e

2M √ γ

.

ˆ then ft = ct + ft 0 , where ft 0 is a Corollary 7.1.4 If ft is a c`adl`ag process in H c`adl`ag process in H and ct is a c`adl`ag function of time. Thus if ct is a nonnegative process then the processes Pt , Pˆ t , t ∈ [0, T ∗ ], are c`adl`ag processes in G as well. Forward rates taking values in H also generate a regular bank account process. Proposition 7.1.5 If ft is a c`adl`ag process in H then e

t

0 ft (u)du

t

=e

0

R(u)du

,

is c`adl`ag as well. In particular, the bank account process is c`adl`ag. The proof is a consequence of the following lemma. If, for 0 ≤ s ≤ t ≤ T ∗ , | fs |H , | ft |H ≤ M, then for a constant c   t s √ I :=| e 0 ft (T)dT − e 0 fs (T)dT |≤ c | ft − fs |H + t − s . (7.1.32)

Lemma 7.1.6

Proof

Let k be such that | ex − ey |≤ k | x − y |,

for | x |, | y |≤ M.

The assertion follows from the following estimation  t    s (ft (T) − fs (T))dT | + | ft (T)dT | I≤k | 0

M ≤ k √ | ft − fs |H + γ

s



 M ≤ k √ | ft − fs |H + γ

t

γ

γ

| e− 2 T e 2 T ft (T) | dT

0

 s

t

e−γ T dT

1 2

| ft |H .

7.2 Portfolios and Strategies

161

7.2 Portfolios and Strategies 7.2.1 Portfolios Investor trades on the bond market by changing her/his portfolio in time. By a simple portfolio one usually means a sequence: p = (b, (T1 , a1 ), . . . , (Tn , an )), where b is a real number, interpreted as an amount of money kept on the bank account, positive numbers T1 < T2 < · · · < Tn are maturities of bonds possessed by the investor in quantities a1 , a2 , . . . , an . The value X(t) of the portfolio at time t, when the bond prices are P(t, T), T ≥ 0, is then equal to: X(t) = b +

n 

ak P(t, Tk ).

k=1

Note that if Tk ≤ t then ak is the number of Tk -bonds that have already matured and its value P(t, Tk ), according to the convention introduced earlier, is t

P(t, Tk ) = e

Tk

R(s)ds

t

=e

Tk

f (s,s)ds

.

In fact, the investment in bank account can be interpreted in terms of bonds that matured at time 0 and have value B(t) = P(t, 0). Their number, at time t, is equal to a0 , where t

b = a0 e

0

R(s)ds

= a0 P(t, 0).

Thus the simple portfolio can be redefined as p = ((0, a0 ), (T1 , a1 ), . . . , (Tn , an )). From now on we use this interpretation of the investment in the bank account. Let us notice that if, at time t > 0, there are in the portfolio Ti -bonds with Ti ≤ t then the amount of money held on the bank account equals   t R(s)ds e Ti ai , Ti ≤t

so is described not only by a0 but also in terms of bonds that matured before t. Of course, we could modify a0 such that it would contain also money invested in these bonds and simply neglect them. This idea, however, would make the concept of portfolio dependent on the running time, which we would like to avoid. If we identify the simple portfolio p with the discrete measure ϕ ϕ = a0 δ0 +

n  k=1

ak δTk ,

162

Fundamentals

then its value can be written in a compact way as the integral  +∞ X(t) := (ϕ, P(t, ·)) = P(t, T)ϕ(dT).

(7.2.1)

0

The concept of simple portfolios can be extended to portfolios being general finite signed measures on [0, +∞). The space of such measures equipped with weak topology will be denoted by M or M([0, +∞)). For ϕ ∈ M, ϕ([a, b]),

0 ≤ a ≤ b,

can be interpreted as the number of bonds in the portfolio with maturities in the interval [a, b]. The value of the portfolio, at time t, is then given by (7.2.1). Writing (7.2.1) in the form X(t) = (ϕ, P(t, ·)1[0,t] (·)) + (ϕ, P(t, ·)1(t,+∞] (·))  +∞  t  t e T R(s)ds ϕ(dT) + P(t, T)ϕ(dT), = 0

t

allows us to separate money really saved on the bank account from money invested in bonds that mature in the future. Let us stress that the expression (μ, h), where h is a function and μ a measure, will always denote the integral of the function h with respect to the measure μ.

7.2.2 Strategies and the Wealth Process To some extent, mathematical finance is concerned with the problem of how to choose trading strategies, i.e. dynamically select portfolios, to achieve financial aims. Let us start with the concept of simple self-financing strategies. Suppose that an investor, at time t0 = 0, starts with an initial capital X(0) = x and forms a portfolio   ϕs = (0, a00 ), (T10 , a01 ), . . . , (TN0 0 , a0N0 ) , which is constant for s ∈ [0, t1 ) with t1 > 0 and positive natural number N0 . At the starting point the portfolio necessarily should satisfy the budget condition: x = (ϕ0 , P(0, ·)) =

N0 

a0k P(0, Tk0 ),

k=0

where T00 := 0. More generally, the investor is selecting, at times t1 , . . . , tM new portfolios m m m m ϕtm = ((0, am 0 ), (T1 , a1 ), . . . , (TNm , aNm )),

m = 1, . . . , M,

7.2 Portfolios and Strategies

163

which are constant on the intervals [tm , tm+1 ), and their values are equal to X(tm ) = (ϕtm , P(tm , ·)) =

Nm 

m am k P(tm , Tk ),

m = 1, 2, . . . , M,

(7.2.2)

k=0

with T0m := 0. The new portfolios are required to be completely financed from the capitals arrived at tm . In particular, at t1 we have X(t1 ) = (ϕ0 , P(t1 , ·)), which, in view of (7.2.2), yields the budget constraint (ϕ0 , P(t1 , ·)) = (ϕt1 , P(t1 , ·)), at time t1 . In general, for k = 1, 2, . . . , M, we obtain that (ϕtk−1 , P(tk , ·)) = (ϕtk , P(tk , ·)).

(7.2.3)

Consequently, by (7.2.2) and (7.2.3) in each point of the trading grid the capital can be written in the form X(tm ) = X(0) +

m−1 

(X(tk ) − X(tk−1 ))

k=1

= X(0) +

m−1 

((ϕtk−1 , P(tk , ·)) − ((ϕtk−1 , P(tk−1 , ·)))

k=1

= X(0) +

m−1 

(ϕtk−1 , P(tk , ·)),

(7.2.4)

k=1

where P(tk , T) := P(tk , T) − P(tk−1 , T). Notice that the sum in (7.2.4) can be interpreted as an integral over the interval [0, tm ):  tm (ϕs , dP(s, ·)). X(tm ) = X(0) + 0

Since the strategy is constant on the interval [tm , tm+1 ), the wealth process X(t), t ∈ [tm , tm+1 ), can be written in two ways X(t) = (ϕt , P(t, ·))  t = X(0) + (ϕs , dP(s, ·)).

(7.2.5)

0

In the second identity, the integrator is the bond price process and the integrand, the trading strategy. It can be also written in the differential form dX(t) = (ϕt , dP(t, ·)), t ≥ 0,

(7.2.6)

164

Fundamentals

which means financially that the change of the investor’s capital is a result of the movements of the bond prices only. By a general trading strategy, called simply a strategy in the sequel, one understands a predictable process ϕt , t ≥ 0 with values in the space M([0, +∞)). A strategy is self-financing if the wealth process satisfies  t (7.2.7) X(t) = (ϕt , P(t, ·)) = X(0) + (ϕs , dP(s, ·)), t ≥ 0, 0

where in the stochastic integral in (7.2.7) strategy ϕ and bond prices P are integrand and integrator, respectively. General stochastic integration will be discussed in Section 7.2.3. For a given strategy ϕ with initial capital x = (ϕ0 , P(0, ·)) define by  the socalled discrepancy function of ϕ by the formula  t (t) = x + (ϕs , dP(s, ·)) − (ϕt , P(t, ·)), t ≥ 0, 0

which measures how much the self-financing condition is violated. It is obvious that (0) = 0 and that ϕ is self-financing if and only if (t) = 0 for all t ≥ 0. For theoretical considerations it is important that, by modifying investment on the bank account, one can arrive at a self-financing strategy with the same initial capital. Proposition 7.2.1 If, for a given strategy (ϕt ), taking values in M, with initial capital x = (ϕ0 , P(0, ·)), the discrepancy is locally integrable, then the strategy   t (s) (t) (7.2.8) ϕ˜ t := + R(s)ds δ0 + ϕt , t ≥ 0 B(t) 0 B(s) is self-financing and (ϕ˜0 , P(0, ·)) = x. Proof

Define (t) b(t) := + B(t)

One has to check that



t 0



t

b(t)B(t) + (ϕt , P(t, ·)) = x + 0

(s) R(s)ds, B(s)

t ≥ 0.



t

b(s)dB(s) +

(ϕs , dP(s, ·))

(7.2.10)

0

for t ≥ 0. Set y(t) := B(t)b(t). Then (7.2.10), is equivalent to:  t 1 y(t) = b(s)B(s) · dB(s) + (t), B(s) 0  t y(s)R(s)ds + (t). = 0

(7.2.9)

(7.2.11) (7.2.12)

7.2 Portfolios and Strategies

165

Thus, for the function z(t) := y(t) − (t), one gets the equation z (t) = R(t)z(t) + (t)R(t), z(0) = 0. The preceding initial condition follows from (7.2.9), which yields b(0) = (0) = 0. Consequently,  t (s) R(s)ds, z(t) = B(t) 0 B(s) and the required formula easily follows. Self-financing strategies can be characterized with the use of discounted bond ˆ prices. It is easy to check that the discounted wealth process X: ˆ = X(t)/B(t) = X(t)/P(t, 0), X(t)

t ≥ 0,

for a simple self-financing strategy, can be written as the integral with respect to the discounted bond prices:  t ˆ ·)), t ≥ 0. ˆ = X(0) ˆ (7.2.13) X(t) + (ϕs , dP(s, 0

This important identity extends to general trading strategies. Proposition 7.2.2 Let (ϕt ) be an M-valued strategy such that the wealth process is a semimartingale. Then (ϕt ) is self-financing if and only if ˆ ·)), ˆ = (ϕt , dP(t, dX(t) Proof

t ≥ 0.

(7.2.14)

By the Itˆo product formula we obtain  d −1 ˆ ·)) = (ϕt , dP(t, B (t) (ϕt , P(t, ·))dt + B−1 (t)(ϕt , dP(t, ·)). dt

ˆ = B−1 (t)(ϕt , P(t, ·)), so However, since X(t)  d −1 ˆ = B (t) (ϕt , P(t, ·))dt + B−1 (t)d(ϕt , P(t, ·)). dX(t) dt Hence (7.2.14) is satisfied if and only if d(ϕt , P(t, ·)) = (ϕt , dP(t, ·)), which means that ϕ is self-financing. Corollary 7.2.3 It follows from Proposition 7.2.1 and Proposition 7.2.2 that for any strategy ϕ with locally integrable discrepancy there exists a self-financing strategy ϕ˜ such that  t  t ˆ ·)) = ˆ ·)), t ≥ 0. (ϕs , dP(s, (ϕ˜s , dP(s, 0

0

166

Fundamentals

This follows from the fact that, for any t ≥ 0, the measure ϕ˜t given by Proposiˆ 0) ≡ 1. tion 7.2.1 may differ from ϕt only at zero and that P(t,

7.2.3 Wealth Process as Stochastic Integral The wealth process corresponding to an M-valued strategy ϕ starting from an initial capital x was defined by  t (7.2.15) X(t) = x + (ϕs , dP(s, ·)), t ≥ 0. 0

Here we discuss this definition in more detail. If a strategy ϕt =

M 

ak (t)δTk ,

k=0

is based on a finite number of bonds with maturities 0 = T0 < T1 < · · · < TM , then (7.2.15) simplifies to  t M  t  (ϕs , dP(s, ·)) = ak (s)dP(s, Tk ), t ≥ 0. 0

k=0 0

The preceding integral has a well-defined meaning provided that for each k = 0, 1, . . . , M the processes P(s, Tk ), s ∈ [0, T ∗ ] are semimartingales, the processes ak , k = 1, 2, . . . , M are locally bounded and predictable and a0 is adapted and locally integrable (see Protter [102]). If now ϕ is a general strategy that involves an infinite number of bonds it is convenient to assume that the bond price process takes values in the Hilbert space G = W 1,γ introduced in Section 7.1.2, and ϕ takes values in its dual G∗ . We refer to Proposition 7.2.6 for important properties of G∗ . If (ϕt ) is a piecewise constant strategy, say ϕs = ϕsk , s ∈ (sk , sk+1 ], where s0 = 0 < s1 < · · · < sm , and ϕ(sk ) are Fsk -measurable random variables, then one has  t  (ϕs , dP(s, ·)) = (ϕ(sk ), P(t ∧ sk+1 ) − P(t ∧ sk )), t ≥ 0. 0

k

The concept of stochastic integral for more general integrands one extends by following classical procedure (see e.g. Protter [102, pp. 134–135] and Peszat and Zabczyk [100], chapter 8). One first defines the integral under additional conditions on the integrator and integrands. They allow the extension due to convenient isometric-type formulas. The final step is based on localization. The stochastic integration in infinite dimension is well developed especially when the integrator is a square integrable martingale (see M´etivier [92], or also Peszat and Zabczyk [100] where the largest class of admissible integrands is described). However, for models studied in the book the general theory is not very convenient,

7.2 Portfolios and Strategies

167

as for instance, the description of the operator Qt (see (4.2.5)) is rather cumbersome. Therefore we use a direct approach taking into account that the bond prices, studied in the book, can be represented in the following way  t  t 0 (s, T)ds + 1 (s, T), dW(s) P(t, T) = P(0, T) + 0 0  t  t + 2 (s, T, y)π˜ (ds, dy) + 3 (s, T, y)π(ds, dy), (7.2.16) 0 U ∗ [0, T ] with

U

0

T∗

fixed > 0 and T > 0. In the decomposition W is a Qwhere t ∈ Wiener process, π and π˜ are the jump measure and the compensated jump measure with intensity ν corresponding to the process Z. The coefficients in (7.2.16) viewed as functions of T, i.e. 0 (s) = 0 (s, T), 1 (s) = 1 (s, T), 2 (s, y) = 2 (s, T, y), 3 (s, y) = 3 (s, T, y), with s ∈ [0, T ∗ ] are predictable G-valued processes such that  T∗   |0 (s)|G + |1 (s)|2G ds < +∞, P − a.s.,

(7.2.17)

0



T∗

0

  U

 |2 (s, y)|2G + |3 (s, y)|G ds ν(dy) < +∞,

P − a.s.

(7.2.18)

Under those conditions the stochastic integrals in (7.2.16) are well defined and the bond price process has c`adl`ag trajectories in G. Now we can define in a rigorous way the stochastic integral (7.2.15) for M-valued integrands by formulating conditions for their G∗ -norm. Theorem 7.2.4 Assume that conditions (7.2.17), (7.2.18) hold and ϕ is an M-valued predictable process such that   T∗  |ϕ(s)|G∗ |0 (s)|G + |3 (s, y)|G ν(dy) ds < +∞ P − a.s., (7.2.19) 

0

0

T∗

 |ϕ(s)|2G∗

 |1 (s)|2G

U

+ U

|2 (s, y)|2G ν(dy)

ds < +∞

P − a.s.

(7.2.20)

Then the following statements are true. (i) The process (ϕt ) is stochastically integrable over (Pt ) and, for t ∈ [0, T ∗ ],  t  t  t (ϕs , dP(s, ·)) = (ϕs , 0 (s, ·))ds + (ϕs , 1 (s, ·)), dW(s) (7.2.21) 0 0 0  t  t (ϕs , 2 (s, ·, y))π˜ (ds, dy)+ (ϕs , 3 (s, ·, y))π(ds, dy). + 0 U

0 U

(ii) All predictable processes ϕ whose trajectories are bounded, almost surely in total variation, are integrable.

168

Fundamentals

We establish first the result under more restrictive conditions. Proposition 7.2.5 Assume that  ∗   T  2 E |0 (s)|G + |1 (s)|G ds < +∞,

(7.2.22)

0



T∗

E 0

  U

|2 (s, y)|2G





+ |3 (s, y)|G ds ν(dy) < +∞.

If (ϕt ) is an M-valued, predictable process for which the conditions  ∗    T E |ϕ(s)|G∗ |0 (s)|G + |3 (s, y)|G ν(dy) ds < +∞, 0

 0

(7.2.24)

U T∗

E

(7.2.23)

 |ϕ(s)|2G∗



 |1 (s)|2G

+ U

|2 (s, y)|2G ν(dy)

 ds < +∞,

(7.2.25)

t are satisfied, then the stochastic integral 0 (ϕs , dP(s, ·)), t ∈ [0, T ∗ ] is well defined and the identity (7.2.21) holds. Moreover, if 0 = 0 and 3 = 0 then the isometric formula holds  t   t 1 (ϕs , dP(s, ·)) |2 = E | (ϕs , 1 (s, ·)Q 2 |22 ds E | 0

0

+E

 t 

| (ϕs , 2 (s, ·, y)) |2 ds ν(dy) ,

t ∈ [0, T ∗ ]

U

0

(compare (5.4.4) and (5.4.6)). Proof

We consider only the case when  t 2 (s, T, y)π˜ (ds, dy), t ∈ [0, T ∗ ], P(t, T) = P(0, T) + 0

(7.2.26)

U

as the other cases can be checked similarly. If (7.2.23) is satisfied then an arbitrary simple integrand with deterministic ϕ k satisfies condition (7.2.25). For (ϕt ) satisfying  ∗    T 2 2 E |ϕ(s)|G∗ |2 (s, y)|G ν(dy) ds < +∞, 0

U

ϕn

one can construct a sequence of simple integrands such that  ∗    T n 2 2 E |ϕ(s) − ϕ (s)|G∗ |2 (s, y)|G ν(dy) ds → 0, 0

(7.2.27)

U

as n → +∞. To do it, let us choose a countable set M0 = {μ1 , μ2 , . . .} of measures from M dense in G∗ and define ϕn (s) for every natural n and s ∈ [0, T ∗ ] as the closest

7.2 Portfolios and Strategies

169

to ϕ(s) element in {μ1 , μ2 , . . . , μn }. If there are several such elements one chooses the one with the smallest index. The process ϕn (s), s ∈ [0, T ∗ ] is predictable, takes a finite number of values and for every s ∈ [0, T ∗ ] the sequence |ϕ(s) − ϕn (s)|G∗ tends monotonically to 0. Therefore (7.2.27) holds. Although the constructed processes take a finite number of values and are predictable they are not, in general, simple ones. It is, however, well known (see e.g. Da Prato and Zabczyk [33, p. 98]) that for an arbitrary predictable set A ∈  × [0, T ∗ ] and arbitrary  > 0 there exists a finite number of disjoint predictable rectangles S1 , . . . , Sk of the form B × (s, t], B ∈ Fs such that their sum S approximates A up to  > 0, i.e. PT ∗ ((A \ S) ∪ (S \ A)) < . Here PT ∗ denotes the product of the measure P and the Lebesgue measure on [0, T ∗ ]. This easily allows approximating the sequence ϕ n by simple integrands and complete the required construction. By Doob’s inequality, as m, n → +∞:   t  t m m 2 E supt≤T ∗ | (ϕs , dP(s, ·)) − (ϕs , dP(s, ·))| (7.2.28) 0  ∗0    T m n 2 ≤ 4E |ϕs − ϕs |G∗ |2 (s, y)|2G ν(dy) ds → 0. (7.2.29) 0

U

t n Therefore for a subsequence nk the integrals 0 (ϕs k , dP(s, ·)) converge, almost surely, uniformly on [0, T ∗ ] to a stochastic process, which is the required integral  t  t (ϕs , dP(s, ·)) = lim (ϕsnk , dP(s, ·)). k→+∞ 0

0

It is a c`adl`ag process. It is also clear that the identity (7.2.21) is true. Proof of the Theorem 7.2.4 (i) By Proposition 7.2.5 it is enough to define the integral by localization. To simplify notation we restrict again to processes Pt of the form (7.2.26). Thus let us define, for each natural n, m, the stopping times Tn , Sm as follows   t Tn = inf{t ∈ [0, T ∗ ] : |ϕs |2G∗ |2 (s, y)|2G ds ν(dy) ≥ n }, (7.2.30) U 0  t |2 (s, y)|2G ds ν(dy) ≥ m }, (7.2.31) Sm = inf{t ∈ [0, T ∗ ] : 0

U

where the infimum of the empty set is, by definition, equal to T ∗ . Then both sequences converge to T ∗ . Moreover, the processes ϕ n (s) := ϕ(t ∧ Tn ) and 2m (t, T, y) := 2 (t ∧ Sm , T, y) satisfy the conditions of Proposition 7.2.5 and therefore the stochastic integrals  t (ϕsn , 2m (s, ·, y))π˜ (ds, dy), t ∈ [0, T ∗ ], 0

U

170

Fundamentals

are well defined. They can be identified with the integrals  t∧Tn ∧Sm (ϕs , dP(s, ·)), 0

for which the limit, as n, m → +∞, is the required integral. (ii) It is enough to show that the process |ϕ(t)|G∗ has bounded trajectories. This follows, however, from the following proposition of independent interest. Proposition 7.2.6 The G∗ -norm of an arbitrary finite measure μ on [0, +∞) viewed as a functional on G = W 1,γ is given by the formula,   |μ|G∗ = (μ(R+ ))2 +

+∞

1/2 e−γ u (μ([u, +∞))2 du ,

0

so |μ|G∗ ≤ |μ|(1 + 1/γ )1/2 , where |μ| denotes the total variation of μ, i.e., |μ| := (μ+ + μ− )(R+ ). Proof

Let h be an arbitrary element of G. Then  +∞  +∞  s (μ, h) = h(s)μ(ds) = (h(0) + h (u)du)μ(ds) 0 0 0  +∞  +∞ = h(0)μ(R+ ) + 1[0,s] (u)h (u)du μ(ds) 0 0  +∞ = h(0)μ(R+ ) + h (u)μ([u, +∞))du 0  +∞ = h(0)μ(R+ ) + eγ u (h (u))(e−γ u μ([u, +∞))du 0

= < h, g >G , where

 g(u) = μ(R+ ) +

u

e−γ s μ([s, +∞))ds, u ≥ 0.

0

Thus |μ|G∗ = |g|G and the result follows. Remark 7.2.7 The following formula can be useful if one wants to approximate strategies by simpler ones, for instance based on a finite number of bonds, without losing much from the gains. Assume that μ1 − μ2 are two finite measures on R+ and let Fμ1 , Fμ2 , be their cumulative distribution functions: Fμi (u) = μi ([0, u]), u ≥ 0, i = 1, 2.

7.3 Non-arbitrage, Claims and Their Prices It follows from the proposition that   2 ∗ |μ1 − μ2 |G = (μ1 (R+ ) − μ2 (R+ )) +

+∞

−γ u

e 0

171

1/2 (Fμ1 (u)) − Fμ2 (u))) du . 2

Thus, in particular, an arbitrary measure can be approximated, as close as one wishes, by atomic measures with a finite number of atoms.

7.3 Non-arbitrage, Claims and Their Prices A self-financing strategy ϕ is called an arbitrage opportunity if for the corresponding wealth process starting from zero:  t (ϕs , dP(s, ·)), t ∈ [0, T ∗ ], X(t) = 0

one has P(X(s) ≥ 0, s ∈ [0, T ∗ ]) = 1,

and, for some t ∈ [0, T ∗ ],

P(X(t) > 0) > 0.

We say that the bond market is arbitrage free, if there is no arbitrage opportunity in the class of self-financing strategies based on a finite number of bonds. This means that for self-financing strategies of the form ϕt =

M 

ak (t)δTk ,

t ∈ [0, T ∗ ],

(7.3.1)

k=0

the condition



t

X(t) =

(ϕs , dP(s, ·)) ≥ 0,

t ∈ [0, T ∗ ]

(7.3.2)

(ϕs , dP(s, ·)) = 0,

t ∈ [0, T ∗ ].

(7.3.3)

0

implies that

 X(t) =

t

0

In view of Proposition 7.2.2 and Corollary 7.2.3 one can reformulate conditions (7.3.2) and (7.3.3) with the use of discounted quantities and forget the self-financing requirement. So, the market is arbitrage free if, for arbitrary strategy (7.3.1) such that 

t

ˆ ·)) ≥ 0, (ϕs , dP(s,

t ∈ [0, T ∗ ],

(7.3.4)

ˆ ·)) = 0, (ϕs , dP(s,

t ∈ [0, T ∗ ].

(7.3.5)

0

one has

 0

t

172

Fundamentals

Important criteria for non-arbitrage uses the idea of local martingale. It is based on the fact that the stochastic integral, with respect to a local martingale, is a local martingale, under very general conditions on the integrand. In particular, this is the case when the integrand is locally bounded. ˆ T), Proposition 7.3.1 Let, for each T > 0, the discounted bond price process P(t, ∗ t ∈ [0, T ] be a local martingale on a filtered probability space (, F, (Ft ), P). Then there are no arbitrage opportunities in the class of locally bounded strategies of the form (7.3.1). Proof

For ϕ given by (7.3.1), the corresponding discounted wealth process ˆ = X(t)



t 0

ˆ ·)) = (ϕs , dP(s,

M  

t

ˆ Tk ), ak (s)dP(s,

t ∈ [0, T ∗ ]

k=1 0

is a local martingale. For the localizing sequence τn we have thus ˆ ˆ ∧ τn )), 0 = X(0) = E(X(t

t ∈ [0, T ∗ ],

n = 1, 2, . . . .

ˆ ∧ τn ) = 0 for arbitrary n, t ∈ [0, T ∗ ] and thus Xˆ If (7.3.4) holds then we obtain X(t disappears. So, (7.3.5) is satisfied and the market is arbitrage free. An important sufficient condition for non-arbitrage is based on the concept of martingale measure. A probability measure Q is a martingale measure for the bond market defined on (, F, (Ft ), P) if it is equivalent to P and for arbitrary T > 0 the process ˆ T), P(t,

t ∈ [0, T ∗ ]

is a local martingale on (, F, (Ft ), Q). It is clear that each step in the proof of Proposition 7.3.1 remains true if P is replaced by a martingale measure Q. Thus if there exists a martingale measure for a bond model then the model is arbitrage free. For markets with a finite number of trading assets this sufficient condition is, under rather weak assumptions, also a necessary condition and constitutes the First Fundamental Theorem of Asset Pricing. An extensive study of this issue can be found in Delbaen and Schachermayer [36] and Delbaen and Schachermayer [37]. A contingent claim executed at time T ∗ > 0 is an arbitrary FT ∗ -measurable random variable X. It is interpreted as an amount of money that the writer of the contract, or agent, agrees to pay to the buyer of the contract at time T ∗ . A selffinancing strategy ϕ, taking values in M, with the initial capital x is a replicating strategy, or hedging strategy for X if  T∗ X =x+ (ϕs , dP(s, ·)). (7.3.6) 0

7.4 HJM Modelling

173

Then X is called attainable. Equivalently, using the discounted bond prices we see that the discounted attainable claim, Xˆ = e−

 T∗ 0

R(s)ds

X,

admits the representation Xˆ = x +



T∗

ˆ ·)). (ϕs , dP(s,

(7.3.7)

0

Representation (7.3.6) means that the writer of the contract, can recover the claim starting with the initial capital x and cleverly investing in the bond market. The capital x is then, rightly called, the price of the claim X. If Q is a martingale measure then, under proper integrability requirement, we have by (7.3.7) the following formula for the price p(X) of X ˆ p(X) = EQ [X],

(7.3.8)

where EQ [·] stands for the expectation under Q.

7.4 HJM Modelling One of the most important ways of describing bond markets starts from models where the dynamics of forward rates consists of two parts, the drift term, reflecting basic market trends and the noise term, corresponding to chaotic fluctuations. In particular, in the Heath–Jarrow–Morton (HJM) model the field f (t, T) is described as a family of stochastic processes f (·, T) parametrized by T > 0 and assumed to be of the following form. For fixed T, one requires that  t  t f (t, T) = f (0, T) + α(s, T)ds + σ (s, T), dZ(s) , t ≤ T, (7.4.1) 0

0

where Z is a L´evy process taking values in U = Rd , α(·, T) a real-valued process and σ (·, T) a U-valued predictable process. In the preceding ·, · stands for the scalar product in U. This modelling was introduced by Heath, Jarrow and Morton in [67] with Z being a one-dimensional Wiener process and afterwards developed by many authors by considering more general processes including discontinuous ones. The HJM model with forward rates (7.4.1) will be denoted by (α, σ , Z). We introduce conditions on the model (α, σ , Z) that guarantee that the bond prices P(t, T) = e−

T t

f (t,s)ds

,

0 ≤ t ≤ T,

(7.4.2)

as well as the bank account process t

B(t) = e

0

R(s)ds

t

=e

0

f (s,s)ds

,

t ≥ 0,

(7.4.3)

174

Fundamentals

are well defined on some finite time interval [0, T ∗ ]. They should be regarded as instructive examples rather than a universal set of sufficient conditions. Recall that the definition of P(t, T) can be extended also for t > T by cashing the bond at time T and putting 1 on the savings account. This can be achieved in the model (7.4.1) by assuming that α(t, T) = 0, Then



T

f (t, T) = f (0, T) +

σ (t, T) = 0 

α(s, T)ds +

0

T

for t > T.

σ (s, T), dZ(s) = f (T, T) = R(T),

(7.4.4)

t ≥ T,

0

(7.4.5) and consequently (see (7.1.12) and (7.1.13)) one obtains P(t, T) = e

t T

R(s)ds

= e−

T t

f (t,s)ds

,

t > T.

Proposition 7.4.1 Assume that in the HJM model (α, σ , Z), for arbitrary S > 0,  S | f (0, T) | dT < +∞, (7.4.6) 

T∗

0

0



S



T∗

| α(t, T) | dt dT < +∞, 0

0



S

| σ (t, T) |2 dt dT < +∞,

(7.4.7)

0

and (7.4.4) holds. Then the family of bond prices P(t, T) = e−

T t

f (t,u)du

t ∈ [0, T ∗ ], T ∈ (0, +∞),

,

(7.4.8)

and the bank account process t

B(t) = e

0

R(s)ds

,

t ∈ [0, T ∗ ],

are well defined. Moreover, the discounted bond prices are given by ˆ T) := e− P(t, Proof

T 0

f (t,s)ds

,

t ∈ [0, T ∗ ], T ∈ (0, +∞).

We have to show that the integrals  T f (t, u)du,

(7.4.9)

(7.4.10)

t

which define the bond prices by (7.4.8), are well defined for any t ∈ [0, T ∗ ], T ∈ (0, +∞). It follows from (7.4.1) that  T  T  T t  T t f (t, u)du = f (0, u)du + α(s, u)ds du + σ (s, u), dZ(s) du. t

t

t

0

t

0

7.4 HJM Modelling

175

Taking S so large that T < S we see that (7.4.6) and (7.4.7) are sufficient conditions for applicability of the Fubini and stochastic Fubini theorem (see Theorem 5.4.5). In particular, the preceding double integrals are then well defined. In view of (7.4.1), for almost all t ∈ [0, T ∗ ], the short-rate process is given by  t  t α(s, t)ds + σ (s, t), dZ(s) . R(t) = f (t, t) = f (0, t) + 0

The integrals  t  t  t  R(u)du = f (0, u)du + 0

0

0

u

0

α(s, u)ds du +

 t 

0

0

u

σ (s, u), dZ(s) du,

0

(7.4.11) [0, T ∗ ].

are, however, well defined for each t ∈ To see this we write the double integrals in the form  t  u  t  t α(s, u)ds du = α(s, u)1[0,u] (s)ds du,  t  0

0

0

0

0

0

0

 t  t u σ (s, u), dZ(s) du = σ (s, u)1[0,u] (s), dZ(s) du,

0

and use again (7.4.6)–(7.4.7). Consequently, the bank account process (7.4.3) is well defined. In view of (7.4.11) one can show that the discounted bond prices ˆ T) := e− P(t,

t 0

R(s)ds

P(t, T),

admit the representation (7.4.9). Under additional conditions forward curves can be regular functions of maturities. Proposition 7.4.2 Let us assume that in the HJM model (α, σ , Z) the conditions (7.4.6), (7.4.7) and (7.4.4) are satisfied. If f (0, ·) is differentiable on [0, +∞), α(t, ·), σ (t, ·) are differentiable on [0, +∞) for any t ∈ [0, T ∗ ], and for arbitrary S > 0,   ∗  T∗ T ∂α ∂σ 2 sup | sup | (s, T) | ds < +∞, E (s, T) | ds < +∞, 0 T∈(0,S) ∂T 0 T∈(0,S) ∂T (7.4.12) then f (t, T) is differentiable over T and the following formulas hold 5  t  t4 ∂f ∂α ∂f ∂σ (t, T) = (0, T) + (s, T)ds + (s, T), dZ(s) , t ∈ [0, T ∗ ], T > 0, ∂T ∂T 0 ∂T 0 ∂T (7.4.13)  R(t) = R(0) + 0

t

∂f (s, s)ds, ∂T

t ∈ [0, T ∗ ].

In particular, the short-rate process has differentiable paths.

(7.4.14)

176

Fundamentals

Proof We choose S greater than T. By (7.4.12), it follows from the dominated convergence theorem that  t  t ∂ ∂α α(s, T)ds = (s, T)ds, t ∈ [0, T ∗ ]. ∂T 0 0 ∂T Similarly, since for any s ∈ [0, T ∗ ] and small ε > 0 we have 2    σ (s, T + ε) − σ (s, T) ∂σ ∂σ  − (s, T) ≤ 2 sup | (s, T) |2 ,  ε ∂T ∂T T∈(0,S) it follows that (7.4.12) implies that  ∗  2  T  σ (s, T + ε) − σ (s, T)  ∂σ  − (s, T) ds −→ 0. E  ε→0 ε ∂T 0 The significance of the preceding convergence is that in the L´evy–Itˆo decomposition of Z we can apply the square integrability criteria and show that 5  t4  t ∂σ ∂ σ (s, T), dZ(s) = (s, T), dZ(s) , t ∈ [0, T ∗ ]. ∂T 0 0 ∂T Hence, if f (0, ·) is differentiable and (7.4.12) holds then f (t, T) is also differentiable over T and (7.4.13) holds. Since R(t) = f (t, t), the short-rate process admits the following representation  t  t α(s, t)ds + σ (s, t), dZ(s) R(t) = f (0, t) + 0

0

 t  t ∂f ∂α (0, u)du + (s, u)du ds = f (0, 0) + α(s, s) + 0 ∂T 0 s ∂T 5  t  t4 ∂σ (s, u)du, dZ(s) , t ∈ [0, T ∗ ]. σ (s, s) + + ∂T 0 s 

t

By (7.4.12) we can apply the Fubini and stochastic Fubini theorem and change the order of integration. This yields 5  t  u  u4 ∂α ∂f ∂ R(t) = f (0, 0) + (0, u)du + (s, u)ds + σ (s, u), dZ(s) du ∂T ∂T 0 0 ∂T 0  t  t α(s, s)ds + σ (s, s), dZ(s) , t ∈ [0, T ∗ ], + 0

0

which, in view of (7.4.13), (7.4.4), and the continuity of α(t, T) and σ (t, T) over T, yields (7.4.14).

7.4 HJM Modelling

177

7.4.1 Bond Prices Formula In this section we establish the formula for bond prices in terms of the representation of forward rates (7.4.1) in the HJM model. In fact, we consider a generalized form of (7.4.1) where the volatility is splitted into three parts and integrated separately over components of the L´evy process Z. So, (7.4.1) is replaced by  0 σ 1 (t, T, y)π˜ (dt, dy) df (t, T) = α(t, T)dt + σ (t, T), dW(t) + |y|≤1

 +

|y|>1

σ 2 (t, T, y)π(dt, dy),

t ∈ [0, T ∗ ], T > 0,

(7.4.15)

where W, π, π˜ come from the L´evy–Itˆo decomposition of the L´evy process Z and f (0, T) = f0 (T) is a given function. The HJM model with forward rates (7.4.15) will be denoted (α, σ 0 , σ 1 , σ 2 , Z). Let us notice that for a given process σ (t, T) and σ 0 (t, T) := σ (t, T),

σ 1 (t, T, y) := σ (t, T)y,

σ 2 (t, T, y) := σ (t, T)y,

the dynamics (7.4.15) reduces to (7.4.1) with slightly modified α(t, T). Similarly, as in the model (α, σ , Z), we assume that α(t, T) = 0,

σ 0 (t, T) = 0,

σ 1 (t, T, y) = 0,

σ 2 (t, T, y) = 0 for t > T, (7.4.16)

and to enable application of the Fubini type theorems we need that for arbitrary S > 0, 

S

 0

0 T ∗ S

 0

T∗

| f (0, T) | dT < +∞,

S

| α(t, T) | dt dT < +∞,

(7.4.17)

0 T∗  S 

 | σ 0 (t, T) |2 dt dT < +∞, 0

0



0

|y|≤1

| σ 1 (t, T) |2 dt dT ν(dy) < +∞, (7.4.18)



T∗ 0



S 0

|y|>1

| σ 2 (t, T) | dt dT ν(dy) < +∞.

(7.4.19)

Proposition 7.4.3 In the HJM model (α, σ 0 , σ 1 , σ 2 , Z) satisfying (7.4.16), (7.4.17)–(7.4.19) the discounted bond prices are given by  ˆ T) dP(t, 0  1 (t, T, y)π˜ (dt, dy) = D(t, T)dt −  (t, T), dW(t) + ˆ P(t−, T) |y|≤1   2 (t, T, y)π(dt, dy), t ∈ [0, T ∗ ], T > 0, (7.4.20) + |y|>1

178

Fundamentals

and the bond prices by ˆ T) dP(t, T) dP(t, = f (t, t)dt + , ˆ P(t−, T) P(t−, T) where



T

D(t, T) := −  + 

0

t ∈ [0, T ∗ ], T > 0,

 T 1 1 α(t, s)ds + | Q 2 σ 0 (t, s)ds |2 2 0    T T 1 − 0 σ (t,s,y)ds 1 −1+ σ (t, s, y)ds ν(dy), e

|y|≤1

T

0



 1 (t, T, y) := e− 0  T 2   2 (t, T, y) := e− 0 σ (t,s,y)ds − 1 .  0 (t, T) :=

σ 0 (t, s)ds,

Let us write Pˆ in the form

Proof

ˆ T) = e−X(t,T) , P(t,

(7.4.21)

where



T 0

σ 1 (t,s,y)ds

T

X(t, T) :=

 −1 ,

t ∈ [0, T ∗ ], T > 0.

f (t, s)ds, 0

Application of the stochastic Fubini theorem yields X(t, T) =

 T 0

+ =

0

 T  t  0

 T 0

+

 T  t

f (0, s)ds +

0

|y|≤1

0

|y|≤1

0



T

0



0

σ 1 (v, s, y)π(dv, ˜ dy) ds +

 t  T

f (0, s)ds +

 t

0

 T  t α(v, s)dv ds + σ 0 (v, s), dW(v) ds

0

 α(v, s)ds dv +

0

 T  t  0

 t 6 T 0

 σ 1 (v, s, y)ds π(dv, ˜ dy) +

|y|>1

0

σ 2 (v, s, y)π(dv, dy) ds 7

σ 0 (v, s)ds, dW(v)

0



 t |y|>1

0

Hence X(t, T), T > 0 satisfies

T

 σ 2 (v, s, y)ds π(dv, dy).

0



dX(t, T) = A(t, T)dt +  0 (t, T), dW(t) +   2 (t, T, y)π(dt, dy), +

|y|≤1

 1 (t, T, y)π˜ (dt, dy)

|y|>1

with





T

α(t, s)ds,

A(t, T) := 

0

0

σ 0 (t, s)ds,

0 T

 1 (t, T, y) :=

T

 0 (t, T) =

σ 1 (t, y, s)ds,



T

 2 (t, T, y) = 0

σ 2 (t, y, s)ds.

7.4 HJM Modelling

179

The Itˆo formula yields

 t ˆP(t, T) = e−X(t,T) = e−X(0,T) − e−X(s−,T) dX(s, T) 0  1 t −X(s−,T) e d[X(s, T), X(s, T)]c + 2 0    e−X(s,T) − e−X(s−,T) + eX(s−,T) X(s, T) ,

t ∈ [0, T ∗ ].

s∈[0,t]

(7.4.22) Since

  X(s, T) =  1 (s, T, Z(s)) +  2 (s, T, Z(s)) 1{Z(s)=0} ,

and 1

d[X(s, T), X(s, T)]c =| Q 2  0 (s, T) |2 ds, the jump part in (7.4.22) equals I(t, T) :=

   e−X(s,T) − e−X(s−,T) + eX(s−,T) X(s, T) s∈[0,t]

=



e−X(s−,T) (e−X(s,T) − 1 + X(s, T))

s∈[0,t]

=

  1 2 e−X(s−,T) e− (s,T,Z(s))− (s,T,Z(s)) − 1 + ( 1 (s, T, Z(s)) +  2 (s, T, Z(s))

 s∈[0,t]

=

 t 0

=

 t

U

 t 0

+



|y|≤1 U

0

+

  1 2 e−X(s−,T) e− (s,T,y)− (s,T,y) − 1 + ( 1 (s, T, y) +  2 (s, T, y) π(ds, dy)

 t 0

  1 e−X(s−,T) e− (s,T,y) − 1 +  1 (s, T, y) π˜ (ds, dy)



|y|≤1 U

|y|>1

  1 e−X(s−,T) e− (s,T,y) − 1 +  1 (s, T, y) dsν(dy

  2 e−X(s−,T) e− (s,T,y) − 1 +  2 (s, T, y)) π(ds, dy).

Thus dI(t, T) = e−X(t−,T) +e +e





|y|≤1

−X(t−,T)

e−





|y|>1

−X(t−,T)

1 (t,T,y)



|y|≤1

e− 

 − 1 +  1 (t, T, y) π˜ (dt, dy)

2 (t,T,y)

 − 1 +  2 (t, T, y) π˜ (dt, dy)

− 1 (t,T,y)

e

− 1 +  (t, T, y) ν(dy) dt. 1



180

Fundamentals

Consequently, by (7.4.22) we obtain   ˆ T) = − P(t−, T) A(t, T)dt +  0 (t, T), dW(t) + dP(t, 

|y|≤1

 1 (t, T, y)π˜ (dt, dy)

 1 1 +  2 (t, T, y)π(dt, dy) + P(t−, T) | Q 2  0 (t, T) |2 dt + dI(t, T), 2 |y|>1 which yields ˆ T) dP(t, = ˆP(t−, T)



 1 1 1 | Q 2  0 (t, T) |2 + (e− (t,T,y) − 1 +  1 (t, T, y))ν(dy) dt 2 |y|≤1   1 2 −  0 (t, T), dW(t) + (e− (t,T,y) −1)π˜ (dt, dy) + (e− (t,T,y) − 1)π(dt, dy), −A(t, T) +

|y|≤1

|y|>1

and (7.4.20) follows. Formula (7.4.21) is a consequence of (7.1.23).

7.4.2 Forward Curves in Function Spaces The HJM modelling with forward rates  t  t α(s, T)ds + σ (s, T), dZ(s) , f (t, T) = f (0, T) + 0

t ∈ [0, T ∗ ], T > 0,

0

(7.4.23) can be regarded as a model where forward curves ft := {f (t, T), T > 0} live in a Hilbert space H, for instance L2,γ defined by (7.1.28). Let us define function valued processes αt (T) := α(t, T),

σti (T) := σ j (t, T),

j = 1, 2, . . . , d,

and assume that they take values in L2,γ , are adapted and predictable and such that with probability 1  T∗ d  T∗  |αs |L2,γ ds < +∞, |σsj |L2,γ ds < +∞. (7.4.24) 0

j=1

0

Then the integrals 

t

αs ds, 0

d   j=1

t 0

σsj dZj (s),

t ∈ [0, T ∗ ],

(7.4.25)

are well-defined L2,γ - valued stochastic processes. Proposition 7.4.4 Setting f0 (T) := f (0, T), T ≥ 0, the process ft given by  t d  t  f t = f0 + αs ds + σsj dZj (s), t ∈ (0, T ∗ ] (7.4.26) 0

j=1

0

7.4 HJM Modelling

181

is such that for each t ∈ [0, T ∗ ] the function T → f (t, T) is a representation of the element ft in L2,γ . Proof Assume that P-a.s. (7.4.23) holds for arbitrary T > 0 and t ∈ [0, T ∗ ]. Let h be a continuous function on [0, +∞) with bounded support. Then for each t ∈ [0, T ∗ ], by changing the order of integration, we obtain  +∞  +∞  +∞  t  f (t, T)h(t)eγ T dT = f (0, T)h(T)eγ T dT + α(s, T)ds h(T)eγ T dT 0

0

0 d  +∞   t 

+

0

j=1



+∞

=

 σ j (s, T)dZj (s) h(T)eγ T dT

0

 t  f (0, T)h(T)eγ T dT +

0

+

0

0 d  t   j=1

0

+∞

+∞

 α(s, T)h(T)eγ T dT ds

0

 σ j (s, T)h(T)eγ T dT dZj (s), t ∈ [0, T ∗ ].

0

The preceding relation can be written in the Hilbert space setting as  t f (t, ·), h(·) L2,γ = f (0, ·), h(·) L2,γ + α(s, ·), h(·) L2,γ ds 0

+

d   j=1

t

0

σ j (s, ·), h(·) L2,γ dZj (s),

t ∈ [0, T ∗ ].

It follows from (7.4.26) that  t d  t  σsj , h L2,γ dZi (s), ft , h L2,γ = f0 , h L2,γ + αs , h L2,γ ds + 0

j=1

t ∈ [0, T ∗ ].

0

Since f0 , h L2,γ = f (0, ·), h(·) L2,γ ,

αs , h L2,γ = α(s, ·), h(·) L2,γ ,

σsj , h L2,γ = σ j (s, ·), h(·) L2,γ ,

s ∈ [0, T ∗ ],

s ∈ [0, T ∗ ],

j = 1, 2, . . . , d,

we see that P-a.s. ft , h L2,γ = f (t, ·), h(·) L2,γ ,

t ∈ [0, T ∗ ],

and therefore ft = f (t, ·), as required. Note also that, f (t, T) = f (T, T) for T ≤ t and therefore ft (T) = f (T, T) for almost all T ≤ t. Consequently, P(t, T) = e−

T t

ft (s)ds

,

ˆ T) = e− P(t,

T 0

ft (s)ds

t ∈ [0, T ∗ ], T ≥ 0.

(7.4.27)

182

Fundamentals

In general, if one starts from (7.4.26) and αt (s) = 0, σt (s) = 0,

for almost all s ≤ t,

then for arbitrary 0 ≤ t1 ≤ t2 ft2 (s) = ft1 (s),

for s ≤ t1 .

One therefore defines R(t) = fT (t), t ≤ T. Instead of the spaces L2,γ one can arrive at similar results for other Hilbert spaces such as Lˆ 2,γ (see Rusinek [113]).

7.5 Factor Models and the Musiela Parametrization In many situations it is convenient to regard forward rates and bond prices as curves from sets of functions given in advance. In particular, one can assume that the forward curves T → f (t, T),

T ∈ [t, +∞)

are of the form f (t, T) = G(T − t, X(t)),

0 ≤ t ≤ T,

(7.5.1)

where G is a real-valued function defined on [0, +∞) × E and X(t), t ≥ 0 is a stochastic process. It is called a factor process, takes values in a closed subset E of Rm and could be interpreted as a description of economical environment. So, for fixed value of the factor process X(t), the function u → G(u, X(t)),

u≥0

yields the forward rate as a function of time to maturity u = T − t. Therefore the short-rate process is given by R(t) = G(0, X(t)),

t ≥ 0.

This very convenient way of parametrizing forward rates in terms of time to maturity was introduced by Musiela and is called the Musiela parametrization. From the definition of the bond prices we have that P(t, T) = e−

T t

f (t,s)ds

= e−

T t

G(u−t,X(t))du

0 ≤ t ≤ T,

,

and therefore the bond curves have the form P(t, T) = F(T − t, X(t)), where F(T − t, x) := e−

T t

G(u−t,x)du

= e−

0 ≤ t ≤ T,

 T−t 0

G(s,x)ds

,

0 ≤ t ≤ T.

7.5 Factor Models and the Musiela Parametrization

183

For F and G we have the relations ∂F (u, x) G(u, x) = − ∂u , F(u, x)

F(u, x) = e−

u

G(v,x)dv

0

,

u ≥ 0, x ∈ E.

(7.5.2)

If the function F has the form F(u, x) = e−C(u)− D(u),x , and the factor X(t) = R(t) is the short rate then the resulting model is called affine term structure or briefly affine model. If, additionally, the short rate R(t) is given by the stochastic equation dR(t) = A(R(t))dt + B(R(t−)), dZ(t) ,

R(0) = R0 ,

(7.5.3)

with some functions A, B, then the affine model can be viewed as a particular HJM model. Proposition 7.5.1 Let C and D be twice differentiable functions on [0, +∞). Then the affine model with the bond prices P(t, T) = e−C(T−t)− D(T−t),R(t) ,

T ≥ t ≥ 0,

and the short rate (7.5.3) is the HJM model (α, σ , Z) with forward rate df (t, T) = α(t, T)dt + σ (t, T), dZ(t) , where α(t, T) := A(R(t))D (T − t) − C (T − t) − D (T − t)R(t), σ (t, T) := D (T − t)B(R(t−)), Proof yields

T ≥ t ≥ 0, (7.5.4)

T ≥ t ≥ 0.

(7.5.5)

Writing the bond prices in the affine model with the use of forward rates e−C(T−t)− D(T−t),R(t) = e−

T t

f (t,s)ds

,

T ≥ t.

Consequently, f (t, T) = C (T − t) + D (T − t)R(t). Using Itˆo’s formula and (7.5.3) we obtain that, for each T > 0, df (t, T) = −C (T − t)dt + D (T − t)dR(t) − R(t)D (T − t)dt = α(t, T)dt + σ (t, T), dZ(t) where α and σ are given by (7.5.4) and (7.5.5).

8 Arbitrage-Free HJM Markets

In this chapter we characterize arbitrage-free HJM models by describing the set of all martingale measures. In deriving the corresponding Heath– Jarrow–Morton drift conditions we use in an essential way the martingale representation theorem and Girsanov’s formula for equivalent measures.

8.1 Heath–Jarrow–Morton Conditions In this section we work with the HJM model (α, σ , Z) given, for each T > 0, by the forward rate dynamics  t  t f (t, T) = f (0, T) + α(s, T)ds + σ (s, T), dZ(s) , t ∈ [0, T ∗ ], (8.1.1) 0

0

Rd

with characteristic triplet (a, Q, ν). For any driven by a L´evy process Z in U = T > 0, α(·, T) is real-valued and σ (·, T) is a U-valued predictable process. In view of Theorem 7.4.1, the assumptions  S | f (0, T) | dT < +∞, S > 0, (8.1.2) 0

 0

T∗



S



T∗

| α(t, T) | dt dT < +∞, 0

0



S

0

| σ (t, T) |2U dt dT + ∞,

S > 0, (8.1.3)

α(t, T) = 0,

σ (t, T) = 0 for

t>T

(8.1.4)

guarantee that the bond prices P(t, T) = e−

T t

f (t,u)du

t ∈ [0, T ∗ ],

,

T>0

and the bank account process B(t) = e

t 0

R(s)ds

t

=e

0

f (s,s)ds

,

t ∈ [0, T ∗ ]

8.1 Heath–Jarrow–Morton Conditions

185

are well defined. Moreover, the discounted bond prices are given then by ˆ T) := e− P(t,

T 0

f (t,s)ds

t ∈ [0, T ∗ ],

,

T > 0.

Our aim now is to characterize models that satisfy the following (MM) condition. There exists a measure Q ∼ P such that for each T > 0 the discounted bond price process ˆ T) := e− P(t,

T 0

f (t,s)ds

t ∈ [0, T ∗ ],

,

(MM)

is a Q-local martingale. Recall that the measure Q appearing in (MM) is called a martingale measure and its existence is sufficient for the model to be arbitrage free (see Section 7.3 for details). Our analysis leads to a generalized version of the famous HJM drift condition from Heath, Jarrow and Morton [67] that yields a special type of dependence between α and σ in (8.1.1). They are formulated in terms of the processes  T  T α(t, v)dv, (t, T) := σ (t, v)dv, t ∈ [0, T ∗ ], T > 0. A(t, T) := t∧T

t∧T

(8.1.5) Let us recall that a predictable process φ(t) = (φ1 (t), . . . , φd (t)), resp. ψ(t, y) belongs to (U) resp. ψ ∈ 1,2 if and only if  0

T∗

d 

φi2 (t)dt < +∞,

P − a.s.,

i=1

resp. 

T∗ 0

 

 | ψ(s, y) |2 ∧ | ψ(s, y) | dsν(dy) < +∞,

P − a.s.

U

(see Section 5.4.1 and Section 5.4.2 for details). The Laplace exponent J of Z is given by    1 (8.1.6) e− u,y − 1 + 1{|y|≤1} u, y ν(dy) J(u) = − a, u + Qu, u + 2 U (see Section 5.2). Theorem 8.1.1 Let (α, σ , Z) be an HJM model satisfying (8.1.2)–(8.1.4), where Z is a L´evy process with characteristic triplet (a, Q, ν) and the processes A(t, T), (t, T) are given by (8.1.5). Let us assume that  S | σ (t, v) | dv < +∞, S > 0. (8.1.7) sup t∈[0,T ∗ ] 0

186

Arbitrage-Free HJM Markets

(a) If the model satisfies (MM), then there exist processes (φ, ψ) such that φ ∈ (U), eψ − 1 ∈ 1,2 , which satisfy in addition  T∗  | eψ(s,y) − 1 | ds ν(dy) < +∞, P − a.s., (8.1.8) {|y|≤1}

0

and, for each T ∈ (0, +∞),  T∗  e− (s,T),y · eψ(s,y) ds ν(dy) < +∞, {|y|>1}

0

P − a.s.

(8.1.9)

(b) If (8.1.8) and (8.1.9) are satisfied for some processes (φ, ψ) with φ ∈ (U), eψ − 1 ∈ 1,2 then the model satisfies (MM) if and only if for each T ∈ (0, +∞) almost all s ∈ [0, T ∗ ], P-a.s.: 1 A(s, T) = − (s, T), a + Q(s, T), (s, T) − Qφ(s), (s, T) 2   ψ(s,y) − (s,T),y  e (e − 1) + 1{|y|≤1} (s, T), y ν(dy). (8.1.10) + U

(c) If (8.1.8) and (8.1.9) are satisfied for some processes (φ, ψ) with φ ∈ (U), eψ − 1 ∈ 1,2 and for each T > 0,  T∗  e− (s,T),y dsν(dy) < +∞, P − a.s., (8.1.11) 0

{|y|>1}

then the model satisfies (MM) if and only if for each T > 0 almost all s ∈ [0, T ∗ ], P-a.s.:  A(s, T) = J((s, T))− Qφ(s), (s, T) + (eψ(s,y) − 1)(e− (s,T),y − 1)ν(dy). U

(8.1.12) Theorem 8.1.1 constitutes a fundamental tool for the study of arbitrage-free HJM models. Conditions (8.1.10) and (8.1.12), which are called HJM conditions, enable, at least theoretically, to check if a given HJM model (α, σ , Z) is arbitrage free and to construct models that are arbitrage free. We comment now on the direct consequences of Theorem 8.1.1. Remark 8.1.2 For a given HJM model (α, σ , Z) the processes (φ, ψ) appearing in the formulation of Theorem 8.1.1 are the generating pair of a martingale measure Q. The problem of the existence of Q is thus equivalent to the solvability of the equation (8.1.10) or (8.1.12) over (φ, ψ). The existence of such (φ, ψ) in the general HJM model is, however, an open problem. Even particular examples of models for which it is known if (φ, ψ) exist are hardly found in the literature. In Section 8.2.2,

8.1 Heath–Jarrow–Morton Conditions

187

Section 8.2.3 and Section 8.2.4 we partially fill this gap by presenting a couple of models for which the required pair (φ, ψ) exists and some for which it does not exist. Remark 8.1.3 If the pair φ ≡ 0, ψ ≡ 0 satisfies conditions of Theorem 8.1.1 then the original measure P is a martingale measure. In this case, the necessary conditions (8.1.8), (8.1.9) amount to  t e− (s,T),y dsν(dy) < +∞. (8.1.13) 0

{|y|>1}

This implies that, for each T > 0, (s, T) belongs to the domain of the Laplace exponent J of Z almost ω-surely for almost all s. In particular, this means that Z has finite some exponential moments. Moreover, (8.1.10) and (8.1.12) reduce to A(t, T) = J((t, T)), which leads to the condition α(t, T) = J  ((t, T))σ (t, T).

(8.1.14)

The last relation opens the door to construction of arbitrage-free models. Starting from any L´evy process Z and volatility process σ (t, T), such that (8.1.13) is satisfied, one can define the drift α(t, T) by (8.1.14). Then the resulting model is clearly arbitrage free and P is a martingale measure. Remark 8.1.4 If Z is a Q-Wiener process then the HJM condition has the form A(t, T) =

1 Q(t, T), (t, T) − Qφ(t), (t, T) , 2

which was originally obtained by Heath, Jarrow and Morton in [67] in the onedimensional case. Remark 8.1.5 It follows from the inequality (8.1.22), which will be obtained in the proof of Theorem 8.1.1, that condition (8.1.8) together with the fact that eψ − 1 ∈ 1,2 is equivalent to 

T∗ 0

 | eψ(t,y) − 1 | dtν(dy) < +∞,

P − a.s.

U

This means that eψ − 1 ∈ 1 , where  T∗  1 := {g = g(s, y) − predictable : | g(s, y) | dsν(dy) < +∞, 0

P − a.s.}.

U

Remark 8.1.6 If the volatility σ of the model (α, σ , Z) is such that (t, T), y ≥ 0 for | y |> 1 then both conditions (8.1.9) and (8.1.11) are automatically satisfied.

188

Arbitrage-Free HJM Markets

8.1.1 Proof of Theorem 8.1.1 For the proof of Theorem 8.1.1 we need the representation of the discounted bond price process in the HJM model. It can be deduced from Proposition 7.4.3. Proposition 8.1.7 In the HJM model (α, σ , Z) satisfying (8.1.2)–(8.1.4) the discounted bond price process ˆ T) = e− P(t,

T 0

f (t,s)ds

t ∈ [0, T ∗ ], T > 0

,

admits the following representation   ˆ T) dP(t, 1 Q(t, T), (t, T) − A(t, T) − (t, T), a dt = ˆ 2 P(t−, T)  (t, T), y π˜ (dt, dy)

− (t, T), dW(t) − U

 + U

! e− (t,T),y − 1 + 1{|y|≤1} (t, T), y π(dt, dy).

(8.1.15)

Proof of Theorem 8.1.1 (a) In view of Lemma 6.2.2 we examine the process ˆ T) under the measure P and use the integral representation of P(·, ˆ T) given ρ(·)P(·, by Proposition 8.1.7. Recall that  T ˆ T) = e−X(t,T) , with X(t, T) := f (t, s)ds, t ∈ [0, T ∗ ], T > 0, P(t, 0

ˆ T) = e ρ(t)P(t,

VtT

t ∈ [0, T ∗ ], T > 0,

with VtT := Y(t) − X(t, T),

,

where Y(t), defined by ρ(t) = eY(t) , is equal to  t   t 1 1 t φ(s), dW(s) − | Q 2 φ(s) |2U ds + (eψ(s,y) − 1)π˜ (ds, dy) Y(t) = 2 0 0 0 U −

 t

(eψ(s,y) − 1 − ψ(s, y))π(ds, dy), t ∈ [0, T ∗ ]

U

0

(see formula (6.2.6) in Section 6.2). Moreover,  T  T  t  X(t, T) = f0 (s)ds + α(v, s)dv ds + 0

 =

0 T

0



t

f0 (s)ds + 0

0

0



t

A(v, T)dv +

T



t

σ (v, s), dZ(v) ds

0

(v, T), dZ(v) ,

t ∈ [0, T ∗ ], T > 0,

0

or, equivalently, dX(t, T) = A(t, T)dt + (t, T), dZ(t) ,

t ∈ [0, T ∗ ], T > 0.

(8.1.16)

8.1 Heath–Jarrow–Morton Conditions

189

Since VtT = Y(t) − X(t, T) = ψ(t, Z(t)) − (t, T), Z(t) , it follows from Itˆo’s formula that 

T

T

t

eVt = eV0 + 0



T

t

= eV0 + 0

T

eVs− dVsT +

1 2



t

0

T

eVs− d[VsT , VsT ]c +



T

T

eVs− (eVs − 1 − VsT )

s≤t

 & 1 1 T eVs− φ(s), dW(s) − | Q 2 φ(s) |2 ds + (eψ(s,y) − 1)π˜ (ds, dy) 2 U

 −

(eψ(s,y) − 1 − ψ(s, y))π(ds, dy) − (A(s, T) + (s, T), a )ds − (s, T), dW(s) U

 −



{|y|≤1}



1 2

+

t

T

{|y|>1}

' (s, T), y π(ds, dy)

1

eVs− | Q 2 (φ(s) − (s, T)) |2 ds

0

 t

+

(s, T), y π˜ (ds, dy) −

0

   T eVs− eψ(s,y)− (s,T),y − 1 − ψ(s, y) − (s, T), y π(ds, dy).

U

Finally, we have ˆ T) = eVt = eV0 + M(t) + A(t) + B(t), ρ(t)P(t, T

T

(8.1.17)

where 

t

M(t) :=

T

eVs− φ(s), dW(s) +

0

0

 t

−

e



0

 t

T Vs−

T

eVs− (eψ(s,y) − 1)π˜ (ds, dy) U

(s, T), y π˜ (ds, dy),

B

 1  1 1 1 T eVs− − | Q 2 φ(s) |2 −A(s, T) − (s, T), a + | Q 2 (φ(s) − (s, T)) |2 ds, 2 2 0  t   T B(t) := eVs− eψ(s,y)− (s,T),y − eψ(s,y) + 1B (s, T), y π(ds, dy), t

A(t) :=

0

U

and B := {y :| y |≤ 1}. Let us notice that M(t) is a local martingale, A(t) and ˆ T) is also a local B(t) have finite variation and A(t) is predictable. Since ρ(t)P(t, martingale, it follows from (8.1.17) that A(t) + B(t) is a local martingale as well and thus, in view of Proposition 4.2.9, that it is a process of locally integrable variation. In view of Proposition 4.2.8 the process A(t) has also locally integrable variation, so does B(t). Hence, it follows from Theorem 4.4.3 that the compensator of B(t) exists and is of the form

190 Bp (t) =

Arbitrage-Free HJM Markets

 t 0

  T eVs− eψ(s,y)− (s,T),y − eψ(s,y) + 1B (s, T), y dsν(dy), U

t ∈ [0, T ∗ ]. (8.1.18) Since the function s → VsT is c`adl`ag and thus bounded on finite intervals, the preceding integral exists if and only if  T∗   ψ(s,y)− (s,T),y  e − eψ(s,y) + 1B (s, T), y dsν(dy) < +∞, (8.1.19) 0

U

for each T almost ω-surely. In view of (8.1.7) the function s → (s, T) is bounded and it follows that  T∗   − (s,T),y  e − 1 + (s, T), y 1B (y)dsν(dy) < +∞, (8.1.20) 0

U

and further from (8.1.20) and (8.1.19) that  T∗     − (s,T),y  − 1)(eψ(s,y) − 1B (y))dsν(dy) < +∞. (e 0

U

Let us split the preceding integral into two integrals over B and Bc := U \ B. Since the function (s, y) → (e− (s,T),y − 1) is bounded on [0, T ∗ ] × B, (8.1.8) follows. To show that (8.1.9) is satisfied, we show that  T∗  eψ(s,y) dsν(dy) < +∞. (8.1.21) 0

Bc

Let us denote A := {(s, y) ∈ [0, T ∗ ] × U :| eψ(s,y) − 1 |≤ 1}, A¯ := {(s, y) ∈ [0, T ∗ ] × U :| eψ(s,y) − 1 |> 1}. It follows from the fact that eψ − 1 ∈ 1,2 that  T∗  | eψ(s,y) − 1 | dsν(dy) 0 Bc   = | eψ(s,y) − 1 | ν(dy) +

[0,T ∗ ]×Bc ∩A¯

[0,T ∗ ]×Bc ∩A

 ≤

[0,T ∗ ]×Bc ∩A

1  2 1dsν(dy)

[0,T ∗ ]×Bc ∩A

|e

| eψ(s,y) − 1 | dsν(dy)

ψ(s,y)

1 2 − 1 | dsν(dy) 2

8.2 Martingale Measures 191 1   2  ∗ 1 ψ(s,y) c 2 ψ(s,y) 2 |e − 1 | dsν(dy) ≤ T ν(B ) |e − 1 | dsν(dy) +  +

A¯

A¯

A

| eψ(s,y) − 1 | dsν(dy) < +∞.

(8.1.22)

Condition (8.1.21) follows from the preceding estimation and the fact that ν(Bc ) < +∞. (b) If (8.1.8) and (8.1.9) are satisfied then the process B(t) admits a compensator Bp (t) of the form (8.1.18) and ˆ T) − eV0 − M(t) − (B(t) − Bp (t)) = A(t) + Bp (t), ρ(t)P(t, T

t ∈ [0, T ∗ ].

ˆ T) is a local martingale if and only if A(t) + Bp (t) is a local The process ρ(t)P(t, martingale. But A(t) + Bp (t) is a predictable process of finite variation, so, in view of Proposition 4.2.10, it is a local martingale if and only if A(t) + Bp (t) = 0,

t ∈ [0, T ∗ ].

Using formulas for A(t) and Bp (t) and rearranging terms yields 1 A(s, T) = − (s, T), a + Q(s, T), (s, T) − Qφ(s), (s, T) 2   ψ(s,y) − (s,T),y  e (e − 1) + 1B (s, T), y ν(dy), (8.1.23) + U

for each T > 0 almost all s almost ω-surely. (c) If (8.1.11) is satisfied then (s, T) belongs to the domain of J for almost all s and we can rearrange terms in (8.1.23) in the following way 1 A(s, T) = − (s, T), a + Q(s, T), (s, T) 2   − (s,T),y  e − 1 + 1B (s, T), y ν(dy) − Qφ(s), (s, T) + U





 eψ(s,y)− (s,T),y − eψ(s,y) − e− (s,T),y + 1 ν(dy).

+ U

Taking into account (8.1.6) and simplifying the last term yields  A(s, T) = J((s, T))− Qφ(s), (s, T) + (eψ(s,y) −1)(e− (s,T),y − 1)ν(dy). U

8.2 Martingale Measures In view of Theorem 8.1.1 martingale measures in the HJM model (α, σ , Z) are described in terms of their generating pairs (φ, ψ), which are supposed to satisfy the relation:

192

Arbitrage-Free HJM Markets 1 A(t, T) = − (t, T), a + Q(t, T), (t, T) − Qφ(t), (t, T) 2   ψ(t,y) − (t,T),y  e (e − 1) + 1{|y|≤1} (t, T), y ν(dy). +

(8.2.1)

U

The relation (8.2.1) simplifies considerably in the case when the process Z is real valued. Proposition 8.2.1 Let (α, σ , Z) be an HJM model where Z is a one-dimensional L´evy process with characteristics (a, q = 1, ν), i.e. the Gaussian part is a standard Wiener process. Assume that (8.1.2)–(8.1.4), (8.1.7) are satisfied and for each t ∈ [0, T ∗ ] the function T →

α(t, T) , σ (t, T)

is right continuous at point t. (a) Assume that (MM) holds. If (φ, ψ) is the generating pair of some martingale measure, then for almost all t ∈ [0, T ∗ ], P − a.s.:    α(t, t) −a− eψ(t,y) − 1{|y|≤1} y ν(dy). (8.2.2) φ(t) = − σ (t, t) R (b) Assume that there exists ψ with eψ −1 ∈ 1 such that φ given by (8.2.2) belongs to  and E[ρT ∗ ] = 1 with ρ given by (6.2.1). Then ψ determines a martingale measure if and only if for each T > 0, almost all t ∈ [0, T ∗ ], P-a.s.:    α(t, t) 1 = eψ(t,y) e−(t,T)y −1+(t, T)y ν(dy). A(t, T)− ((t, T))2 −(t, T) 2 σ (t, t) R (8.2.3) Proof

(a) Dividing both sides of (8.2.1) by (t, T) yields    e−(t,T)y − 1 1 A(t, T) = −a + (t, T) − φ(t) + + 1{|y|≤1} y ν(dy). eψ(t,y) (t, T) 2 (t, T) R (8.2.4)

Since lim T↓t

A(t, T) α(t, T) α(t, t) = lim = , T↓t σ (t, T) (t, T) σ (t, t)

letting T ↓ t in (8.2.4) yields α(t, t) = −a − φ(t) + σ (t, t) which is (8.2.2).

  R

 − eψ(t,y) + 1{|y|≤1} yν(dy),

8.2 Martingale Measures

193

(b) Inserting (8.2.2) into (8.2.1) yields 1 A(t, T) = − a(t, T) + ((t, T))2  2  α(t, t) ψ(t,y) + (t, T) − 1{|y|≤1} )yν(dy) + a + (e σ (t, t) R +

  R

 eψ(t,u) (e−y(t,T) − 1) + 1{|y|≤1} (t, T)y ν(dy).

Rearranging terms provides (8.2.3). It follows from Proposition 8.2.1 that if a given HJM model (α, σ , Z) admits a martingale measure Q then the process φ of its generating pair (φ, ψ) is determined by ψ, drift and volatility in a unique way. In particular, if ψ satisfying (8.2.3) is unique, so is φ, and the related martingale measure is unique as well. With the use of Proposition 8.2.1 we can characterize martingale measures in the case in which Z is continuous. Corollary 8.2.2 Let us assume that the HJM model (α, σ , at + W(t)) with a ∈ R and a Wiener process W satisfies the assumptions of Proposition 8.2.1. Then the following statements are true. (a) (necessity) If there exists a martingale measure then its generating process φ is given by φ(t) = −

α(t, t) − a, σ (t, t)

t ∈ [0, T ∗ ].

(8.2.5)

This follows from (8.2.2) with ν ≡ 0. (b) (sufficiency) Assume that φ(t) given by (8.2.5) belongs to  and that E[ρT ∗ ] = 1 with ρ defined by (6.2.1). If for each T > 0, almost all t ∈ [0, T ∗ ], P-a.s.:  T  α(t, t) α(t, T) = σ (t, T) σ (t, u)du + , t ∈ [0, T ∗ ], T > 0, (8.2.6) σ (t, t) t then there exists a martingale measure and it is generated by φ. This follows from (8.2.3) by putting ν = 0 and differentiation over T. (c) (uniqueness) If there exists a martingale measure then it is unique. This follows from the fact that (8.2.5) uniquely determines φ.

8.2.1 Specification of Drift An interesting question concerned with an HJM model (α, σ , Z) is if the existence of a martingale measure implies that the volatility σ determines the drift α in a unique way. It is easily seen that the answer is positive in the case in which the original

194

Arbitrage-Free HJM Markets

measure P is a martingale measure. Indeed, condition (8.1.10) in Theorem 8.1.1 with φ ≡ 0, ψ ≡ 0 yields A(t, T) = J((t, T)), for each T > 0, almost all t ∈ [0, T ∗ ], P-a.s. Differentiation over T yields the formula for drift, i.e. α(t, T) = DJ((t, T)), σ (t, T) . In the particular case, when Z is a real-valued standard Wiener process we obtain the well-known classical HJM drift condition α(t, T) = (t, T)σ (t, T). In the case when the martingale measure is different from P, the problem becomes more involved and the answer is, in general, negative. We explain this in the onedimensional case with the use of condition (8.2.3) in Proposition 8.2.1. Recall that (8.2.3) yields    1 α(t, t) 2 A(t, T) = ((t, T)) + (t, T) eψ(t,y) e−(t,T)y − 1 + (t, T)y ν(dy), + 2 σ (t, t) R which, by differentiation, yields  T  ! T α(t, t) σ (t, u)du + eψ(t,y) (e−y t σ (t,u)du − 1)yν(dy) . α(t, T) = σ (t, T) + σ (t, t) U t (8.2.7) Now we see that even if Z has no jumps then α(t, T) is determined by the volatility and the boundary value of the drift/volatility ratio, that is, by the transformations α(t, t) (t, T) → σ (t, T), t→ . σ (t, t) So, even in this degenerated case volatility does not determine the drift. It follows from (8.2.7) that in the general case we need additionally a function ψ and then α(t, T) is determined by the triplet α(t, t) , ψ such that eψ − 1 ∈ 1 . (t, T) → σ (t, T), t→ σ (t, t)

8.2.2 Models with No Martingale Measures It is instructive to analyze some examples of HJM models that do not admit martingale measures. Since the conditions for drift and volatility of the model (α, σ , Z) in the following examples are quite general, they throw some light on the necessary conditions for the existence of martingale measures. Proposition 8.2.3 Let the coefficients of the model (α, σ , Z), where Z is an arbitrary L´evy process, satisfy with positive dP × dt-measure (on  × [0, T ∗ ]) the following two conditions

8.2 Martingale Measures

195

lim σ (t, T) > 0,

(8.2.8)

T→+∞

and for each T > 0, α(t, T) ≤ H(σ (t, T)),

(8.2.9)

where H is a concave function of sublinear growth, i.e. H(x) ≤ cx,

x ∈ R,

with some c ∈ R. Then the model does not satisfy (MM). Proposition 8.2.4 Let the coefficients in the model (α, σ , Z) satisfy with positive dP × dt-measure the following conditions lim α(t, T) > 0,

(8.2.10)

| σ (t, u) | du < +∞,

(8.2.11)

T→+∞

and



+∞ t

and let Z be a L´evy process with bounded jumps. Then the model does not satisfy (MM). The next example is concerned with the affine model P(t, T) = e−C(T−t)−D(T−t)R(t) ,

t ∈ [0, T ∗ ],

T > 0,

(8.2.12)

t ∈ [0, T ∗ ],

(8.2.13)

with positive short rate of the form R(t) = F(R(t))dt + G(R(t−))dZ(t),

where Z is a L´evy process. This model was introduced in Section 7.5 where we also showed that it can be viewed as an HJM model (α, σ , Z) with relevant coefficients. Proposition 8.2.5 Let in the affine model (8.2.12)–(8.2.13) Z be an arbitrary L´evy process. (a) If G(x) > 0, x > 0, F(x) ≤ G(x), x ≥ 0, C(·), D(·) are convex functions and D (u) ≥ 0, u ≥ 0,

lim D (u) > 0,

u→+∞

then the model does not satisfy (MM). (b) If C(·), D(·) are twice differentiable and  +∞ | D (u) | du < +∞,

(8.2.14)

(8.2.15)

0

lim C (u) < 0,

u→+∞

lim D (u) < 0

u→+∞

then the model, with any function F, does not satisfy (MM) .

(8.2.16)

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Arbitrage-Free HJM Markets

The proofs of Proposition 8.2.3 and Proposition 8.2.4 are based on Proposition 8.2.1. Proposition 8.2.5 can be deduced from Proposition 8.2.3 and Proposition 8.2.4. Proof of Proposition 8.2.3 We prove that there are no processes ψ solving the equation (8.2.3) in Proposition 8.2.1. With the use of (8.2.9), the concavity of H and Jensen’s inequality we obtain, for t < T,  T  T A(t, T) = α(t, u)du ≤ H(σ (t, u))du t



≤ (T − t)H

t

1 (t, T) , T −t

t ∈ [0, T ∗ ].

Consequently, since H is sublinear, we can estimate the left side of (8.2.3) as follows A(t, T) −

α(t, t) 1 (8.2.17) ((t, T))2 − (t, T) 2 σ (t, t)  1 α(t, t) 1 ≤ (T − t)H (t, T) − ((t, T))2 − (t, T) T −t 2 σ (t, t) α(t, t) 1 ((t, T))2 − (t, T) 2 σ (t, t)    1 α(t, t) ≤ − (t, T) (t, T) − 2 c − , t ∈ [0, T ∗ ], T > t. 2 σ (t, t) ≤ c(t, T) −

For (ω, t) satisfying (8.2.8) the function T → (t, T) is unbounded and consequently, for large T the expression (8.2.17) becomes strictly negative. Since e−z + z − 1 ≥ 0,

z ∈ R,

the right side of (8.2.3) is nonnegative. So, (8.2.3) is not satisfied for any L´evy measure ν. Proof of Proposition 8.2.4 Let Z be a L´evy process with bounded jumps, i.e. | Z(t) |≤ y¯ , t ∈ [0, T ∗ ], for some y¯ > 0. Using the inequality ex − 1 − x ≤ 1 |x| 2 2 e x , x ∈ R we can estimate the right side of (8.2.3), for (ω, t) such that (8.2.10) and (8.2.11) are satisfied, as follows     1 eψ(t,y) e−(t,T)y − 1 + (t, T)y ν(dy) ≤ eK(t)¯y K 2 (t)¯y2 eψ(t,y) ν(dy), 2 U R (8.2.18)  +∞ where K(t) := t | σ (t, u) | du. Hence for any ψ solving (8.2.3),  1 α(t, t) 1 A(t, T) − ((t, T))2 − (t, T) eψ(t,y) ν(dy). (8.2.19) ≤ eK(t)¯y K 2 (t)¯y2 2 σ (t, t) 2 R

8.2 Martingale Measures

197

By (8.2.10) and (8.2.11) the left side of (8.2.19) tends to +∞ for T → +∞ while the right side is finite, so we arrive at a contradiction. Proof of Proposition 8.2.5 In view of Proposition 7.5.1 the affine model can be written in the HJM parametrization t ∈ [0, T ∗ ], T > 0,

df (t, T) = α(t, T)dt + σ (t, T)dZ(t), with

α(t, T) := F(R(t))D (T − t) − C (T − t) − D (T − t)R(t), σ (t, T) := D (T − t)G(R(t−)). (a) By the positivity of G and (8.2.14) we have lim σ (t, T) = G(R(t−)) lim D (T − t) > 0,

T→+∞

T→+∞

dP × dt − a.s.

Since F ≤ G, the convexity of C and D yields α(t, T) ≤ G(R(t))D (T − t) − C (T − t) − D (T − t)R(t) ≤ σ (t, T),

T > 0, dP × dt − a.s.

The assertion follows from Proposition 8.2.3. (b) In view of (8.2.15) we have  +∞  | σ (t, u) | du =| G(R(t−)) | · t

+∞

| D (u) | du < +∞,

0

and consequently lim α(t, T) = F(R(t)) lim D (T − t) − lim C (T − t) − lim D (T −t)

T→+∞

T→+∞

T→+∞

T→+∞

= − lim C (u) − lim D (u)R(t). u→+∞

u→+∞

By (8.2.16) the last term in the preceding is strictly positive and the assertion follows from Proposition 8.2.4.

8.2.3 Invariance of L´evy Noise Our aim now is to construct an HJM model that admits a martingale measure Q different from the original measure P. Additionally, we require that Z remains a L´evy process under Q. Although this narrows the class of examined models, it also offers a possibility of treating the model after measure change with the use of the HJM conditions. First we prove that for an HJM model (α, σ , Z) for which Z is still a L´evy process under Q, the drift α is determined by volatility σ in a deterministic manner.

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Arbitrage-Free HJM Markets

Using this property we construct a class of possible drifts in the HJM model driven by a compound Poisson process such that the model admits a martingale measure. Theorem 8.2.6 Let the HJM model (α, σ , Z) satisfy the assumptions (8.1.2)–(8.1.4) and (8.1.7) and let Z be a L´evy process with characteristic triplet (a, Q, ν) under the measure P. If (MM) is satisfied and Z is a L´evy process under some martingale measure Q, then   α(t, T) = H σ (t, T),

T

σ (t, u)du ,

t ∈ [0, T ∗ ],

T > 0,

t

where H is a deterministic function: H(x, z) = x, Q(z − φ) − a −

  U

 eψ(y)− z,y x, y + 1{|y|≤1} x, y ν(dy),

x, z ∈ U,

where φ ∈ U and ψ(y) is a deterministic function such that  | eψ(y) − 1 | ν(dy) < +∞. U

Proof Let Q be a martingale measure under which Z remains a L´evy process. In view of Theorem 6.2.1 (d) its generating pair (φ, ψ) is such that φ(ω, t) = φ and ψ(ω, t, y) = ψ(y), where φ is a deterministic vector in U and ψ(·) a deterministic function. Then (see (8.1.10)) we have 1 A(t, T) = − (t, T), a + Q(t, T), (t, T) − Qφ, (t, T) 2   ψ(y) − (t,T),y  e (e − 1) + 1{|y|≤1} (t, T), y ν(dy). + U

Differentiation over T yields 1 1 α(t, T) = − σ (t, T), a + Qσ (t, T), (t, T) + Q(t, T), σ (t, T) − Qφ, σ (t, T) 2 2    eψ(y) e− (t,T),y σ (t, T), y + 1{|y|≤1} σ (t, T), y ν(dy) − U

= − σ (t, T), a + Qσ (t, T), (t, T) − Qφ, σ (t, T)    eψ(y) e− (t,T),y σ (t, T), y + 1{|y|≤1} σ (t, T), y ν(dy) − U

  = H σ (t, T), (t, T) .

8.2 Martingale Measures

199

Proposition 8.2.7 Let Z be a compound Poisson process with L´evy measure ν. Let us consider the class of functions Gν satisfying  g(y) > 0, g(y)ν(dy) < +∞, R

and define  Fg (z) := −

R

ye−zy g(y)ν(dy),

providing that the right side is well defined. If in the model (α, σ , Z) the drift is determined by  α(t, T) = Fg

T

σ (t, u)du σ (t, T),

t ∈ [0, T ∗ ], T > t,

(8.2.20)

t

for some function g ∈ Gν , then the model satisfies (MM) and Z is a compound Poisson process under some martingale measure. Proof

The characteristic triplet of the process Z equals   1

yν(dy), 0, ν(dy) . 0

For g ∈ Gν let us consider the measure Qg ∼ P with generating pair (φ ≡ 0, ψ(t, y) ≡ ln g(y)). Since ν is finite and g ∈ Gν , it is clear that  0

T∗

 R

| eψ(t,y) − 1 | dtν(dy) = T ∗

 R

| g(y) − 1 | ν(dy) < +∞,

so the measure Qg satisfies the necessary condition to be a martingale measure (see (8.1.8) and Remark 8.1.5). Moreover, the process Z remains under Qg a L´evy process and its characteristic triplet under Qg equals 



1

yg(y)ν(dy), 0, g(y)ν(dy) 0

(see Theorem 6.2.1). Hence its Laplace exponent under Qg equals  J Qg (z) = (e−zy − 1)g(y)ν(dy) R

∂ Qg and ∂z J (z) = Fg (z). In view of Theorem 8.1.1, condition (8.2.20) means that Qg is a martingale measure for the model (α, σ , Z).

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Arbitrage-Free HJM Markets

8.2.4 Volatility-Based Models In this section we continue our discussion from Section 8.2.3 on the construction of HJM models that admit a martingale measure Q different from P, the original one. We focus, however, on a more special dependence of the drift on volatility. To motivate our idea, let us recall that the existence of a martingale measure for the model (α, σ , Z) is equivalent to the following drift condition    α(t, T) = σ (t, T) − a + (t, T) − φ(t) − (eψ(t,y) e−(t,T)y − 1{|y|≤1 })yν(dy) R

[0, T ∗ ], T

> 0 (see formula (8.1.10) for a one-dimensional noise in for t ∈ Theorem 8.1.1). This means that any volatility σ (t, T) together with corresponding (t, T) determine α(t, T):   (8.2.21) α(t, T) = H σ (t, T), (t, T) , t ∈ [0, T ∗ ], T > 0, where H = H(u, v), u, v ∈ R is a random field. The randomness of H is hidden in the generating pair (φ, ψ) of a martingale measure. In the sequel we focus on the particular class of HJM models where drift is some deterministic function of volatility only, that is, α(t, T) = g(σ (t, T)),

t ∈ [0, T ∗ ], T > 0.

(8.2.22)

We formulate the problem as follows. For a given L´evy process, find a volatility σ (t, T) and a function g such that the HJM model df (t, T) = g(σ (t, T))dt + σ (t, T)dZ(t)

(8.2.23)

admits a martingale measure. Moreover, we focus on time homogeneous volatilities σ (t, T) = σ (T − t) that are given by the ordinary differential equation σ  (v) = h(σ (v)),

σ (0) = σ0 ,

v ≥ 0,

(8.2.24)

where h : R → R is a given function. The function g is assumed to be differentiable and such that g(σ (v))/σ (v) is also differentiable on [0, +∞). This setting comprises a reasonably wide subclass of arbitrage-free HJM models. We determine g such that at least one martingale measure exists. Theorem 8.2.8 Let the volatility σ (t, T) = σ (T − t) be given by (8.2.24). Let ψ(y) be a deterministic function satisfying  | eψ(y) − 1 | ν(dy) < +∞. (8.2.25) R

If the function g in the HJM model (8.2.23) is given by  x K(z) dz, x ∈ R, g(x) := c1 x + x 2 c2 z

(8.2.26)

8.2 Martingale Measures where c1 , c2 are constants, K(z) = K(h, ψ)(z) :=



z3  1+ h(z)

R

201

 eψ(y) e{w(σ0 )−w(z)}y y2 ν(dy) ,

z ∈ R, (8.2.27)

and



z

u du, h(u)

w(z) := 0

z ∈ R,

(8.2.28)

then the model admits a martingale measure. Proof In the first part of the proof we show that the model admits a martingale measure if and only if, for t ∈ [0, T ∗ ], u ≥ 0,      eψ(t,y) e−(u)y y2 ν(dy) , σ  (u) g (σ (u))σ (u) − g(σ (u)) = σ (u)3 1 + R

(8.2.29) with some process ψ(t, y) such that eψ(t,y) − 1 ∈ 1 . Since σ (t, T) = σ (T − t) is time homogeneous, so α(t, T) = g(σ (T − t)) and  T  T−t A(t, T) = α(u − t)du = α(u)du = A(T − t), 0 ≤ t ≤ T, t



0 T

(t, T) = t

 σ (u − t)du =

T−t

σ (u)du = (T − t),

0≤t≤T

0

for some functions A(·) and (·). Since lim T↓t

α(T − t) g(σ (T − t)) α(0) g(σ0 ) = = lim = T↓t σ (T − t) σ (T − t) σ0 σ (0)

is finite, we can use Proposition 8.2.1. Setting v := T − t we see from (8.2.3) that the model admits a martingale measure if and only if    α(0) 1 eψ(t,y) e−(v)y − 1 + (v)y ν(dy), = A(v) − ((v))2 − (v) 2 σ (0) (8.2.30) R ∗ t ∈ [0, T ], v ≥ 0, with some function ψ(t, y) such that eψ(t,y) − 1 ∈ 1 . Differentiation of (8.2.30) yields  ! α(0) eψ(t,y) (e−(v)y − 1)yν(dy) , t ∈ [0, T ∗ ], v ≥ 0, − α(v) = σ (v) (v) + σ (0) R and further  α(0) α(v) eψ(t,y) (e−(v)y − 1)yν(dy), − = (v) − σ (v) σ (0) R

t ∈ [0, T ∗ ], v ≥ 0.

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Arbitrage-Free HJM Markets

By (8.2.22), we obtain g(σ (v)) g(σ0 ) = (v) − − σ (v) σ0

 R

eψ(t,y) (e−(v)y − 1)yν(dy),

t ∈ [0, T ∗ ], v ≥ 0.

The preceding left side can be written as the integral of the derivative of the function (v)) v → g(σ σ (v) . In the same way we represent the right side. This yields  v   v σ (u)(g (σ (u))σ (u) − g(σ (u))) σ (u)du du = σ 2 (u) 0 0  v eψ(t,y) e−(u)y σ (u)y2 du ν(dy), t ∈ [0, T ∗ ], v ≥ 0. + 0

R

Comparison of the integrands yields (8.2.29). In the second part of the proof, with the use of (8.2.29), we determine g. It can be checked that (8.2.24) is equivalent to the equation σ  (v) =

σ (v) , w (σ (v))

v ≥ 0,

where w is given by (8.2.28). It follows that σ (v) = w (σ (v))σ  (v), and



v

(v) =

 σ (u)du =

0

v

v ≥ 0,

w (σ (u))σ  (u)du = w(σ (v)) − w(σ0 ),

v ≥ 0.

0

Using this and putting any ψ(t, y) which does not depend on t, i.e. ψ(t, y) = ψ(y) satisfying (8.2.25) into (8.2.29), yields     σ (v)   3 ψ(y) {w(σ0 )−w(σ (v))}y 2 (σ (v))σ (v) − g(σ (v)) = σ (v) 1 + e e y ν(dy) , g w (σ (v)) R v ≥ 0. Setting x := σ (v) leads to     x   3 ψ(y) {w(σ0 )−w(x)}y 2 (x)x − g(x) = x e e y ν(dy) , g 1 + w (x) R which yields 

2 



g (x)x − g(x) = x w (x) 1 +

 R

 eψ(y) e{w(σ0 )−w(x)}y y2 ν(dy) ,

x ∈ R,

x ∈ R.

Finally we obtain (see (8.2.27)) that g (x)x − g(x) = K(x),

x ∈ R.

(8.2.31)

Since g(0) = 0 (see (8.1.4)) the solution of the equation above is given by (8.2.26).

8.2 Martingale Measures

203

Remark 8.2.9 In the HJM model in Proposition 8.2.8 there exist many functions g that guarantee that the model (8.2.23) admits a martingale measure. Indeed, the constants c1 and c2 in (8.2.26) are arbitrary. Moreover, the function K in (8.2.26) depends on the function ψ, which can be chosen freely. Remark 8.2.10 If Z is a Wiener process with drift then the integral in (8.2.29) disappears and it follows from the proof of Theorem 8.2.8 that (8.2.29) is equivalent to (8.2.31). Consequently, the model (8.2.23) admits a martingale measure if and only if  x z dz, x ∈ R. g(x) = c1 x + x h(z) c2

8.2.5 Uniqueness of the Martingale Measure Our first result on the uniqueness of the martingale measure in the HJM model is concerned with a finite activity L´evy process Z in Rd , which additionally has a finite number of jump sizes. This means that the support of the L´evy measure ν is concentrated on a finite set, i.e. supp{ν} = {y1 , y2 , . . . , yn },

yi ∈ U = Rd , i = 1, 2, . . . , n.

We will assume that the original measure P is a martingale measure. In the analysis we use the following functions: (t, T) := ( 1 (t, T), . . . ,  d (t, T)), e− (t,T),y1 − 1, . . . , e− (t,T),yn − 1. (8.2.32) To simplify notation their dependence on ω is not indicated. For fixed (ω, t) they will be viewed as functions of T ∈ [0, +∞). Theorem 8.2.11 Let P be a martingale measure for an HJM model, where Z is a finite activity L´evy process in Rd with a finite number of jump sizes. Then P is a unique martingale measure if and only if functions (8.2.32) are linearly independent dP × dt-a.s. Proof Since P is a martingale measure, the drift is determined by the volatility in the following way A(t, T) = J((t, T)),

(8.2.33)

where J stands for the Laplace exponent of Z. Let us now formulate conditions for the generating pair (φ, ψ) of an equivalent measure Q that ensure that Q is also a martingale measure. Recall that by (8.1.12) we know that φ, ψ satisfy A(t, T) = J((t, T)) − Qψ(t), (t, T)  + (eψ(t,y) − 1)(e− (t,T),y − 1)ν(dy), U

t ∈ [0, T ∗ ].

(8.2.34)

204

Arbitrage-Free HJM Markets

By (8.2.33) and (8.2.34) we have  − Qφ(t), (t, T) + (eψ(t,y) − 1)(e− (t,T),y − 1)ν(dy) = 0, U

which, in the finite activity setting, can be written as n  Qφ(t), (t, T) = (eψ(t,yi ) − 1)(e− (t,T),yi − 1)δi ,

(8.2.35)

i=1

where δi := ν({yi }), i = 1, 2, . . . , n. So, Q is a martingale measure if and only if (8.2.35) is satisfied. If (8.2.32) are linearly independent then one can find maturities T1 , . . . , TK such that the vectors ⎛ ⎛ ⎞ ⎞  i (t, T1 ) e−(t,T1 )yj − 1 ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ .. .. wi := ⎜ := , v ⎜ ⎟ ⎟ , i = 1, 2, . . . , d, j = 1, 2, . . . , n j . . ⎝ ⎝ ⎠ ⎠  i (t, TK ) e−(t,TK )yj − 1 (8.2.36) are linearly independent. For these maturities (8.2.35) has the form ⎤ ⎡ ⎤⎡  1 (t, T1 ) . . .  d (t, T1 ) (Qφ(t))1 ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ .. .. .. ⎥ ⎢ ⎥ ⎢ . . . ⎦ ⎣ ⎦⎣  1 (t, TK ) . . .  d (t, TK ) (Qφ(t))d ⎡ ψ(t,y1 ) − 1) ⎡ − (t,T1 ),y1 ⎤ − 1 . . . e− (t,T1 ),yn − 1 ⎢ δ1 (e e ⎢ ⎥⎢ .. .. .. =⎣ ⎦⎢ . . . ⎣ e− (t,TK ),y1 − 1 . . . e− (t,TK ),yn − 1 δ (eψ(t,yn )−1)

⎤ ⎥ ⎥ ⎥, ⎦

n

(8.2.37) where (Qφ(t))i , i = 1, 2, . . . , d are the coordinates of the vector Qφ(t). It follows from the linear independence of {wi , vj }, i = 1, 2, . . . , d; j = 1, 2, . . . , n that (8.2.37) may hold if and only if φ(t) ≡ 0 and ψ(t, yi ) ≡ 0, i = 1, 2, . . . , n. So, Q = P. Conversely, if (8.2.32) are linearly dependent then there are constants αi = αi (ω, t), i = 1, 2, . . . , d and βj = βj (ω, t), j = 1, 2, . . . , n such that α1  1 (t, T) + · · · + αd  d (t, T) = β1 (e− (t,T),y1 − 1) + · · · + βn (e− (t,T),yn − 1), Moreover, one can choose these constants such that αi , βj i = 1, 2, . . . , d; j = 1, 2, . . . , n, and define (φ(t), ψ(t, yi )) by (Qφ(t))i := αi ,

δj (eψ(t,yj ) − 1) := βj ,

T ∈ [t, +∞). ∈

(−1, 1),

i = 1, 2, . . . , d; j = 1, 2, . . . , n.

8.2 Martingale Measures

205

Then (8.2.35) is satisfied for each T > 0, and φ ∈ (U), eψ − 1 ∈ 1,2 . This means that (φ, ψ) is a generating pair of some martingale measure. The problem of uniqueness of the martingale measure was studied in Eberlein, Jacod and Raible [47] in a slightly different framework. The bond market model in [47] consists of bonds with maturities forming a dense subset J of [0, T ∗ ], that is, only bonds P(t, T), t ∈ [0, T ∗ ],

T ∈ J,

are tradeable.

(8.2.38)

In the forward-rate dynamics the process Z is an Rd -valued process with independent but not necessarily stationary increments and the coefficients are assumed to satisfy the condition   | α(t, T) | + | σ (t, T) |Rd < +∞, P − a.s. (8.2.39) sup t,T∈[0,T ∗ ]

Now we present the main results of [47] adopting them to processes Z with stationary increments. The first result is concerned with the case d = 1. Theorem 8.2.12 Assume that (8.2.38) holds and Z is a one-dimensional L´evy process with a characteristic triplet (a, q, ν). (a) If q + ν(R) > 0 then the set of all martingale measures contains at most one element if and only if for almost all t ∈ [0, T ∗ ]:  T∗ | σ (t, T) | dT > 0. (8.2.40) t

(b) If q + ν(R) = 0 then the set of all martingale measures contains at most one element. The condition q + ν(R) > 0 means that the L´evy process is not trivial, i.e. it contains a Wiener part or has nonzero jumps. Condition (8.2.40) means that σ is not degenerated. In concrete examples it can easily be fulfilled. The next result shows that condition (8.2.40) can be omitted if the model coefficients are deterministic. Theorem 8.2.13 Assume that (8.2.38) holds and Z is a one-dimensional L´evy process. If the processes α(t, T),

σ (t, T),

t ∈ [0, T ∗ ],

T∈J

are deterministic then the set of all martingale measures is either empty or a singleton. The preceding result can be generalized to the case d ≥ 2.

206

Arbitrage-Free HJM Markets

Theorem 8.2.14 Assume that J is a dense subset in [0, T ∗ ] and Z is a L´evy process in Rd . If the processes α(t, T),

σ (t, T),

t ∈ [0, T ∗ ],

T∈J

are deterministic and the spaces Et := span{(t, T), T ∈ J} are such that dim Et ≤ 1,

t ∈ [0, T ∗ ],

(8.2.41)

then the set of all martingale measures is either empty or a singleton. Condition (8.2.41) is automatically satisfied if d = 1 and then Theorem 8.2.14 reduces to Theorem 8.2.13. If d ≥ 2 then (8.2.41) is very restrictive. It tells that the vectors (t, T),

T∈J

in Rd for each t ∈ [0, T ∗ ] satisfy (t, T) = c(t, S),

T, S ∈ J

for some c = c(t), which means that it has an extremely degenerated structure.

9 Arbitrage-Free Forward Curves Models

We investigate the non-arbitrage problem for forward curves models moved by a Markov process. The main result of the chapter is the term structure equation. Some applications to special factor processes like multiplicative or Ornstein– Uhlenbeck processes are presented.

9.1 Term Structure Equation We now study the problem of absence of arbitrage in the model given by forward curves of the form f (t, T) = G(T − t, X(t)),

0 ≤ t ≤ T,

(9.1.1)

where G is a real-valued function defined on [0, +∞) × E and X(t), t ≥ 0 is a factor process taking values in a closed subset E of Rn . Recall that the short rate is given by R(t) = G(0, X(t)), and consequently

t

B(t) = e

0

t ≥ 0,

G(0,X(s))ds

t≥0

,

defines the bank account process. The corresponding bond curves P(t, T) = F(T − t, X(t)),

0≤t≤T

(9.1.2)

are described by the function F, defined on [0, +∞) × E, by F(T − t, x) := e−

T t

G(u−t,x)du

= e−

 T−t 0

G(s,x)ds

,

0 ≤ t ≤ T.

Thus for F and G, we have the important relations ∂F (u, x) G(u, x) = − ∂u , F(u, x)

(see also Section 7.5).

F(u, x) = e−

u 0

G(v,x)dv

,

u ≥ 0, x ∈ E

(9.1.3)

208

Arbitrage-Free Forward Curves Models

We start with a given process X on E and our aim is to determine forward curves G(·, x), x ∈ E and bond curves F(·, x), x ∈ E, for which the bond prices satisfy the following condition: For each T > 0 the discounted bond price process ˆ T) = P(t,

P(t, T) , B(t)

t ∈ [0, T]

(MP)

is a martingale. As a first approximation of the financial reality one regards the case of X being a Markov process say, with transition semigroup (Pt ) acting on the space Cb (E) of continuous bounded functions on E equipped with the supremum norm. If E is unbounded the elements of Cb (E) are required to have finite limits at infinity, and then Cb (E) is separable. The infinitesimal generator of (Pt ) and its domain will be denoted by A and D(A). The (MP) condition should hold for processes X x starting from an arbitrary x ∈ E. Let (Qu , u ≥ 0) be the discounted semigroup   u x Qu ϕ(x) := E ϕ(X x (u))e− 0 G(0,X (s))ds , u ≥ 0, x ∈ E ϕ ∈ Cb (E), also called the pricing semigroup. Theorem 9.1.1 (a) Condition (MP) is satisfied if and only if the function F satisfies one of the following two relations: Qu (F(r, ·))(x) = F(r + u, x),

r ≥ 0, u ≥ 0, x ∈ E,

(9.1.4)

or Qu 1(x) = F(u, x),

u ≥ 0, x ∈ E.

(9.1.5)

(b) If 1 is in the domain of the infinitesimal generator of Qu and G(0, ·) ∈ Cb (E), then the bond curves F(·, x), x ∈ E are solutions of the following evolution equation ∂ F(u, x) = AF(u, ·)(x) − G(0, x)F(u, x), ∂u

F(0, x) = 1(x),

u > 0, x ∈ E. (9.1.6)

Moreover, they have the following stochastic representation: F(u, x) = E[e− Proof

u 0

G(0,X x (s))ds

].

(9.1.7)

(a) Property (MP) means that, for any T > 0, the processes ˆ T) = e− P(t,

T t

G(u−t,X x (t))du

· e−

= F(T − t, X x (t)) · e−

t 0

t 0

G(0,X x (u))du

G(0,X x (u))du

,

0≤t≤T

9.1 Term Structure Equation

209

are martingales. So, for s < t,   t s x x E F(T − t, X x (t))e− 0 G(0,X (u))du | Fs = F(T − s, X x (s))e− 0 G(0,X (u))du . Equivalently, setting r = T − t,   t x E F(r, X x (t))e− s G(0,X (u))du | Fs = F(T − s, X x (s)).

(9.1.8)

Applying to the left side of (9.1.8) the Markov property of X, we get that Qt−s (F(r, ·))(X x (s)) = F(T − s, X x (s)),

(9.1.9)

or Qt−s (F(r, ·))(X x (s)) = F(r + t − s, X x (s)), which implies (9.1.4). Conversely, if (9.1.4) holds, then (9.1.9) is true as well. But (9.1.9) is equivalent to (9.1.8), so the martingale property takes place. Since F(0, x) = 1, x ∈ E, it follows from (9.1.4) that Qu 1(x) = F(u, x),

u ≥ 0, x ∈ E.

Conversely, if (9.1.5) holds, then by the semigroup property of Qt , t ≥ 0 (9.1.4) holds as well. (b) Since ϕ = Qt 1 is in the domain of the infinitesimal operator of the pricing semigroup, the limit Qh ϕ(x) − ϕ(x) h→0 h lim

(9.1.10)

exists uniformly in x. However, uniformly in x, E[ϕ(Xhx )(e−

h 0

G(0,Xsx )ds

− 1)]

h

−→ −ϕ(x)G(0, x), h→0

and therefore E[ϕ(Xhx )e− Qh ϕ(x) − ϕ(x) = h =

h

E[ϕ(Xhx )(e−

0

G(0,Xsx )ds

] − ϕ(x)

h h 0

G(0,Xsx )ds

− 1)]

h

+

E[ϕ(Xhx )] − ϕ(x) h

−→ −ϕ(x)G(0, x) + Aϕ(x). h→0

Since the semigroup (Qu ) defines a solution to the Cauchy problem (9.1.6), the assertion follows.

210

Arbitrage-Free Forward Curves Models To show the final part of the theorem, recall that the short-rate process Rx (t), ˆ u) is a martingale for any u > 0, we t ≥ 0, is exactly G(0, Xtx ), t ≥ 0. Since P(·, have the following relation u

ˆ u) = E[P(u, ˆ u)] F(u, x) = e− 0 G(v,x)dv = P(0,  u x   u  x = E e− 0 R (s)ds = E e− 0 G(0,X (s))ds ,

u ≥ 0, x ∈ E.

Remark 9.1.2 The equation (9.1.6) is called a term structure equation and can serve as a starting point for the construction of models satisfying (MP). Here it was established under stringent assumptions using the semigroup concepts. In specific cases, when, for instance, the process X is given by a stochastic equation, it can be derived directly under less restrictive conditions on G(0, ·) and with operators A extended to much greater sets. Theorem 9.1.1 is of great heuristical value as, in many cases, it allows to obtain formulae for F and G. We illustrate the strength of the term structure equation by deriving, formally, several explicit formulae for forward and bond curves.

9.1.1 Markov Chain and CIR as Factor Processes We start by characterizing bond curves in the case in which the factor process is a Markov chain or Cox–Ingersoll–Ross (CIR) process. Example 9.1.3 Let X be a Markov process with a finite state space E = {1, 2, . . . , M} and the generator matrix A = (aij )i,j∈E ,

where aij ≥ 0 for i = j and aii = − j=i ai,j . Denoting Fj (u) := F(u, j), j = 1, 2, . . . , M, we see that (9.1.6) amounts to the system ⎧

⎨F  (u) = M k=1 ajk Fk (u) + bj Fj (u), j ⎩F (0) = 1, j

with bj := down to

Fj (0) Fj (0)

= Fj (0), j = 1, 2, . . . , M. For M = 2 the preceding system boils ⎧  ⎪ ⎪ ⎪F1 (u) = (a11 + b1 )F1 (u) − a11 F2 (u), ⎨

F2 (u) = −a22 F1 (u) + (a22 + b2 )F2 (u), ⎪ ⎪ ⎪ ⎩ F1 (0) = F2 (0) = 1.

9.1 Term Structure Equation

211

The form of the solution (F1 , F2 ) depends on the roots of the characteristic polynomial ) * a11 + b1 − λ −a11 det = λ2 − λ(a11 + a22 + b1 + b2 ) + a11 b2 −a22 a22 + b2 − λ + a22 b1 + b1 b2 . If γ := (a11 + a22 + b1 + b2 ) − 4(a11 b2 + a22 b1 + b1 b2 )2 > 0, then the roots are given by   1 λ1 = a11 + a22 + b1 + b2 − (a11 + a22 + b1 + b2 )2 −4(a11 b2 + a22 b1 + b1 b2 ) , 2   1 λ2 = a11 + a22 + b1 + b2 + (a11 +a22 + b1 + b2 )2 −4(a11 b2 + a22 b1 + b1 b2 ) , 2 and corresponding eigenvectors of the system are  ∗  a11 −a22 + b1 −b2 − (a11 +a22 +b1 +b2 )2 −4(a11 b2 +a22 b1 +b1 b2 ) v1 = − ,1 , 2a22  ∗  a11 −a22 + b1 −b2 + (a11 +a22 +b1 +b2 )2 −4(a11 b2 +a22 b1 +b1 b2 ) ,1 . v2 = − 2a22 Consequently,



F1 (u) F2 (u)

= c1 v1 eλ1 u + c2 v2 eλ2 u ,

where the constants c1 , c2 are such that F1 (0) = F2 (0) = 1. If γ = 0 then there is only one eigenvalue λ0 of multiplicity 2. Then either there are two linearly independent eigenvectors w1 , w2 or there is only one eigenvector w0 . In the first case  F1 (u) = c1 w1 eλ0 u + c2 w2 eλ0 u , F2 (u) while in the second



F1 (u) F2 (u)

= c1 w0 ueλ0 u + c2 wˆ 0 eλ0 u ,

where w ˆ 0 is a certain vector. If γ < 0 then there are two complex eigenvalues λ1 = α+βi and λ2 = α−βi with corresponding complex vectors w1 and w¯1 - the conjugate ¯ 1 ) and w3 := 2i (w1 − w¯ 1 ) are real vectors and of w1 . Then w2 := 12 (w1 + w      F1 (u) = c1 w2 cos(βu) + w3 sin(βu) eαu + c2 w3 cos(βu) − w2 sin(βu) eαu . F2 (u)

212

Arbitrage-Free Forward Curves Models

Example 9.1.4 Let G(0, x) = x and the Markov process X(t) = R(t) be the short rate with CIR dynamics, i.e.  dX(t) = (aX(t) + b)dt + X(t)dW(t), t > 0. Then the term structure equation is of the form: ⎧ ⎨ ∂ F(t, x) = (ax + b) ∂F (t, x) + 1 x ∂ 2 F2 (t, x) − xF(t, x), ∂t ∂x 2 ∂x ⎩F(0, x)

= 1,

x ≥ 0.

One easily checks that its solution is F(t, x) = e−C(t)−D(t)x ,

t, x ≥ 0,

where the functions C and D are solutions of the equations C (t) = bD(t), D (t) = aD(t) − 1/2D2 (t) + 1, C(0) = D(0) = 0,

t ≥ 0.

The preceding explicit formulae will be rederived in the context of the general affine term structure model (see Section 10.2.2).

9.1.2 Multiplicative Factor Process Here we extend Proposition 2.4.15 to the continuous time setting. Namely, we determine bond curves in the model with random factor of the form dX(t) = aX(t)dt + bX(t)dZ(t),

X(0) = x > 0,

t > 0,

(9.1.11)

where Z is a real L´evy process, a, b ∈ R. By Theorem 4.4.6 we have that 1 2 )t+bZ(t)

X(t) = xe(a− 2 b where

$

S(t) :=

· S(t),

t > 0,

(1 + Z(s))e−Z(s) .

s∈[0,t]

Proposition 9.1.5 Let us assume that the forward curve model with factor X given by (9.1.11) and G(0, x) = −γ ln x, x > 0, γ = 0, satisfies (MP). Then the bond curves are given by γ

1 2 2 )u

F(u, x) = xγ u e 2 (a− 2 b

E[eγ (b

u 0

u

Z(s)ds+

0

ln(S(s))ds)

].

(9.1.12)

In particular, if Z is a Wiener process then γ

1 2 2 γ2 2 3 )u + 6 b u

F(u, x) = xγ u e 2 (a− 2 b

,

and if Z is a compound Poisson process with jumps greater than −1 then

(9.1.13)

9.1 Term Structure Equation

213

 +∞

−uν(R)+ γ2 (a− 21 )u2 −

F(u, x) = xγ u e

1 −γ u ln(1+y) −1)ν(dy) −∞ γ ln(1+y) (e

,

(9.1.14)

providing that b = 1 and γ < 0. Proof

By (9.1.7), F(u, x) = E[eγ = E[eγ

u 0

u 0

ln(X(s))ds

]

{ln x+(a− 21 b2 )s+bZ(s)+ln(S(s))}ds

],

which yields (9.1.12). If Z is a standard Wiener process, then the integral  u W(s)ds 0

has a zero mean Gaussian distribution. Its second moment equals ) *      2

u

E

W(s)ds

u

=E

u

W(s)ds

0

0



0

=

W(v)dv = E 

0

u  s

0

0

E[W(s)W(v)]dvds = 

u

vdv +

0

s

Hence E eγ b

u

W(s)ds

0

u u

0

u3 sdv ds = . 3

!

=e



u

W(s)W(v)dvds

0

u u

 =

u

0

(s ∧ v)dvds

0

γ 2 b2 u3 6

and (9.1.13) follows. If Z is a compound Poisson process Z(t) =

N(t) 

Yi ,

t ≥ 0,

Y=1

with Yi > −1 and b = 1, then 1

X(t) = xe(a− 2 )t

$

1

˜

(1 + Z(s)) = xe(a− 2 )t eZ(s) ,

s∈[0,t]

where ˜ := Z(t)

N(t) 

ln(1 + Yi ),

t≥0

i=1

is also a compound Poisson process. Then F(u, x) = E[eγ

u 0

ln(X(s))ds

γ

1

2

] = xγ u e 2 (a− 2 )u E eγ

u 0

˜ Z(s)ds

! .

214

Arbitrage-Free Forward Curves Models

To determine the latter expectation, we write the integral in exponent in the form  u  u  s  +∞ ˜Z(s)ds = ln(1 + y)π(dr, dy) ds 0

0

 =

0

 =

0

−1

0

+∞  u

+∞  +∞ −∞

0 +∞

−∞

ln(1 + y)1[0,u] (s)1[0,s] (r)π(dr, dy) ds

ln(1 + y)(u − r)π(dr, dy),

where π stands for the jump measure of Z. Now with the use of Theorem 6.6 in Peszat and Zabczyk [100], for γ < 0, we obtain ! !  u  +∞ u ˜ E eγ 0 Z(s)ds = E eγ 0 −∞ ln(1+y)(u−r)π(dr,dy) = e−

 u  +∞  0

−∞

 1−eγ (ln(1+y)(u−r)) drν(dy)

 +∞ −uν(R)− −∞

=e

1 −γ u ln(1+y) −1)ν(dy) γ ln(1+y) (e

.

Finally, we obtain (9.1.14). One can combine the results from Proposition 9.1.5 to characterize bond curves in the case in which Z is a finite activity process.

9.1.3 Affine Term Structure Model In this section we are concerned with the affine model P(t, T) = e−C(T−t)−D(T−t)R(t) ,

0 ≤ t ≤ T,

with the jump diffusion short rate dR(t) = F(R(t))dt + G(R(t−), dZ(t) ,

t ≥ 0,

(9.1.15)

where Z stands for a d-dimensional L´evy process with characteristic triplet (a, Q, ν). We derive conditions on C, D, F, G implied by the term structure equation. These conditions will be derived again and studied in detail in Chapter 10. Proposition 9.1.6 Let the bond curve F(·, ·) be of the form F(u, x) = e−C(u)−D(u)x ,

u, x ≥ 0,

and the short rate be given by (9.1.15). If (MP) is satisfied then   J G(x)D(u) = −C (u) − [D (u) − 1]x + D(u)F(x), u, x ≥ 0,

(9.1.16)

where J stands for the Laplace exponent of Z, providing that all terms in (9.1.16) are well defined.

9.1 Term Structure Equation

215

Proof We stress that in the proof the functions F(x), G(x) should be distinguished from the functions F(u, x), G(u, x). The bond curves satisfy   ∂F (u, x) = e−C(u)−D(u)x − C (u) − D (u)x . (9.1.17) ∂u Recall that the generator of R given by (9.1.15) is F(u, x) = e−C(u)−D(u)x ,

1 Af (x) = f  (x)F(x) + f  (x) G(x), a + f  (x) QG(x), G(x) 2  & ' f (x + G(x), y ) − f (x) − 1{|y|≤1} (y)f  (x) G(x), y ν(dy) + U

(see Section B.1.1). Let us now determine Afγ (x) for fγ (x) := e−γ x . Since fγ (x) = −γ e−γ x ,

fγ (x) = γ 2 e−γ x ,

we obtain

  1 Afγ (x) = −γ e−γ x F(x) + G(x), a + γ 2 e−γ x QG(x), G(x) 2  & ' e−γ (x+ G(x),y ) − e−γ x + 1{|y|≤1} (y)γ e−γ x G(x), y ν(dy). + U

Using the formula for J (see (5.2.9)), we have ! 1 Afγ (x) = e−γ x − γ F(x) − γ G(x), a + e−γ x Qγ G(x), γ G(x) 2  & ' e− γ G(x),y − 1 + 1{|y|≤1} (y) γ G(x), y ν(dy) + U

! = e−γ x − γ F(x) + J(γ G(x)) .

(9.1.18)

Since G(0, x) = x, the term structure equation (9.1.6) has the form AF(u, ·)(x) − xF(u, x) =

∂F (u, x). ∂u

(9.1.19)

By (9.1.18),

! AF(u, ·)(x) = e−C(u)−D(u)x −D(u)F(x) + J(D(u)G(x)) .

(9.1.20)

Taking into account (9.1.17) and (9.1.20), we can write (9.1.19) in the form !   e−C(u)−D(u)x −D(u)F(x) + J(D(u)G(x)) − x = e−C(u)−D(u)x −C (u) − D (u)x , which yields (9.1.16). Notice that (9.1.16) is identical with (10.2.4) in Theorem 10.2.1.

216

Arbitrage-Free Forward Curves Models

9.1.4 Ornstein–Uhlenbeck Factors In this section we characterize bond curves for factor models where X is the Ornstein–Uhlenbeck process driven by a L´evy process Z. Explicit formulae can be obtained when Z is a general real L´evy process or a multidimensional Wiener process. Proposition 9.1.7 Let G(0, x) = x and the short-rate process, identical with the factor process, be an Ornstein–Uhlenbeck process, dRx (t) = (a + bRx (t))dt + dZ(t),

Rx (0) = x,

t ≥ 0,

where a ∈ R+ , b ∈ R and Z is a real-valued L´evy process with Laplace exponent J. If (MP) is satisfied then     bu bu 2 bu 2 (9.1.21) F(u, x) = e− x(e −1)/b+a(e −1)/b −au/b eJ (e −1)/b −u/b . Proof

Since R is given by



dR (t) = xe + x

bt



t

b(t−s)

e

t

ads +

0

eb(t−s) dZ(s),

0

we have  t  t  t  s  t  s Rx (s)ds = x ebs ds + a eb(s−u) du ds + eb(s−u) dZ(u) ds, 0

0

0

0

0

0

and, by the stochastic Fubini theorem,  t  t  t  s  t  t x bs b(s−u) b(s−u) R (s)ds = x e ds + a e du ds + e ds dZ(u). 0

0

0

It follows from (9.1.7) that F(u, x) = E e−

u 0

Rx (s)ds

0

0

u

!

 u  !  u  s b(s−v)  u  u b(s−v)  bs dv)ds ds dZ(v) E e− 0 v e = e− x 0 e ds+a 0 ( 0 e  u  u u   u  s b(s−v) bs dv)ds J 0 ( v eb(s−v) ds)dv e . = e− x 0 e ds+a 0 ( 0 e Above we used the following elementary formula valid for an arbitrary continuous function h E[e−

t 0

h(s)dZ(s)

t

]=e

0

J(h(s))ds

,

t > 0.

This implies the result. Proposition 9.1.8 Let the factor process X x be an Ornstein–Uhlenbeck process dX x (t) = (a + AX x (t))dt + dW(t),

X x (0) = x ∈ Rd ,

(9.1.22)

9.1 Term Structure Equation

217

where, a ∈ Rd , A is a d × d-matrix and W is a Wiener process in Rd with identity covariance matrix. Let us assume that G(0, x) := Kx, x , where K is a positive d × d-matrix. The corresponding bond curve is of the form F(u, x) = e− M(u)x,x − b(u),x −c(u) , u ≥ 0, x ∈ Rd ,

(9.1.23)

where M(u) is a matrix solution of the matrix Riccati equation M  (u) = K + 2A∗ M(u) − 2M 2 (u),

u ≥ 0,

(9.1.24)

with M(0) = 0, b(u) is a solution of the equation b (u) = (A∗ − 2M(u))b(u) + 2M(u)a,

u ≥ 0,

(9.1.25)

with b(0) = 0 and c(u) satisfies 1 | b(u) |2 +TraceM(u), 2 The corresponding forward curve has the form c (u) = b(u), a −

u ≥ 0,

G(u, x) = M  (u)x, x + b (u), x + c (u), In particular, if a = 0 then

u

F(u, x) = e− M(u)x,x −

0

TraceM(s)ds

c(0) = 0.

(9.1.26)

u ≥ 0, x ∈ Rd .

, u ≥ 0, x ∈ Rd ,

and, for u ≥ 0, x ∈ Rd , G(u, x) = TraceM(u) + (K + 2A∗ M(u) − 2M 2 (u))x, x ). Proof

The generator of the process X x (t) acts on regular functions ϕ as follows:

1 ϕ(x) + a + Ax, Dϕ(x) (9.1.27) 2 (see Section B.1.1). Here , D denote the Laplacian and the gradient operator, respectively. We show that the term structure equation Aϕ(x) =

∂ F(u, x) = AF(u, x) − G(0, x)F(u, x), ∂u

F(0, x) = 1(x),

u > 0, x ∈ Rd (9.1.28)

is solved by the function (9.1.23) with functions M(u), b(u), c(u) satisfying M(0) = 0, b(0) = 0, c(0) = 0. It follows from (9.1.23) that

and

∂F (u, x) = −F(u, x)[ M  (u)x, x + b (u), x + c (u)] ∂u

(9.1.29)

  ∂F ∂ ∂ (u, x) = −F(u, x) M(u)x, x + b(u), x . ∂xi ∂xi ∂xi

(9.1.30)

218

Arbitrage-Free Forward Curves Models

By (9.1.30) we obtain, for i = 1, 2, . . . , d,   ∂2 ∂F(u, x) ∂ ∂ F(u, x) = − M(u)x, x + b(u), x ∂xi ∂xi ∂xi ∂xi2 ) * ∂2 ∂2 − F(u, x) M(u)x, x + 2 b(u), x ∂xi2 ∂xi ) 2 ∂ ∂ = F(u, x) M(u)x, x + b(u), x ∂xi ∂xi * ∂2 ∂2 − 2 M(u)x, x − 2 b(u), x . ∂xi ∂xi

(9.1.31)

It follows from (9.1.30) that DF(u, x) = −F(u, x)[D M(u)x, x + b(u)] = −F(u, x)[M(u)x + M ∗ (u)x + b(u)],

(9.1.32)

and from (9.1.31) that 

d  2  ∂ ∂ M(u)x, x + 2bi (u) M(u)x, x + b2i (u) ∂xi ∂xi i=1 8 ∂2 − 2 M(u)x, x ∂xi  = F(u, x) | M(u)x + M ∗ (u)x |2 + | b(u) |2 +2 b(u), M(u)x + M ∗ (u)x  − 2TraceM(u) . (9.1.33)

F(u, x) = F(u, x)

Taking into account (9.1.27), (9.1.29), (9.1.32) and (9.1.33), we insert (9.1.23) into (9.1.28). This yields −F(u, x)[ M  (u)x, x + b (u), x + c (u)] 1 = −F(u, x)[ (M(u) + M ∗ (u))x + b(u), a + Ax ]+ F(u, x)[| (M(u)+ M ∗ (u))x |2 2 + | b(u) |2 +2 b(u), (M(u) + M ∗ (u))x − 2TraceM(u)] − F(u, x) Kx, x . (9.1.34) Comparing quadratic terms yields 1 − M  (u)x, x = − (M(u) + M ∗ (u))x, Ax + (M(u) + M ∗ (u))x, (M(u) + M ∗ (u))x 2 − Kx, x .

9.1 Term Structure Equation

219

Since M ∗ (u) = M(u): M  (u)x, x = Kx, x + 2 A∗ M(u), x − 2 M 2 (u)x, x . This implies that M(u) satisfies (9.1.24). Similarly, comparing linear terms and free terms in (9.1.34) yields (9.1.25) and (9.1.26). The formula for G(u, x) follows from (9.1.3). For a = 0 we see from (9.1.25) that b(u) ≡ 0 and, consequently, by (9.1.26), u c(u) = 0 TraceM(s)ds. Remark 9.1.9 As a byproduct we derived the so-called Cameron–Martin formula (see Lipster and Shiryaev [89]), which in the case a = 0 takes the form E[e−

u 0

KX x (s),X x (s) ds

] = e−−

u 0

TraceM(s)ds

.

The matrix Riccati equation (9.1.24) is of great importance in the control theory (see Zabczyk [123]). Example 9.1.10

For d = 1 and K = k, we obtain √ 9 9 √ k k e2 2ku − 1 √ . M(u) = tanh( 2ku) = 2 2 e2 2ku + 1

(9.1.35)

10 Arbitrage-Free Affine Term Structure

This chapter is concerned with affine models of bond prices. Short rates are given as solutions of stochastic equations driven by a L´evy process or they can be general Markov processes. Conditions are found under which the discounted bond prices are local martingales. Generalizations of the Cox– Ingersoll–Ross and the Vasiˇcek short-rate models to the framework with L´evy factors are presented. The discrete time case was solved by Filipovi´c and Zabczyk [59], [58] and is discussed in Section 2.4.

10.1 Preliminary Model Requirements Bond prices in the affine term structure model are required to be of the form P(t, T) = e−C(T−t)−D(T−t)R(t) ,

0 ≤ t ≤ T,

(10.1.1)

where C, D are deterministic continuous functions and R stands for the short rate. To make the model regular (see Section 7.1), we assume that the short rate R(t) is a nonnegative process and that C(u), D(u) ≥ 0,

C(u) ≤ C(v), D(u) ≤ D(v),

0 ≤ u ≤ v.

(10.1.2)

As in the previous chapter, our goal is to figure out which affine models have the following martingale property: For each T > 0 the discounted bond price process ˆ T) = P(t,

P(t, T) , B(t)

t ∈ [0, T]

(MP)

is a local martingale, which implies that the model does not allow arbitrage. We want to find short-rate processes and functions C and D such that (MP) is satisfied. Since P(T, T) = 1, necessarily C(0) = 0,

D(0) = 0.

(10.1.3)

10.2 Jump Diffusion Short Rate

221

Writing the model in terms of forward rates we obtain from (10.1.1) that  T C(T − t) + D(T − t)R(t) = f (t, u)du, 0 ≤ t ≤ T. t

Thus if C, D are absolutely continuous then f (t, T) = C (T − t) + D (T − t)R(t),

0 ≤ t ≤ T.

(10.1.4)

In fact, we will require that C and D are continuously differentiable on [0, +∞). Since R(t) = f (t, t), we have C (0) = 0,

D (0) = 1.

(10.1.5)

We divide our considerations into two parts. First we assume that the short-rate process R solves a stochastic differential equation of the form dR(t) = F(R(t))dt + G(R(t−)), dZ(t) ,

t ≥ 0,

R(0) = x,

(10.1.6)

where Z is a L´evy process and F(·), G(·) are deterministic functions. If there exist required functions C, D such that the resulting affine model satisfies (MP), we say that (10.1.6) generates an affine model with (MP) property. We try to characterize all equations generating affine models with (MP) property. A complete answer is presented in the case in which Z is a one-dimensional martingale. For subordinators Z we consider special cases. For multidimensional noise we provide examples only. In the second part we will merely require that R is a general Markov process and present theory developed by Filipovi´c in [53]. This setting covers, in particular, short rates of the form (10.1.6), but characterization of models satisfying (MP) is not so explicit.

10.2 Jump Diffusion Short Rate Our basic requirement is that (10.1.6) is positive invariant, i.e. has a unique nonnegative strong solution for each initial condition R(0) = x ≥ 0. We assume also that supp{ν} ⊆ Rd+ ,

G(x) ∈ Rd+ ,

The first condition in (10.2.1) implies that  e− λ,y ν(dy) < +∞, |y|>1

for x ≥ 0.

λ ∈ Rd+ ,

and consequently the Laplace exponent J of Z (see (5.2.9)) given by    1 J(λ) = − a, λ + Qλ, λ + e− λ,y − 1 + 1{|y|≤1} λ, y ν(dy) 2 U

(10.2.1)

(10.2.2)

(10.2.3)

222

Arbitrage-Free Affine Term Structure

is well defined on Rd+ . On the other hand, for λ ∈ / Rd+ condition (10.2.2) enforces exponential moments of ν, which is rather restrictive. So, J(λ) does not exist in this case, in general. Our aim now is to find conditions for F, G and Z in (10.1.6) that guarantee that the affine model (10.1.1), with some functions C, D, satisfies (MP).

10.2.1 Analytical HJM Condition Our starting point in examining the (MP) condition for affine models is to show that under (MP) the coefficients F, G in (10.1.6) and the functions C, D in (10.1.1) satisfy a certain analytical equation that involves also the Laplace exponent of the noise process Z. This equation (see (10.2.4)) appears by treating the affine model as a particular form of the HJM model. Let us recall that (10.2.4) has already been derived, in a bit informal way, in the previous chapter as a consequence of the term structure equation (see (9.1.16)). We stress that although the condition (10.2.4) implies that the discounted bond prices are local martingales it does not imply the positivity of the short rate and does not exclude explosions of the rates. A related example will be discussed in Section 10.2.3. d The right directional derivative D+ w J(λ) of J at a point λ along a vector w ∈ R is defined by the formula D+ w J(λ) := lim h↓0

J(λ + hw) − J(λ) . h

Theorem 10.2.1 Assume that (10.1.6) is positive invariant, (10.2.1) is satisfied and that F, G are continuous. Let C, D be twice differentiable functions satisfying (10.1.3) and (10.1.5). Then the affine model (10.1.1) satisfies (MP) if and only if   J G(x)D(v) = −C (v) − [D (v) − 1]x + D(v)F(x), v ≥ 0, x ≥ 0. (10.2.4) If (10.2.4) holds and J is differentiable in the interior of Rd+ , then 2 1   DJ G(x)D(v) , D (v)G(x) = F(x)D (v) − C (v) − D (v)x

(10.2.5)

for v > 0 and x such that G(x) is in the interior of Rd+ . d If (10.2.4) holds and D+ w J(0) exists for any vector w ∈ R+ , then   F(x) = D+ G(x) J(0) + C (0) + D (0)x,

x ≥ 0.

(10.2.6)

Remark 10.2.2 Formula (10.2.5) is obtained by differentiating (10.2.4) over v outside of zero. The case v = 0 must be handled separately, because DJ(0) does not exist for a general L´evy process and therefore the right directional derivative of J in (10.2.6) is introduced.

10.2 Jump Diffusion Short Rate

223

Proof of Theorem 10.2.1 We convert the model to the HJM framework. In view of Proposition 7.5.1 the forward rate satisfies df (t, T) = −C (T − t)dt + D (T − t)dR(t) − R(t)D (T − t)dt = α(t, T)dt + σ (t, T), dZ(t) , where α(t, T) := F(R(t))D (T − t) − C (T − t) − D (T − t)R(t),

(10.2.7)

σ (t, T) := D (T − t)G(R(t−)).

(10.2.8)

It follows from Theorem 8.1.1 and Remark 8.1.3 that (MP) is satisfied if and only if   A(t, T) = J (t, T) . (10.2.9) By (10.2.7) and (10.2.8), for t < T,  T  T  A(t, T) = α(t, s)ds = F(R(t))D (s − t) − C (s − t) − D (s − t)R(t) ds t

t

= F(R(t))[D(T −t)−D(0)]−[C (T −t)−C (0)]− [D (T − t) − D (0)]R(t),  T  T (t, T) = σ (t, s)ds = D (s − t)G(R(t−))ds = G(R(t−))[D(T − t) − D(0)]. t

t

Taking into account (10.1.3) and (10.1.5) we can write (10.2.9) in the form   J G(R(t−))D(T − t) = −C (T − t) − [D (T − t) − 1]R(t) + D(T − t)F(R(t)) (10.2.10) for each T > 0, P-almost surely, for almost all t ∈ [0, T]. In (10.2.10) we can replace R(t−) by R(t) because R(t) = R(t−) for almost all t ≥ 0. Thus it is clear that (10.2.4) is sufficient for (10.2.10) to hold. To see that (10.2.4) is also necessary for (10.2.10) let us assume that for some x¯ ≥ 0 and v¯ > 0, J(G(¯x)D(¯v)) > −C (¯v) − [D (¯v) − 1]¯x + D(¯v)F(¯x). Then, by the continuity of J, F, G and D , there exists δ > 0 such that J(G(x)D(v)) > −C (v) − [D (v) − 1]x + D(v)F(x) for x ∈ ((¯x − δ) ∧ 0, x¯ + δ) and v ∈ (¯v − δ, v¯ + δ). Let us consider the solution R of (10.1.6) starting from x¯ and define τ := inf{t ≥ 0 :| R(t) − x¯ |> δ}.

224

Arbitrage-Free Affine Term Structure

For t ∈ (0, τ ) and T such that T − t ∈ (¯v − δ, v¯ + δ), we have J(G(R(t−))D(T − t)) > −C (T − t) − [D (T − t) − 1]R(t) + D(T − t)F(R(t)), which is a contradiction. Now we prove (10.2.6). From (10.2.4) we obtain   J G(x)D(v) − J(0) −C (v) − [D (v) − 1]x + D(v)F(x) = , v v

v > 0, x ≥ 0. (10.2.11)

But

  J G(x)D(v) − J(0) v

   J G(x)D(v) − J(0) D(v) = · −→ D+ G(x) J(0), v↓0 D(v) v

so taking into account (10.1.3) and (10.1.5) and letting v ↓ 0 in (10.2.11), we arrive at (10.2.6). Remark 10.2.3

Let us consider the short-rate equation

dR(t) = F(R(t))dt + G(R(t−)), dZ(t) ,

t ≥ 0,

R(0) = x,

where Z is a martingale and assume that this short rate generates an affine model satisfying (MP). Then (a) DJ(λ) exists for any λ in the interior of Rd+ and (10.2.5) holds, (b) the drift is a linear function, i.e. F(x) = C (0) + D (0)x,

x ≥ 0.

(10.2.12)

Point (b) follows from (10.2.6) in Theorem 10.2.1. In fact, if Z is a martingale then d D+ G(x) J(0) = 0. This and the differentiability of J in the interior of R+ are shown in the following proposition. Proposition 10.2.4 Let Z be a Rd -valued L´evy martingale with characteristic triplet (a, Q, ν). If the L´evy measure ν is concentrated on Rd+ then (a) the right directional derivative along a vector w, at point zero, exists and D+ w J(0) = 0, for any w ∈ Rd+ , (b) DJ(λ) exists for any λ in the interior Int(Rd+ ) of Rd+ . Proof

The Laplace exponent of Z equals    1 J(λ) = Qλ, λ + e− λ,y − 1 + λ, y ν(dy), 2 y≥0

λ ∈ Rd+

(10.2.13)

10.2 Jump Diffusion Short Rate

225

(see Proposition 5.3.4). Since 12 Qλ, λ is differentiable on Rd+ and disappears at zero, we examine the differentiability and existence of directional derivatives along w ∈ Rd+ at zero for the functions    e− λ,y − 1 + λ, y ν(dy), λ ∈ Rd+ , J 0 (λ) := y≥0,|y|≤1



 e− λ,y − 1 + λ, y ν(dy),



J 1 (λ) :=

λ ∈ Rd+ .

y≥0,|y|>1

(a) For h > 0 we have  J 0 (hw) =



 e−h w,y − 1 + h w, y ν(dy)

y≥0,|y|≤1



 =h

2 y≥0,|y|≤1

 e−h w,y − 1 + h w, y ( w, y )2 ν(dy) h2 ( w, y )2

(10.2.14)

and, by dominated convergence,  −h w,y    e − 1 + h w, y 1 2 ( w, y ) ν(dy) −→ ( w, y )2 ν(dy). h↓0 2 y≥0,|y|≤1 h2 ( w, y )2 y≥0,|y|≤1 So, it follows from (10.2.14) that 0 D+ w J (0) = lim h↓0



J 0 (hw) = 0. h

Since Z is integrable, |y|>1 | y | ν(dy) < +∞, and therefore we can split J 1 (hw):     1 −h w,y J (hw) = − 1 ν(dy) + h w, yν(dy) , h > 0. e y≥0,|y|>1

It follows that J 1 (hw) = h

y≥0,|y|>1



e−h w,y − 1 w, y ν(dy) + w, h w, y y≥0,|y|>1   w, y ν(dy) + w, −→ −

h↓0

y≥0,|y|>1

1 So, D+ w J (0) = 0 and the assertion (a) follows. (b) The following formulae are true  0 y(1 − e− λ,y )ν(dy), DJ (λ) = y≥0,|y|≤1

 DJ (λ) = 1

y≥0,|y|>1

y(1 − e− λ,y )ν(dy),

 yν(dy) y≥0,|y|>1

yν(dy) = 0.

y≥0,|y|>1

λ ∈ Int(Rd+ ), λ ∈ Int(Rd+ ).

226

Arbitrage-Free Affine Term Structure In fact 

− λ,y

| y(1 − e

 ) | ν(dy) ≤

y≥0,|y|≤1

| y λ, y | ν(dy) y≥0,|y|≤1

 ≤| λ |

| y |2 ν(dy) < +∞,

y≥0,|y|≤1

and 

− λ,y

| y(1 − e

λ ∈ Int(Rd+ ),

 ) | ν(dy) ≤

y≥0,|y|>1

y≥0,|y|>1

| y | ν(dy) < +∞, λ ∈ Int(Rd+ ).

10.2.2 Generalized CIR Equations In this section we characterize affine models satisfying (MP) with short rates of the form dR(t) = F(R(t))dt + G(R(t−))dZ(t),

t ≥ 0,

(10.2.15)

driven by a one-dimensional martingale Z. We present the results by formulating separately necessary conditions for equation (10.2.15) in Theorem 10.2.5 and sufficient conditions in Theorem 10.2.7, where also functions C, D from (10.1.1) are described. We consider two cases concerning the diffusion coefficient G, that is, either ∃ x¯ > 0 such that G(¯x) > 0,

G (¯x) = 0,

(10.2.16)

or G(x) ≡ σ

(10.2.17)

is a positive constant. In the first case, we obtain generalization of the well-known CIR equation while the second extends the Vasiˇcek model on nonnegative short rates. The requirement (10.2.16) is very natural; however, to simplify the presentation we will assume in addition that G(0) = 0.

(10.2.18)

The full proof can be found in Barski and Zabczyk [9], see also an earlier version [8]. Let us also notice that condition (10.2.17) is, in a sense, a complement of (10.2.16). Indeed, if there exists a number x¯ > 0 such that G(¯x) > 0, and for all such numbers G (¯x) = 0, then the function G should be a positive constant on [0, +∞). A particular role in the sequel will be played by the α-stable martingale Z α , where α ∈ (1, 2], with positive jumps only. Recall, see Example 5.3.6, that Z α with α ∈ (1, 2) has the form  t  +∞ Z α (t) = y π˜ (ds, dy), t ≥ 0, 0

0

10.2 Jump Diffusion Short Rate

227

and its L´evy measure is given by ν(dy) =

1

1[0,+∞) (y)dy, y1+α

α ∈ (1, 2).

Its Laplace exponent equals J(z) = cα zα , where cα :=

1 α(α−1) (2 − α)

α ∈ (1, 2),

. By Z α with α = 2 we denote the Wiener process.

Theorem 10.2.5 [Necessity] Let us assume that the equation (10.2.15), with functions F, G which are continuous on [0, +∞), is positive invariant. Let C, D be twice differentiable functions satisfying (10.1.3) and (10.1.5) such that the corresponding affine model (10.1.1) has the (MP) property. (I) If G is differentiable on (0, +∞) and G(¯x) > 0, G (¯x) = 0 for some x¯ > 0, then (a) Z = Z α is a stable L´evy process with index α ∈ (1, 2] with positive jumps only, (b) F(x) = ax + b with a ∈ R, b ≥ 0, x ≥ 0, 1 1 (c) G(x) = c α x α , c > 0, x ≥ 0. (II) If G is a positive constant σ , then (d) Z has no Wiener part, i.e. Q in (10.2.3) disappears,  +∞ (e) the martingale Z has positive jumps only and 0 yν(dy) < +∞,  +∞ (f) F(x) = ax + b, x ≥ 0, with a ∈ R, b ≥ σ 0 yν(dy). Remark 10.2.6 Part (II) of Theorem 10.2.5 indicates that in the short-rate equation with constant volatility the process Z must be a martingale of finite variation. Proof of Part (I) of Theorem 10.2.5 We present the proof under additional assumptions that Z has positive jumps only, G(x) ≥ 0, x ≥ 0 and G(0) = 0. Let us recall that positive jumps of Z imply that J(z) is well defined for z ≥ 0 and J  (z) is well defined for z > 0. The proof is divided into four steps. Let us assume that (MP) is satisfied. Step 1: We prove the linear form of F. Since Z is a martingale, we have that J  (0+) = 0 and, by Remark 10.2.3, we conclude that F(x) = C (0) + D (0)x := b + ax,

x ≥ 0.

(10.2.19)

To show that b ≥ 0, assume, by contradiction, that b < 0 and consider a solution R of (10.2.15) starting from 0. Since R is nonnegative, we have:  t  t R(t) = b ea(t−s) ds + ea(t−s) G(R(s−))dZ(s) ≥ 0, t ≥ 0. 0

0

228

Arbitrage-Free Affine Term Structure

Hence



t

−|b|

e−as ds +

0

or equivalently



t



t

e−as G(R(s−))dZ(s) ≥ 0,

t ≥ 0,

0

−as

e

 G(R(s−))dZ(s) ≥ |b|

0

t

e−as ds

t ≥ 0.

0

Since the preceding stochastic integral is a local nonnegative martingale starting from 0, it must be identically 0. Thus the process  t ea(t−s) ds, t ∈ [0, T] R(t) = b 0

is strictly negative and we have a contradiction. 1

Step 2: We prove that G(x) = ρx α , x ≥ 0 with some constants α, ρ. Formula (10.2.19) and (10.2.4) yield J(G(x)D(v)) = −C (v) − [D (v) − 1]x + D(v)[b + ax] = bD(v) − C (v) + [aD(v) − D (v) + 1]x,

x, v ≥ 0.

(10.2.20)

Since G(0) = 0 and J(0) = 0, bD(v) and (10.2.20) can be written in the form J(D(v)G(x)) = [aD(v) − D (v) + 1]x,

x, v ≥ 0.

(10.2.21)

Since Z is a martingale, J  (λ) exists for λ > 0 (see Proposition 10.2.4). Differentiation of (10.2.21) over x and v yields   (10.2.22) J  D(v)G(x) G (x)D(v) = aD(v) − D (v) + 1, x > 0, v > 0,   J  D(v)G(x) G(x)D (v) = [aD (v) − D (v)]x, x > 0, v > 0. (10.2.23) Since D is continuously differentiable and D (0) = 1, we can find ε > 0 such that D(v) > 0,

D (v) > 0,

v ∈ (0, ε).

Let us assume that the right side of (10.2.22) is zero for v ∈ (0, ε). Then D solves D (v) = aD(v) + 1,

D(0) = 0,

v ∈ (0, ε),

and, on the interval (0, ε), D(v) = 1a (eav − 1) if a = 0 or D(v) = v if a = 0. Since the left side of (10.2.22) equals zero and D(v) > 0 for v ∈ (0, ε) and G (¯x) = 0, we obtain J  (G(¯x)D(v)) = 0,

v ∈ (0, ε).

Hence J  disappears on some interval and consequently must disappear on [0, +∞) as an analytic function. Since J(0) = 0, this implies that J(λ) = 0 for λ ∈ [0, +∞),

10.2 Jump Diffusion Short Rate

229

which is impossible. It follows thus that the right side of (10.2.22) is different from zero for some v¯ ∈ (0, ε). This implies that D(¯v) = 0,

G (x) = 0,

J  (G(x)D(¯v)) = 0,

x > 0.

Hence, by (10.2.22), J  (G(x)D(¯v)) =

D(¯v)a + 1 − D (¯v) , G (x)D(¯v)

x > 0.

Putting this into (10.2.23) with v = v¯ yields D(¯v)a + 1 − D (¯v) · G(x)D (¯v) = x[D (¯v)a − D (¯v)], G (x)D(¯v)

x > 0,

and, consequently, x[D (¯v)a − D (¯v)] D(¯v) G(x) = · , G (x) D(¯v)a + 1 − D (¯v) D (¯v)

x > 0.

(10.2.24)

If D (¯v)a − D (¯v) = 0 then G(x) = 0, x > 0, which is impossible. Hence it follows that G (x) A¯ = , x > 0, G(x) x with A¯ := [D(¯v)a + 1 − D (¯v)]D (¯v)/[D (¯v)a − D (¯v)]D(¯v). The last identity can be written in the form d d ln(G(x)) = A¯ ln x, x > 0 dx dx and yields ¯

ln G(x) − ln xA = k,

x>0

with some constant k. Using this and the continuity of G at zero we obtain ¯

G(x) = ek xA ,

x ≥ 0.

Relabelling the constants yields the assertion. Step 3: We prove that the Laplace exponent of Z is of the form J(z) = βzα , z ≥ 0 with some constant β and α from the previous step. Putting v = v¯ from the previous step into (10.2.21) yields J(D(¯v)G(x)) = [aD(¯v) − D (¯v) + 1]x, Setting z := D(¯v)G(x) yields x = equation

z G−1 ( D(¯ v) ).

x ≥ 0.

To find G−1 (u), we have to solve the

G(x) = u,

230

Arbitrage-Free Affine Term Structure

which is equivalent to 1

ρx α = u. Thus we obtain −1

G

 α u (u) = x = , ρ

and consequently 

J(z) = [aD(¯v) − D (¯v) + 1] · G =

−1



z D(¯v)



[aD(¯v) − D (¯v) + 1] α z . (D(¯v)ρ)α

Finally we obtain J(z) = βzα , z ≥ 0 with β :=

[aD(¯v)−D (¯v)+1] . (D(¯v)ρ)α

Step 4: We show that the constant α from the previous steps belongs to (1, 2]. This means that Z is a stable L´evy process. Since J  (0+) = 0, we have that γ > 1. We show that the case γ > 2 can be excluded. Indeed, by Proposition 5.3.4, the Laplace exponent of a L´evy martingale has the form  +∞  −zy  1 2 e − 1 + zy ν(dy) J(z) = qz + 2 0  1  +∞  −zy  −zy   1 = qz2 + e − 1 + zy ν(dy) + e − 1 + zy ν(dy) 2 0 1  1 −zy  +∞  −zy  1 e − 1 + zy 2 = qz2 + z2 e − 1 + zy ν(dy), z ≥ 0. y ν(dy) + 2 2 (zy) 0 1 Since the function x→

e−x − 1 + x , x2

x≥0

is bounded, the measure y2 ν(dy) is finite on [0, 1] and  +∞  +∞  −zy  e − 1 + zy ν(dy) ≤ z yν(dy) 1

1

we see that J(z) ≤ cz2 + d for some positive constants c, d. Proof of Part (II) of Theorem 10.2.5 By elementary arguments, positivity of solutions to the equation (10.2.15) with G(x) ≡ σ implies that Z has no Wiener part and can have only positive jumps. Repeating the arguments from the proof of Part (I) one can show that that F(x) = ax + b, x ≥ 0, and b ≥ 0. We will establish now that



10.2 Jump Diffusion Short Rate  +∞ +∞ yν(dy) < +∞, b ≥ σ yν(dy).

0

231 (10.2.25)

0

Let π˜ be the compensated jump measure corresponding to the martingale Z. Then for  > 0,  t  +∞  t   t  +∞ Z(t) = yπ˜ (ds, dy) = yπ˜ (ds, dy) + yπ˜ (ds, dy) 0

0

0

0

0



(10.2.26)



+∞

= Z (t) + P (t) − t

yν(dy).

(10.2.27)



Here Z is a L´evy martingale with positive jumps bounded by , P is a compound Poisson process with the L´evy measure ν restricted to the interval [, +∞). For the solution R of the stochastic equation, starting from 0, we have for all t ∈ [0, T]:  t  t −at −as e ds + σ e−as dZ(s) e R(t) = b 0 0  +∞  t  t  t −as −as = (b − σ yν(dy)) e ds + σ e dZ (s) + e−as dP (s) ≥ 0. 0



0

0

(10.2.28)  +∞ If 0 yν(dy) = +∞, then by taking  close to 0, the number (b − σ  yν(dy)) can be made arbitrary small negative. The stochastic integrals with respect to Z , P are independent processes. The integral with respect to Z is less on [0, T] than a given number with positive probability. The integral with respect to P is 0 on [0, T] T], with positive with positive probability. Thus e−at R(t) is negative for some t ∈ [0,  +∞ probability, which is a contradiction. Now we show that b ≥ σ 0 yν(dy). In the opposite case we have that the difference  +∞ b−σ yν(dy)  +∞



is negative for sufficiently small  > 0 and decreases as  ↓ 0. It follows from the Chebyshev inequality that for any γ > 0 and t > 0 t t   t σ 2 E( 0 e−as dZ (s))2 σ 2 0 0 e−2as y2 dsν(dy) −as = → 0, P σ e dZ (s) > γ ≤ →0 γ2 γ2 0 and consequently  P(σ 0

t

e−as dZ (s) ≤ γ ) → 1. →0

Since the integral over Pε disappears with positive probability, we have by (10.2.28) that R(t) < 0, which is a contradiction.

232

Arbitrage-Free Affine Term Structure

Theorem 10.2.7

[Sufficiency]

(I) The equation 1

1

dR(t) = (aR(t) + b)dt + c α R(t−) α dZ α (t),

R(0) = x ≥ 0,

a ∈ R, b ≥ 0, c > 0 (10.2.29)

with α ∈ (1, 2] is positive invariant. There exist functions C, D which together with the solution of (10.2.29) form an affine model satisfying (MP). Moreover, D solves the equation D (v) = −ccα Dα (v) + aD(v) + 1,

v ≥ 0,

D(0) = 0

(10.2.30)

and C is given by C (v) = bD(v), v ≥ 0, C(0) = 0. (II) If G is a positive constant σ and (d), (e), (f ) in Theorem 10.2.5, hold, then the equation dR(t) = (aR(t) + b) + σ dZ(t), R(0) = x ≥ 0, t > 0

(10.2.31)

is positive invariant. The solution of (10.2.31) and functions C, D given by D (v) = D(v)a + 1, D(0) = 0,  +∞    C (v) = D(v) b − σ yν(dy) + 0

(10.2.32) +∞

(1 − e−σ D(v)y )ν(dy), C(0) = 0

0

(10.2.33) form an affine model satisfying (MP). Remark 10.2.8 Equation (10.2.29) defines the generalized CIR model of the short rate. For α = 2 one obtains the original CIR model introduced in Cox, Ingersoll and Ross [30]. In this case equation (10.2.30) becomes the Riccati equation, which can be solved explicitly. Properties of solutions of equation (10.2.29) have been an object of many investigations (see Kyprianou and Pardo [87], Kawazu and Watanabe [81], Ikeda and Watanabe [72], Li, Mytnik [88] and Jeanblanc, Yor and Chesney [78, p. 357–365] for a discussion of various topics and derivation of formulas of financial interests). Equation (10.2.31) generalizes the well-known Vasiˇcek model based on the Wiener process (see Vasiˇcek [121]). Passing from the Wiener process to the general L´evy martingale of finite variation in (10.2.31) enables preserving the positivity of the short-rate process that fails in the original Vasiˇcek model. Example 10.2.9 CIR model

Requiring C (0) = 0 and D (0) = 0 one arrives at the simplified dR(t) =

√  2c R(t)dW(t),

t ≥ 0.

10.2 Jump Diffusion Short Rate Then

⎧ ⎨C (v) = 0,

233

v ≥ 0,

⎩D (v) + cD2 (v) − 1 = 0,

v ≥ 0,

with initial conditions C(0) = 0, C (0) = 0, D(0) = 0, D (0) = 1 and some c > 0. The solutions C, D are given by C(v) ≡ 0 and √

1 e2 cv − 1 √ D(v) = √ , c e2 cv + 1

v ≥ 0.

Proof of Theorem 10.2.7 (I) It was shown in Fu and Li [60] that equation (10.2.29) has a unique nonnegative strong solution. Now we use Theorem 10.2.1 with J(λ) = 1 1 cα λα , F(x) = ax + b and G(x) = c α x α . Then (10.2.4) becomes α  1 1 cα c α x α D(v) = −C (v) − [D (v) − 1]x + D(v)[ax + b], x ≥ 0, v ≥ 0. Consequently, cα cxDα (v) = (aD(v) − D (v) + 1)x + bD(v) − C (v),

x ≥ 0,

v ≥ 0. (10.2.34)

Putting x = 0 yields bD(v) − C (v) = 0,

v ≥ 0,

which is the required formula for C. It follows from (10.2.34) that cα cDα (v) = aD(v) − D (v) + 1,

v ≥ 0,

which yields the equation for D. (II) Note that functions C, D should satisfy, for all x ≥ 0, v ≥ 0, the equation J(σ D(v)) = −C (v) − (D (v) − 1)x + D(v)(ax + b) = x(D(v)a − D (v) + 1) + D(v)b − C (v). Consequently, D (v) = D(v)a + 1,

D(0) = 0,



C (v) = D(v)b − J(σ D(v)),

C(0) = 0.

However, D(v)b − J(σ D(v))   = D(v) b − σ 0

and the proof is complete.

+∞

  yν(dy) + 0

+∞

(1 − e−σ D(v)y )ν(dy),

234

Arbitrage-Free Affine Term Structure

Remark 10.2.10 Both models described in Parts (I) and (II) of Theorem 10.2.7 are regular because the short rate R is positive and the functions C and D turn out to be positive and increasing. Moreover, D can be shown to be bounded. This follows by an elementary analysis of the differential equations for C and D.

10.2.3 Exploding Short Rates In this section we present an affine model satisfying (MP) with exploding short rates. In particular, after some time, bonds might be of value 0. If, for some t0 > 0, R(t) −→ +∞, t→t0

then the discounted bond prices with maturities T > t0 satisfy t

ˆ T) = e−C(T−t)−D(T−t)R(t)− P(t,

0

R(s)ds

−→ 0. t→t0

ˆ T) are zero for t ≥ t0 . As nonnegative local martingales reaching zero, P(t, Consequently, P(t, T) = 0, t ≥ t0 for T > t0 . We focus on the model of the form 1

dR(t) = (aR(t) + b)dt + (cR(t−) + d) β dZ(t),

R(0) = x ≥ 0,

t ≥ 0, (10.2.35)

where Z is a stable subordinator with index β ∈ (0, 1) and a, b, c, d are nonnegative 1 constants. Recall that the L´evy measure of Z is given by ν(dy) = y1+β 1(0,+∞) (y) and its Laplace exponent equals J(z) = −cβ zβ ,

z>0

(10.2.36)

, see Example 5.3.3. Since J  (0+) is infinite in this case, the with cβ := (1−β) β analysis differs from the case in which Z was a martingale (see Remark 10.2.3) and this is why we assume a specific form of drift and volatility in (10.2.35). Since both drift and volatility in (10.2.35) are locally Lipschitz functions, local solutions exist. As we will see, in this case, the affine models satisfying (MP) are regular but the short-rate process explodes in finite time. Proposition 10.2.11

Let the short rate be given by (10.2.35).

(a) Then R generates an affine model satisfying (MP) and the corresponding functions C, D are given by ⎧ ⎨C (v) = D(v)b + cβ dD(v)β , C(0) = 0, v ≥ 0, ⎩D (v) = c cD(v)β + aD(v) + 1, β

D(0) = 0,

v ≥ 0.

(b) The short rate R explodes in finite time with positive probability. Remark 10.2.12 By elementary analysis of the differential equations for C and D one gets that the resulting model is regular.

10.2 Jump Diffusion Short Rate Proof of Proposition 10.2.11

235

(a) Condition (10.2.4) takes the form

1 β

J((cx + d) D(v)) = −C (v) − [D (v) − 1]x + D(v)(ax + b),

x, v ≥ 0.

By (10.2.36): −cβ (cx + d)D(v)β = −C (v) − (D (v) − 1)x + D(v)(ax + b),

x, v ≥ 0, (10.2.37)

or equivalently, (cβ cD(v)β + aD(v) − D (v) + 1)x − C (v) + D(v)b + cβ dD(v)β = 0,

x, v ≥ 0.

As a consequence we obtain differential equations for C and D. (b) By a comparison argument for equation (10.2.35) driven by Z without small jumps (see e.g. the proof of Theorem 13.1.1) one can show that the solution of the equation (10.2.35), starting from x ≥ 0, dominates the solution of the equation 1

dRx (t) = (cRx (t−)) β dZ(t),

t ≥ 0, Rx (0) = x.

(10.2.38)

It is therefore sufficient to show explosion for the latter equation. In the proof of the following Lemma 10.2.13 we follow Ikeda and Watanabe [72]. Lemma 10.2.13 Assume that Rx is a solution of the equation (10.2.38) where Z is a stable subordinator with index β ∈ (0, 1) and c > 0. Then, for fλ (z) := e−λz , 1   x 1−β 1−β Pt fλ (x) := E e−λR (t) ; Rx (t) < +∞ = e−x(λ +tccβ (1−β)) , (10.2.39) where cβ = β1 (1 − β). In particular, P(Rx (t) < +∞) = lim Pt fλ (x) = e−x(tccβ (1−β))

1 1−β

λ↓0

Proof

< 1.

The generator L of Z is of the form  +∞   1 f (x + y) − f (x) 1+β dy, Lf (x) = y 0

and therefore the generator A of the process Rx is  +∞   1 1 f (x + (cx) β y) − f (x) 1+β dy Af (x) = y 0  +∞   1 1 dz f (x + z) − f (x) = z 1 1 1+β ( c1/β x1/β ) 0 cβ xβ  +∞   1 f (x + z) − f (x) 1+β dz = cx z 0

236

Arbitrage-Free Affine Term Structure

(see Section B.1.1). Fix x ≥ 0 and let us denote h(t, λ) := Pt fλ (x) = E(e−λR

x (t)

) = E(e−λR

x (t)

; Rx (t) < +∞).

Then ∂h (t, λ) = APt fλ (x) = Pt Afλ (x). ∂t

(10.2.40)

Since 

+∞ 

e−λ(x+z) − e−λx

Afλ (x) = x 0

 +∞   1  1 −λx −λz dz = cxe − 1 dz e z1+β z1+β 0

= −cβ λβ cxe−λx , so Pt Afλ (x) = −ccβ λβ E(Rx (t)e−λR

x (t)

) = ccβ λβ

∂ x E(e−λR (t) ). ∂λ

(10.2.41)

Therefore, by (10.2.40) and (10.2.41), ∂ ∂ h(t, λ) = ccβ λβ h(t, λ) ∂t ∂λ

(10.2.42)

h(0, λ) = e−λx .

(10.2.43)

The solution of the preceding equation is h(t, λ) = e−x(λ

1−β +tcc

β (1−β))

1 1−β

.

(10.2.44)

In fact, for h given by (10.2.44) we have  1 −1 !  ∂ 1−β ccβ , h(t, λ) = h(t, λ) − x λ1−β + tccβ (1 − β) ∂t  1 −1  ! ∂ 1−β λ−β , h(t, λ) = h(t, λ) − x λ1−β + tccβ (1 − β) ∂λ which shows that h solves (10.2.42). This way, formula (10.2.39) was established. The final part follows from (10.2.39).

10.2.4 Multidimensional Noise We pass now to affine models with multidimensional noise. We restrict our attention to a four-dimensional L´evy processes Z := (W, Z1 , Z2 , L) where W is a Wiener process, Z1 is a stable martingale with index γ1 ∈ (1, 2), Z2 is a stable subordinator with index γ2 ∈ (0, 1) and L is some subordinator. The noise processes are assumed

10.2 Jump Diffusion Short Rate

237

to be independent. Arbitrary dimensions can be treated similarly. We formulate conditions for the short-rate dynamics dR(t) = F(R(t))dt + G0 (R(t))dW(t) + G1 (R(t−))dZ1 (t) + G2 (R(t−))dZ2 (t)+dL(t), (10.2.45) to generate an affine model satisfying (MP). The independence of (W, Z1 , Z2 , L) implies that the Laplace exponent of Z has the form J(z0 , z1 , z2 , z3 ) = J0 (z0 ) + J1 (z1 ) + J2 (z2 ) + J3 (z3 ), where

 +∞ 1 2 γ (e−z1 y − 1 + z1 y)ν1 (dy) = c1 z11 , J0 (z0 ) = z0 , J1 (z1 ) = 2 0  +∞  +∞ γ2 −z2 y J2 (z2 ) = (e − 1)ν2 (dy) = c2 z2 , J3 (z3 ) = (e−z3 y − 1)m(dy). 0

0

where c1 , c2 are some constants. Moreover, ν1 (dy) =

1 y1+γ1

1[0,+∞) (y)dy, ν2 (dy) =

1 y1+γ2



+∞

(y ∧1)m(dy) < +∞.

1[0,+∞) (y)dy, 0

Our aim is to discuss the solvability of the equation (10.2.4), which is equivalent to (MP), i.e.   J G(x)D(v) = −C (v) − [D (v) − 1]x + D(v)F(x), v ≥ 0, x ≥ 0. (10.2.46) To simplify considerations we assume additionally that the drift F is linear. Theorem 10.2.14 Assume that F(x) = ax + b, x ≥ 0 with a ∈ R, b ≥ 0, G(x) := (G0 (x), G1 (x), G2 (x), 1) has nonnegative coordinates and that the equation (10.2.45) is positive invariant. Let C(·), D(·) be twice differentiable functions on [0, +∞). If R generates an affine model satisfying (MP) then √  1 1 G0 (x) = 2 e0 x + f0 , G1 (x) = (e1 x + f1 ) γ1 , G2 (x) = (e2 x + f2 ) γ2 x ≥ 0, (10.2.47) with some nonnegative constants e0 , e1 , e2 , f0 , f1 , f2 . Moreover, D (v) = 1 + D(v)a − D2 (v)e0 − c1 e1 Dγ1 (v) − c2 e2 Dγ2 (v),

D(0) = 0, (10.2.48)

C (v) = −J3 (D(v)) + D(v)b − D2 (v)f0 − c1 f1 Dγ1 (v) − c2 f2 Dγ2 (v),

C(0) = 0. (10.2.49)

Conversely, if G is of the form (10.2.47) and f0 = f1 = f2 = 0 then equation (10.2.45) generates an affine model satisfying (MP).

238

Arbitrage-Free Affine Term Structure

Proof It follows from (10.2.46) that the function         J0 D(v)G0 (x) + J1 D(v)G1 (x) + J2 D(v)G2 (x) + J3 D(v) ,

x, v ≥ 0,

is linear over x, hence its second derivative disappears. Thus we obtain 1      γ γ D2 (v) G20 (x) + c1 Dγ1 (v) G11 (x) + c2 Dγ2 (v) G22 (x) = 0 x, v ≥ 0. 2 Denoting z := D(v) we write the preceding relation in the form    1   γ γ (10.2.50) z2 G20 (x) + c1 zγ1 G11 (x) + c2 zγ2 G22 (x) = 0, x ≥ 0, 2 true for all z from an open interval. Since functions z2 , zγ1 , zγ2 are linearly independent, the coefficients in (10.2.50), should be functions of x identically equal zero. Hence 1 2 γ γ G (x) = e0 x + f0 , G11 (x) = e1 x + f1 , G22 (x) = e2 x + f2 , 2 0 which yields (10.2.47). Plugging the obtained formulas for coordinates of G into (10.2.46) and comparing coefficients of both sides we obtain (10.2.48) and (10.2.49). Remark 10.2.15 The equation for D has always a positive increasing solution. For C to be positive and increasing it is enough that f0 = f1 = f2 = 0. Then the resulting bond market is regular. It is also clear that if fi , i = 0, 1, 2 are large then C might be decreasing for large times. We leave aside the specification of those fi , i = 0, 1, 2 for which the model is regular.

10.3 General Markovian Short Rate It is of interest to characterize general Markov processes R and functions C, D such that the affine model (10.1.1) satisfies (MP) condition for all initial values R(0) = x ≥ 0. This problem was solved by Filipovi´c [53] in the class of nonnegative Markov Feller processes on [0, +∞) with c`adl`ag trajectories. We refer also to a seminal paper by Duffie, Filipovi´c and Schachermeyer [42] where a more general problem is treated. The short-rate process starting from x ≥ 0 will be often denoted by Rx (t), t ≥ 0 and (Pt ) stands for its transition semigroup, i.e. Pt ϕ(x) := E(ϕ(Rx (t))), x ≥ 0, t ≥ 0.

10.3.1 Filipovi´c’s Theorems The following theorems are due to Filipovi´c [53], who derived them using the results of Kawazu and Watanabe [81] on branching processes.

10.3 General Markovian Short Rate

239

Characterization of positive short-rate processes Rx , which generate affine models satisfying (MP), is formulated in terms of the generator of its transition semigroup (Pt ) (see Section B.1). The semigroup is regarded on the space C0 ([0, +∞)) of continuous functions on [0, +∞), vanishing at +∞. In the following theorem  is the linear hull of the set of all functions fλ , λ > 0, where fλ (x) := e−λx , x ≥ 0. Theorem 10.3.1 Assume that A is the infinitesimal operator of the transition semigroup (Pt ) on C0 ([0, +∞)) corresponding to a Feller process R that generates an affine term structure satisfying (MP). Then  is a core of the operator A and for f ∈  +∞ (f (x + y) − f (x))m(dy) Af (x) = αxf  (x) + (βx + b)f  (x) + 0  +∞   +x f (x + y) − f (x) − 1[0,1] (y)f  (x)y μ(dy), (10.3.1) 0

where μ and m are measures on (0, +∞) such that  +∞  +∞ (1 ∧ y) m(dy) + (1 ∧ y2 ) μ(dy) < +∞ 0

(10.3.2)

0

and α, b ∈ R+ , β ∈ R. In addition, either



+∞

(10.3.3)

yμ(dy) < +∞,

(10.3.4)

1

or



+∞

 yμ(dy) = +∞ and

1

for small  > 0, where

0



R0 (λ) = −αλ2 + βλ +

+∞



1 dy = +∞ R0 (y)

(1 − e−λy − λy1[0,1] (y)) μ(dy),

(10.3.5)

λ ≥ 0. (10.3.6)

0

Condition (10.3.5) implies that the Markov process R does not explode. The preceding theorem has the following converse, which in addition describes the form of the functions C, D. Theorem 10.3.2 If the infinitesimal operator of the transition semigroup (Pt ) of the process R is of the form (10.3.1) with parameters satisfying (10.3.2), (10.3.3) and (10.3.4) or (10.3.5) then the process R generates an affine term structure satisfying (MP) with functions C, D, determined by the following formulas:

240

Arbitrage-Free Affine Term Structure  t  D (t) = R(D(t)), D(0) = 0, C(t) = F(D(s))ds,

t ≥ 0,

(10.3.7)

0

where



+∞

R(λ) := −αλ2 + βλ + 1 +

(1 − e−λy − λy1[0,1] (y)) μ(dy)

(10.3.8)

0

= R0 (λ) + 1, λ ≥ 0,  +∞ F(λ) := bλ + (1 − e−λy ) m(dy),

λ ≥ 0.

(10.3.9)

0

We will not prove the theorems but provide some explanatory arguments in the following subsection.

10.3.2 Comments on Filipovi´c’s Theorems We give now some hints to the proofs of Theorem 10.3.1 and Theorem 10.3.2. Complete proofs can be found in [53]. Denote the discounted transition semigroup of the process R by (Qt )   t x Qt ϕ(x) = E ϕ(Rx (t))e− 0 R (s) ds , ϕ ∈ C0 ([0, +∞)), t ≥ 0, x > 0. The first part of the following result shows that an affine model satisfies (MP) if and only if the discounted semigroup transforms exponential functions on positive multiples of exponential functions. Proposition 10.3.3 For a Feller process R the requirement (MP) is satisfied with some continuous and increasing functions C, D, C(0) = D(0) = 0, if and only if there exist functions ϕ(·, ·), ψ(·, ·) such that Qt fλ (x) = e−ϕ(t,λ)−ψ(t,λ)x ,

t, λ, x ≥ 0.

(10.3.10)

If (10.3.10) is satisfied then for 0 ≤ t ≤ T, C(T) = C(T − t) + ϕ(t, D(T − t)),

(10.3.11)

D(T) = ψ(t, D(T − t)).

(10.3.12)

Conversely, if (10.3.11) and (10.3.12) hold for some functions ϕ, ψ then (MP) is satisfied. Proposition 10.3.4 1) Functions R, F, given by (10.3.8), (10.3.9) are related to ψ and φ, defined by (10.3.10), by the formulae: ∂ψ (0, λ) = R(λ), ∂t

∂φ (0, λ) = F(λ), ∂t

λ ≥ 0.

(10.3.13)

10.3 General Markovian Short Rate

241

2) For each λ ≥ 0 the functions ψ(·, λ) and ϕ(·, λ) satisfy the following ordinary differential equation: dψ (t, λ) = R(ψ(t, λ)), dt

t ≥ 0,

ψ(0, λ) = λ, ≥ 0,

(10.3.14)

dϕ (t, λ) = F(ψ(t, λ), t ≥ 0, ϕ(0, λ) = 0, λ ≥ 0. (10.3.15) dt Proof of Proposition 10.3.3 For each T > 0 the discounted bond price process ˆ T) := e− P(t,

t 0

Rx (s) ds

P(t, T),

0 ≤ t ≤ T,

x ∈ [0, +∞)

is, by (MP) condition, a martingale with respect to the underlying filtration. In particular, for t ∈ [0, T], ˆ T) = E(P(t, ˆ T) | F0 ). P(0, Thus, taking into account that Rx is Markovian and Rx (0) = x, we obtain   t x e−C(T)−D(T)x = E P(t, T)e− 0 R (s) ds | F0   t x x = E e−C(T−t)−D(T−t)R (t) e− 0 R (s) ds | F0 = e−C(T−t) Qt fD(T−t) (x).

(10.3.16)

Therefore, for λ = D(T − t), and x ≥ 0, Qt fλ (x) = e−ϕ(t,λ)−ψ(t,λ)x for some ϕ(t, λ) and ψ(t, λ). In fact let [0, λ0 ] := D([0, u0 ]), and D−1 : [0, λ0 ] → [0, u0 ] is such that D(D−1 (λ)) = λ,

λ ∈ [0, λ0 ],

and thus ψ(t, λ) = D(t + T − t) = D(t + D−1 (λ)), ϕ(t, λ) = C(t + D−1 (λ)) − C(D−1 (λ)),

λ ∈ [0, λ0 ].

However, for fixed t > 0 and x, the left side of (10.3.10) is the Laplace transform of a finite measure and therefore is an analytic function of λ > 0. Since ln Qt fλ (0) = −ϕ(t, λ), ln Qt fλ (1) = −ϕ(t, λ) − ψ(t, λ),

(10.3.17) λ ∈ [0, λ0 ],

(10.3.18)

242

Arbitrage-Free Affine Term Structure

functions ϕ(t, ·) and ψ(t, ·) have unique, analytic extensions to the whole interval [0, +∞). It is also clear that (10.3.10) must hold for all t ≥ 0, λ ≥ 0 and x ≥ 0. By (10.3.16), the functions C and D necessarily should be such that e−C(T)−D(T)x = e−C(T−t) e−ϕ(t,D(T−t))−ψ(t,D(T−t))x and therefore (10.3.11) and (10.3.12) are satisfied. Conversely, assume that (10.3.11) and (10.3.12) hold. We show that (MP) condition holds as well. In fact, if 0 ≤ s ≤ t ≤ T then   t E e−C(T−t)−D(T−t)R(t) e− 0 R(u) du | Fs  t  s = e−C(T−t) e− 0 R(u) du E e− s R(u) du e−D(T−t)R(t) | Fs = e−C(T−t) e− = e−C(T−t) e−

s 0

s 0

R(u) du

Qt−s fD(T−t) (R(s))

R(u) du −ϕ(t−s,D(T−t)) −ψ(t−s,D(T−t))R(s)

e

e

,

and we need to show that this expression is equal to e−C(T−s)−D(T−s)R(s) e−

s 0

R(u) du

.

Thus we should have that C(T − s) = C(T − t) + ϕ(t − s, D(T − t)),

(10.3.19)

D(T − s) = ψ(t − s, D(T − t)).

(10.3.20)

Changing, in (10.3.11) and (10.3.12), T to T − s and t to t − s we arrive at (10.3.19) and (10.3.20). Proof of Proposition 10.3.4 1) Let Aˆ be the generator of (Qt ). Then, at least for f in the domain of A, and thus for all fλ , λ > 0, ˆ (x) = Af (x) − xf (x), x ≥ 0. Af By (10.3.10)

 ∂ϕ d ∂ψ Qt fλ (x) = −Qt fλ (x) (t, λ) + (t, λ) x . dt ∂t ∂t

(10.3.21)

(10.3.22)

On the other hand d ˆ t fλ )(x), t ≥ 0, x ≥ 0, Qt fλ (x) = A(Q dt and, in particular, for the derivative at t = 0: d ˆ 0 fλ )(x) = A(f ˆ λ )(x), x ≥ 0. Qt fλ (x)|t=0 = A(Q dt

(10.3.23)

(10.3.24)

10.3 General Markovian Short Rate

243

Combining (10.3.22) and (10.3.24), ˆ λ )(x) = −fλ (x)( ∂ϕ (0, λ) + ∂ψ (0, λ) x). A(f ∂t ∂t

(10.3.25)

However, from the formula for A and (10.3.21)  +∞ ˆ λ )(x) = −fλ (x)[αxλ2 − λ(βx + b) − x + (e−λy − 1)m(dy) A(f 0  +∞ +x (e−λy − 1 + λy1[0,1] (y))(μ(dy))]. (10.3.26) 0

Comparing (10.3.25) and (10.3.26) and coefficients by x one gets the first part of the proposition. 2) The proof is based only on the semigroup property of the discounted semigroup and its specific representation (10.3.10). Indeed, it follows that for all λ ≥ 0 and t, s ≥ 0, Qt (Qs fλ )(x) = e−ϕ(s,λ) Qt fψ(s,λ) (x) = e−(ϕ(s,λ)+ϕ(t,ψ(s,λ)))−ψ(t,ψ(s,λ))x = Qt+s fλ (x) = e−ϕ(t+s,λ)−ψ(t+s,λ)x , and therefore ϕ(t + s, λ) = ϕ(s, λ) + ϕ(t, ψ(s, λ)),

(10.3.27)

ψ(t + s, λ) = ψ(t, ψ(s, λ)).

(10.3.28)

Thus the family of transformation ψ(t, ·), t ≥ 0 is a flow, so it defines a solution of an ordinary differential equation. In fact, for h > 0,  1  1 ψ(h + s, λ) − ψ(s, λ) = ψ(h, ψ(s, λ)) − ψ(s, λ) , h h so, since ψ(0, λ) = λ, we have ∂ψ ∂ψ (s, λ) = (0, ψ(s, λ)), ∂t ∂t

s ≥ 0, λ ≥ 0.

Similarly, since ϕ(0, λ) = 0, ∂ϕ dϕ (s, λ) = (0, ψ(s, λ)), ds ∂s and thus (10.3.13) follows.

s ≥ 0, λ ≥ 0,

244

Arbitrage-Free Affine Term Structure

10.3.3 Examples Theorem 10.3.2 allows to find, at least numerically, functions C and D in the formula for the bond prices. Since R(0) = 1, λ¯ := inf{λ : R(λ) ≤ 0} is a positive number or +∞. Define  x 1 ¯ dy, x ∈ [0, λ), G(x) := 0 R(y)

(10.3.29)

¯t := lim G(λ).

(10.3.30)

λ→λ¯ .

By (10.3.7) and (10.3.30) function D is the inverse of G: D(t) = G −1 (t), Example 10.3.5

Then λ¯ =

−β+

t < ¯t.

Let α > 0, β > 0 and R(λ) = 1 − αλ2 − βλ, λ ≥ 0.  √ −β+ β 2 +4α and ¯t = +∞. For x ∈ 0, , 2α

√

β 2 +4α 2α

⎡⎛

√

⎞

⎤ 2 + 4α β ⎠ ⎦  G(x) =  ln ⎣⎝ √ −β− β 2 +4α β 2 + 4α −β + β 2 + 4α x− 2α α

−β+

and

β 2 +4α 2α

√ D(t) = G

−1

(t) =

2(e

√ t

(e

β 2 +4α α

t

β 2 +4α α

− 1)β+

√

−x

− 1)

β 2 +4α−2

√

β+



,

0 < t < +∞.

β 2 +4α

If, in addition, 

+∞

F(λ) = bλ +

(1 − e−λy )m(dy),

0

then the short-rate process R(t) has generator  +∞ Af (x) = αxf  (x) + (βx + b)f  (x) + (f (x + y) − f (x))m(dy),

(10.3.31)

0

which corresponds to the solution of the following stochastic equation  dR(t) = (βR(t) + b)dt + 2αR(t)dW(t) + dL(t), R(0) = x ≥ 0 (see Section B.1.1). The process L can be an arbitrary subordinator.

10.3 General Markovian Short Rate

245

1 Example 10.3.6 Consider the case in which μ(dy) = y1+γ 1[0,+∞) (y)dy and γ ∈ (1, 2). Since  +∞ 1 1 (1 − e−λy − λy) 1+γ dy = − (2 − γ )λγ γ (γ − 1) y 0

(see Example 5.3.6),



R(λ) = 1 − αλ2 + βλ +

+∞

(1 − e−λy − λy1[0,1] (y))

0

1 y1+γ

dy

 +∞ 1 1 dy (2 − γ )λγ + λ γ (γ − 1) yγ 1  1  1 = 1 − αλ2 + β + λ− (2 − γ )λγ . γ −1 γ (γ − 1)

= 1 − αλ2 + βλ −

(10.3.32)

To calculate function G and its inverse G −1 is not a straightforward task. For instance, 1 , then if α = 0, β = − γ −1 R(λ) = 1 − cγ λγ , cγ =

¯ λ < λ,

1 − 1 (2 − γ ), λ¯ = (cγ ) 1+γ . γ (γ − 1)

Then ¯t defined in (10.3.30) equals +∞ and G −1 , and thus also D, can be found only numerically.

10.3.4 Back to Short-Rate Equations We are returning now to the problem of describing those stochastic equations that generate affine models satisfying (MP) condition, but using Theorem 10.3.1 and Theorem 10.3.2. We rederive some earlier results. We restrict our attention to equations of the form dR(t) = F(R(t)dt + G(R(t−))dZ(t),

R(0) = x ≥ 0,

(10.3.33)

where Z is a real-valued L´evy process with characteristics (a, q, ν). Since processes R should be nonnegative we will require that G is nonnegative and that jumps of Z are nonnegative as well. As before we denote by C0 ([0, +∞)) the space of all continuous functions defined on [0, +∞) and vanishing at infinity, equipped with the supremum norm. Let  be the linear hull of the set of all exponential functions fλ , λ > 0, where fλ (x) = e−λx , x ≥ 0. If ν is a measure concentrated on (0, +∞) and p a positive number then, as before, by νp we denote the image of ν by the linear transformation z → pz, z ≥ 0. We set

246

Arbitrage-Free Affine Term Structure

also ν0 = 0. The generators corresponding to stochastic equations are discussed in Appendix B (see Proposition B.1.2). The conditions that appear in the next proposition play a similar role in the theory of the affine term structure as the analytical HJM conditions of the Proposition 10.2.1. Proposition 10.3.7 Assume that the transition semigroup (Pˆ t ) of the solution of the equation (10.3.33) is strongly continuous and that the domain of its generator Aˆ contains . It coincides with the transition semigroup (Pt ) having the generator specified in Theorem (10.3.1) if and only if xμ + m = νG(x) , 

1

1 2 qG (x) = αx, x ≥ 0, 2

(10.3.34)

¯ ym(dy) = F(x) + F(x) + G(x)a,

(10.3.35)

' 1[0,1] (G(x)y) − 1[0,1] (y) G(x)yν(dy).

(10.3.36)

βx + b + 0

where ¯ F(x) =



& (0,+∞)

Remark 10.3.8

It should be noticed that the function F¯ is well defined. In fact

¯ (a) F(x) = 0 if G(x) = 0,  1 ¯ (b) F(x) = 1G(x) yν(dy) if G(x) ∈ (0, 1], 1 ¯ (c) F(x) = − 1 yν(dy) if G(x) > 1. G(x)

Proof of Proposition 10.3.7 The generator of the solution of the stochastic equation (10.3.33) (see Section B.1.1) has the form ˆ (x) = f  (x)(F(x) + G(x)a) + 1 qf  (x)G2 (x) Af 2  ' & f (x + G(x)y) − f (x) − 1[0,1] (y)G(x)y ν(dy). +

(10.3.37)

(0,+∞)

Equivalently, ˆ (x) = f  (x)(F(x) + G(x)a) + 1 qf  (x)G2 (x) Af 2  ' & f (x + G(x)y) − f (x) − 1[0,1] (G(x)y)f  (x)G(x)y ν(dy) + (0,+∞)

+ f  (x)



&

'

1[0,1] (G(x)y) − 1[0,1] (y) G(x)yν(dy),

(0,+∞)

(10.3.38) (10.3.39)

10.3 General Markovian Short Rate

247

and thus ˆ (x) = f  (x)(F(x) + G(x)a) + 1 qf  (x)G2 (x) Af 2  ' & ¯ f (x + y) − f (x) − f  (x)1[0,1] (y)y νG(x) (dy) + f  (x)F(x). + (0,+∞)

(10.3.40) Note that if G(x) = 0 then νG(x) = 0, the preceding integral makes sense and ¯ F(x) = 0. ˆ Rewriting this formula we arrive at the following formula for A: 1 ˆ (x) = f  (x)(F(x) + F(x) ¯ Af + G(x)a) + qf  (x)G2 (x) 2  ' & (10.3.41) f (x + y) − f (x) − f  (x)(1[0,1] (y)y) νG(x) (dy). + (0,+∞)

On the other hand the operator A is of the form  +∞ (f (x + y) − f (x))m(dy) Af (x) = αxf  (x) + (βx + b)f  (x) + 0  +∞   +x f (x + y) − f (x) − 1[0,1] (y)f  (x)y μ(dy), (10.3.42) 0

and can be written as:

 1 ym(dy))f  (x) Af (x) = αxf  (x) + (βx + b + 0  +∞   + f (x + y) − f (x) − 1[0,1] (y)f  (x)y (xμ(dy) + m(dy)). (10.3.43) 0

The transition semigroups (Pˆ t ) and (Pt ) coincide if their generators coincide on the set , that is, if and only if: ˆ λ (x), λ > 0, x ≥ 0. (10.3.44) Afλ (x) = Af By direct calculations one gets:  1 & ym(dy)] + λ2 αx Afλ (x) = fλ (x) − λ[βx + b + 0  +∞ ' −λy + [e − 1 + λ1[0,1] (y)y](xμ + m)(dy) ,

(10.3.45)

0

& 1 ˆ λ (x) = fλ (x) − λ[F(x) + F(x) ¯ Af + G(x)a] + λ2 qG2 (x) 2  +∞ ' [e−λy − 1 + λ1[0,1] (y)y]νG(x) (dy) . +

(10.3.46)

0

Thus the identity(10.3.44) holds if and only if for each x ≥ 0, the following Laplace exponent:

248

Arbitrage-Free Affine Term Structure  1 ym(dy)] + λ2 αx − λ[βx + b + 0  +∞ + [e−λy − 1 + λ1[0,1] (y)y](xμ + m)(dy), λ > 0,

(10.3.47)

0

coincides, as a function of λ, with the following one: 1 ¯ − λ[F(x) + F(x) + G(x)a] + λ2 qG2 (x) 2  +∞ −λy [e − 1 + λ1[0,1] (y)y]νG(x) (dy), λ > 0. +

(10.3.48)

0

However, the Laplace exponents, say,  +∞ 2 J(λ) = aλ + bλ + [e−λ y − 1 + λ1[0,1] (y)y]ζ (dy), λ ≥ 0, −∞

uniquely determine the parameters a, b, ζ and consequently the specified in the proposition identities hold. Remark 10.3.9 Proposition 10.3.7 implies that if the stochastic equation generates a process for which the (MP) property holds, the functions F, G and the characteristics (a, Q, ν) of the process Z should solve the functional identities (10.3.34), (10.3.35) and (10.3.36). Remark 10.3.10 In a similar way, one can derive analogical conditions in the multidimensional case, when dR(t) = F(R(t)dt + G(R(t−)), dZ(t) ,

R(0) = x ≥ 0

(10.3.49)

and the process Z has characteristics (a, Q, ν). Namely, the conditions (10.3.34), (10.3.35) and (10.3.36), become: 1 1/2 |Q G(x)|2 = αx, x ≥ 0, 2

(10.3.50)

¯ ym(dy) = F(x) + F(x) + G(x), a ,

(10.3.51)

xμ + m = νG(x) , 

1

βx + b + 0

where ¯ F(x) =



' 1[0,1] (| G(x), y |) − 1[0,1] (|y|) G(x)yν(dy),

&

(10.3.52)

(0,+∞)

and νp , where p ∈ Rd is the image of the measure ν by the map that transforms z ∈ Rd onto p, z . A precise description of all solutions of the equations is an open problem. To illustrate the applicability of Proposition 10.3.7 we consider a very special case.

10.3 General Markovian Short Rate

249

Proposition 10.3.11 Assume that ν and μ are locally finite measures on (0, +∞), μ has a continuous density g and G(·) is a continuous, invertible transformation of (0, +∞) onto (0, +∞), such that for all ϕ bounded with compact support in (0, +∞) and all x > 0  +∞  +∞ x ϕ(y)μ(dy) = ϕ(G(x)y)ν(dy). (10.3.53) 0

0

Then ν has also density, say h, and for some γ ∈ R, γ = 0, and positive constants c1 , c2 g(x) = c1

1 x1+γ

h(x) = c2

,

1 x1+γ

1

G(x) = (x/c2 ) γ ,

,

x ≥ 0.

Proof The assumption (10.3.53) is an explicit formulation of the fact that the measure xμ, for any x, is the image of the measure ν by the linear transformation y → G(x)y, y > 0. Thus if one of the measures μ or ν has density, the same follows for the other one. Note that  +∞  +∞ ϕ(G(x)y)ν(dy) = ϕ(G(x)y)h(y)dy 0

0

 =



+∞

ϕ(z)h 0

z G(x)



and we have that for all x > 0,  z 1 h = xg(z), G(x) G(x) In particular, for x = 1 we obtain  1 z = g(z), h G(1) G(1)

1 dz = x G(x)



+∞

ϕ(z)g(z)dz, 0

z > 0.

(10.3.54)

z > 0.

To simplify calculations we assume that G(1) = 1. Then h(z) = g(z), z > 0 and  z 1 g = xg(z), x, z > 0. (10.3.55) G(x) G(x) For given z > 0, let x be such that G(x) = z. Then g(z) = g(1)

1 zG−1 (z)

,

z > 0,

which, by (10.3.55), for any x, z > 0 gives g(1)

1 z −1 z G(x) G ( G(x) )

Hence, for all x, z > 0, G−1

·

1 1 = xg(1) −1 . G(x) zG (z)

 z  1 = G−1 (z), G(x) x

(10.3.56)

250

Arbitrage-Free Affine Term Structure

and, consequently, G−1

z y

=

G−1 (z) , G−1 (y)

y, z > 0.

Rearranging terms yields z z  G−1 (y) = G−1 (z) = G−1 ·y , G−1 y y

y, z > 0.

Finally, we have that for any a, b > 0, G−1 (a)G−1 (b) = G−1 (a · b). It is well known that if ψ is a continuous positive function on (0, +∞) such that ψ(a · b) = ψ(a)ψ(b),

a, b > 0,

then ψ(a) = aγ , a > 0 with some γ ∈ R. It follows that G−1 (a) = aγ , a > 0, γ ∈ R 1 , x > 0. and, by (10.3.56), g(x) = c1 x1+γ Assume that m = 0, the measures μ, ν have densities and G is a one-to-one transformation of R+ and  t 1  t  +∞ Z(t) = yπ˜ (ds, dy) + yπ(ds, dy). 0

0

0

1

Let us see to what stochastic equations we arrive by applying Proposition 10.3.7 and Proposition 10.3.11. We know that, up to positive multiplicative constants, μ(dx) = g(x)dx = Example 10.3.12

1 x1+γ

dx,

ν(dx) = h(x)dx =

1 x1+γ

dx,

G(x) =

1 x1+γ

.

If γ ∈ (0, 1), by direct calculations  1 1 1 ¯ − xγ , F(x) =x γ (1 − γ ) 1−γ  F(x) = x β −

1 γ (1 − γ )



1

+ b + xγ

1 . 1−γ

This leads to the equation   1 1 1 1 + b dt + (R(t)) γ dt + (R(t−)) γ dZ(t). dR(t) = R(t) β − γ (1 − γ ) 1−γ However, ˜ − Z(t) = Z(t)

t , 1−γ

(10.3.57)

10.3 General Markovian Short Rate where Z˜ is the α-stable subordinator, so we arrive at the equation   1 1 ˜ + b dt + (R(t−)) γ dZ(t). dR(t) = R(t) β − γ (1 − γ ) Example 10.3.13

Similarly, if γ ∈ (1, 2),  1 1 1 ¯F(x) = −x + xγ . γ (γ − 1) γ −1

251

(10.3.58)

(10.3.59)

Moreover, ˜ + Z(t) = Z(t)

t , γ −1

(10.3.60)

˜ is the γ -stable martingale. We therefore arrive at the equation where Z(t)   1 1 ˜ (10.3.61) + b dt + (R(t−)) γ dZ(t). dR(t) = R(t) β + γ (γ − 1)

11 Completeness

In this chapter we consider the completeness problem for models where the discounted bond prices are local martingales. Most models turn out not to be complete because completeness forces very strong assumptions on jumps of the underlying L´evy process. Therefore we also describe conditions for approximate completeness, which are less demanding. Our considerations cover the models discussed in the previous sections, i.e. HJM models and factor models including also affine models.

11.1 Problem of Completeness As we already know (see Section 7.3), the fair price of an FT ∗ -measurable contingent claim X is a number x ∈ R such that  T∗ (ϕs , dP(s, ·)), P − a.s. (11.1.1) X =x+ 0

holds for some self-financing strategy (ϕt ). Recall that (ϕt ) is a predictable process taking values in the set M = M([0, +∞)) of finite signed measures on [0, +∞) equipped with the weak topology. Since (ϕ) is assumed to be self-financing, (11.1.1) can be written in the following equivalent form: Xˆ = x +



T∗

ˆ ·)), (ϕt , dP(t,

P − a.s.

(11.1.2)

0

ˆ ·) stand for the discounted claim and (see Section 7.2 for details), where Xˆ and P(t, the discounted bond curve, respectively. In fact, the representation (11.1.2) is more convenient for analysis as it allows one to ignore the self-financing condition. If the self-financing condition is violated, one can modify the measures ϕ(t, ·), t ∈ [0, T ∗ ], at zero so that the modification is self-financing and (11.1.2) still holds for it (see Corollary 7.2.3). So, as we prefer to work with (11.1.2) rather than with (11.1.1),

11.2 Representation of Discounted Bond Prices

253

conditions for the claim X to be replicated will be formulated in terms of its ˆ discounted value X. Recall, if (11.1.2) is satisfied for some x ∈ R and an admissible M-valued strategy ˆ is called attainable (see Section 7.3). Completeness of the bond (ϕt ), then X, or X, market with a given filtration (Ft , t ∈ [0, T ∗ ]) means that each Xˆ ∈ L∞ (, FT ∗ , P) is attainable, where L∞ (, FT ∗ , P) stands for the set of all bounded FT ∗ -measurable random variables. If this is the case, we will also say that the market is complete in the class Xˆ ∈ L∞ (, FT ∗ , P). Complete markets are important because in such markets the price of each bounded claim is defined in a natural way and given by the formula (7.3.8). As we shall see completeness takes place for a rather narrow class of models. One is therefore using a more general concept of approximate completeness where Xˆ is approximated in the mean-square sense. One is looking for a sequence (xn , ϕn ) such that  T∗  2 !  ˆ ·)) − Xˆ (ϕtn , dP(t, = 0. (11.1.3) lim E xn + n→+∞

0

Since bounded random variables are dense in L2 (, FT ∗ , P), we can examine (11.1.3) for Xˆ ∈ L2 (, FT ∗ , P). The limit of {xn }, if exists, can be regarded as a generalized price of X. The completeness problem will be examined for models defined on a probability space (, F, P) equipped with filtration generated by a L´evy process Z in specific classes of admissible strategies (see Section 11.3). We assume that the original measure P is a martingale measure and thus the following assumption holds. For each T > 0 the discounted bond price process P(t, T) ˆ T) = , t ∈ [0, T] (MP) P(t, B(t) is a local martingale. Our aim is to describe conditions for completeness and approximate completeness for HJM models, affine models and factor models.

11.2 Representation of Discounted Bond Prices With the use of (MP) we describe first discounted bond prices and introduce the class of admissible strategies in the following sections. Let (a, Q, ν) be the characteristic triplet of the underlying U = Rd -valued L´evy process Z with the L´evy–Itˆo decomposition:  t  t Z(t) = at + W(t) + y π˜ (ds, dy) + y π˜ (ds, dy), t ≥ 0, 0

|y|≤1

0

|y|>1

254

Completeness

where W is a Wiener process and π˜ the compensated jump measure of Z. Since, by (MP), the discounted price of any T-bond is a local martingale, it can be, by Theorem 6.1.1, represented in the form  ˆ T) = 1 (t, T), dW(t) + dP(t, 2 (t, T, y)π˜ (dt, dy), t ∈ [0, T ∗ ], T > 0 U

(11.2.1) for some processes (1 (t, T)) ∈ (U) and (2 (t, T, y)) ∈ 1,2 , thus satisfying, for each T > 0,  T∗ | 1 (s, T) |2 ds < +∞, 0



T∗ 0

 

 | 2 (s, T, y) |2 ∧ | 2 (s, T, y) | ds ν(dy) < +∞.

U

ˆ T) is constant for t > T, it follows that Since P(t, 1 (t, T) = 0,

2 (t, T, y) = 0,

t > T.

Now we derive explicit forms of 1 (t, T) and 2 (t, T, y) in the decomposition (11.2.1) for models studied in the sequel. Proposition 11.2.1 Let us assume that in the following models (a), (b) and (c) the condition (MP) is satisfied. Let 1 (t, T) and 2 (t, T, y) be given by (11.2.1). (a) For HJM models df (t, T) = α(t, T)dt + σ (t, T), dZ(t) , ˆ 1 (t, T) = −P(t−, T)(t, T),

where (t, T) :=

T t∧T

ˆ 2 (t, T, y) = P(t−, T)[e− (t,T),y − 1], (11.2.2)

σ (t, u)du.

(b) For affine models P(t, T) = e−C(T−t)−D(T−t)R(t) , with short rate dR(t) = F(R(t))dt + G(R(t−)), dZ(t) : ˆ 1 (t, T) = −P(t−, T)D(T − t)G(R(t−)), ˆ 2 (t, T, y) = P(t−, T)[e− D(T−t)G(R(t−)),y − 1].

(11.2.3)

(c) For factor models f (t, T) = G(T − t, X(t)), with factor dX(t) = a(X(t))dt + b(X(t−)), dZ(t) :  T−t ˆ 1 (t, T) = −P(t, T)b(X(t−)) Gx (u, X(t−))du,

(11.2.4)

  T−t  ˆ T) e 0 {G(s,X(t−))−G(s,X(t)+y)}ds − 1 . 2 (t, T, y) = P(t,

(11.2.5)

0

11.2 Representation of Discounted Bond Prices

255

Proof

(a) It follows from Proposition 8.1.7 that    t 1 ˆP(t, T) = P(0, ˆ T) + ˆP(s−, T) Q(s, T), (s, T) − A(s, T) − (s, T), a ds 2 0 

t

−

ˆ P(s−, T) (s, T), dW(s) −



0

+

ˆ P(s−, T) (s, T), dZ0 (s)

0

 t 0

t

U

! ˆ P(s−, T) e− (s,T),y − 1 + 1{|y|≤1} (s, T), y π(ds, dy).

ˆ T) is a local martingale, we can compensate the Since we know that P(t, last integral and then cancel all dt-integrals. This yields directly the required representation. (b) The affine model can be written as an HJM model with coefficients α(t, T) := F(R(t))D (T − t) − C (T − t) − D (T − t)R(t), σ (t, T) := D (T − t)G(t, R(t−)) (see Proposition 7.5.1). It follows that  T  (t, T) = σ (t, u)du = G(R(t−)) t∧T

T

D (u − t)du = G(R(t−))D(T − t∧ T).

t∧T

Now we can use 11.2.2. (c) The required representation can be obtained by repetitive application of the Itˆo formula. Recall that the short rate is given by R(t) = f (t, t) = G(0, X(t)), so ˆ T) = e− P(t,

t 0

R(u)du

P(t, T) = A(t)C(t),

where, for fixed T, A and C are given by A(t) := e−

t 0

R(u)du

= e−

t 0

G(0,X(u))du

,

C(t) := e−

T t

G(u−t,X(t))du

.

By the Itˆo formula ˆ T) = A(t)dC(t) + C(t)dA(t), dP(t,

(11.2.6)

so we need to determine the dynamics of A and C. Clearly, dA(t) = −A(t)G(0, X(t))dt. Since C(t) = e−Y(t) with  T−t G(s, X(t))ds = h(t, X(t)), Y(t) := 0



T−t

where h(t, x) :=

G(s, x)ds, 0

256

Completeness

by the Itˆo formula dY(t) = aY (t)dt + bY (t), dZ(t) + {h(t, X(t)) − h(t, X(t−)) − bY (t), Z(t) }, with bY (t) := hx (t, X(t−))b(t) and some coefficient aY (t). Here we write b(t) := b(X(t−)) for short. It follows, in particular, that Y(t) = h(t, X(t)) − h(t, X(t−)).

(11.2.7)

Consequently, again by the Itˆo formula, 1 dC(t) = −C(t−)dY(t) + C(t−)b2Y (t)dt + C(t−){e−Y(t) − 1 + Y(t)} 2   1 = −C(t−) aY(t) − b2Y (t) dt + bY (t), dZ(t) 2 ! + {− bY (t), Z(t) − e−Y(t) + 1} . Coming back to (11.2.6) we obtain   1 ˆ T) = − C(t)B(t−) aY(t) − b2Y (t) dt + bY (t), dZ(t) dP(t, 2 ! + {− bY (t), Z(t) − e−Y(t) + 1} − C(t)A(t)G(0, X(t))dt. The use of the L´evy–It´o decomposition of Z and (11.2.7) leads to   1 ˆ T) = − P(t−, ˆ dP(t, T) aY(t) − b2Y (t) dt + bY (t), a dt + bY (t), dW(t) 2   bY (t), y π˜ (dt, dy) + bY (t), y π(dt, dy) + 

|y|≤1

|y|>1

! {1 − e{−h(t,X(t−)+y)−h(t,X(t−))} − bY (t), y }π(dt, dy)

+ U

ˆ T)G(0, X(t))dt. − P(t, ˆ T) is a local martingale, we can compensate the integrals over π(dt, dy) Since P(t, on the right side and, after that, cancel all dt-integrals. This gives  !   ˆ T) = −P(t−, ˆ dP(t, T) bY (t), dW(t) + 1 − eh(t,X(t−))−h(t,X(t−)+y) π(dt, ˜ dy) . U

The required representation follows by putting  T−t h(t, x) = G(s, x)ds, bY (t) := hx (t, X(t−))b(t) and hx (t, x) =

 T−t 0

0

Gx (s, x)ds.

11.3 Admissible Strategies

257

11.3 Admissible Strategies Under (MP) we can deduce more specific sufficient conditions defining trading strategies than those described in Section 7.2.3. Since our aim is to give a meaning to the integral  t ˆ ·)), t ∈ [0, T ∗ ], (ϕs , dP(s, 0

we can make use of the decomposition (11.2.1) and define, for t ∈ [0, T ∗ ],  t  t  t ˆ ·)) := (ϕs , dP(s, (ϕs , 1 (s, ·)), dW(s) + (ϕs , 2 (s, ·, y))π˜ (ds, dy), 0

0

0

U

(11.3.1) providing that the right side is well defined. Therefore we define admissible strategies as follows. Definition 11.3.1 Let (MP) be satisfied and the discounted bond prices admit, for any T > 0, the decomposition  ˆ dP(t, T) = 1 (t, T), dW(t) + 2 (t, T, y)π˜ (dt, dy), t ∈ [0, T ∗ ], T > 0, U

(11.3.2) where (1 (t, T)) ∈ (U) and (2 (t, T, y)) ∈ 1,2 . An admissible strategy is an M([0, +∞))-valued predictable process (ϕt ) such that the processes  +∞ (ϕt , 1 (t, ·)) := 1 (t, T)ϕ(t, dT), t ∈ [0, T ∗ ], 

0 +∞

(ϕt , 2 (t, ·, y)) :=

2 (t, T, y)ϕ(t, dT),

t ∈ [0, T ∗ ],

0

belong to (U), 1,2 (see Section 6), respectively, and such that the corresponding discounted wealth process  t ˆ ·)) ˆX(t) = X(0)+ (ϕs , dP(s, 0



t

(ϕs , 1 (s, ·)), dW(s) +

:= X(0)+ 0

 t 0

(ϕs , 2 (s, ·, y))π˜ (ds, dy), t ∈ [0, T ∗ ]

U

(11.3.3) is a martingale. Let us notice that if, for instance, for each s ∈ [0, T ∗ ], T −→ 1 (s, T),

T −→ 2 (s, T, y)

258

Completeness

are continuous and bounded on [0, +∞), then  +∞  (ϕs , 1 (s, ·)) = 1 (s, T)ϕs (dT), (ϕs , 2 (s, ·, y)) = 0

+∞

2 (s, T, y)ϕs (dT)

0

are finite. So, in this case the integrands on the right side of (11.3.1) are well defined. We consider sufficient conditions for ϕ to be an admissible strategy. Recall that if ϕ is admissible then  T∗ | (ϕs , 1 (s, ·)) |2 ds < +∞, (11.3.4) 0



T∗ 0

 

 | (ϕs , 2 (s, ·, y)) |2 ∧ | (ϕs , 2 (s, ·, y)) | ds ν(dy) < +∞.

(11.3.5)

U

It is clear that if there exists a constant K > 0 such that   t    (ϕs , dP(s, ˆ ·)) < K, t ∈ [0, T ∗ ],  

(11.3.6)

0

then ϕ is admissible because a bounded local martingale is a martingale. However, (11.3.6) is very rarely satisfied. More checkable conditioins are formulated below. Proposition 11.3.2 Let the strategy ϕ satisfy (11.3.4), (11.3.5) and 1  ∗ 1  ∗ 2 2 T T 1 2 2 | Q 2 (ϕs , 1 (s, ·)) | ds + E | (ϕs , 2 (s, ·, y)) | dsν(dy) < +∞. E 0

0

U

Then ϕ is admissible. Proof The quadratic variation of the discounted wealth, for t ∈ [0, T ∗ ], equals   t  ·  t 1 ˆ ·) = (ϕs , dP(s, | Q 2 (ϕs , 1 (s, ·)) |2 ds + | (ϕs , 2 (s, ·, y)) |2 π(ds, dy), 0

t

0

0

U

and in this case satisfies  E 0

·

 1 2 ˆ ·)) (ϕs , dP(s, < +∞. T∗

Hence, it follows from the Burkholder–Davies–Gundy inequality (4.3.4) that the · ˆ ·)) is a martingale. integral 0 (ϕs , dP(s, Remark 11.3.3 The requirement that discounted wealth processes are supposed to be martingales, although the integrator is assumed to be only a local martingale, is necessary for further analysis of the completeness problem. It implies that the integrands in (11.3.3) are determined in a unique way. This property fails if we require instead, for instance, that Xˆ is a positive local martingale. We clarify this issue in Proposition 11.3.4 by presenting an example of a positive local martingale

11.3 Admissible Strategies

259

adapted to the filtration generated by a Wiener process, which admits two different integral representations. The proof is based on Example 8, page 237 in Liptser and Shiryaev [89]. Proposition 11.3.4 There exists a positive local martingale M(t), t ∈ [0, 1] adapted to the filtration generated by a one-dimensional Wiener process W such that M(1) is a bounded random variable admitting two different representations  1 M(1) = E[M(1)] + γ (s)dW(s), (11.3.7)  M(1) = 1 +

0 1

ψ(s)dW(s),

(11.3.8)

0

where E[M(1)] = 1, and γ = ψ. Proof

The following stopping time τ := inf{t ∈ [0, 1] : W 2 (t) + t = 1}

satisfies P(0 < τ < 1) = 1. Let us define X(t) := −

2W(t) 1{t≤τ } (1 − t)2

and the Dol´eans-Dade exponent M := E(X) of X. The process X is stochastically integrable with respect to W because  1  τ W(s)2 X(s)2 ds = 4 ds < +∞. 4 0 0 (1 − s) It follows from the Itˆo formula applied to the process W(t)2 /(1 − t)2 that   1 1 t X(s)dW(s) − X(s)2 ds 2 0 0    τ 1 1 1 W 2 (τ ) 2 + − 2W(s) + ds =− (1 − τ )2 (1 − s)3 (1 − s)4 (1 − s)2 0 ≤−

1 + (1 − τ )



τ 0

1 ds ≤ −1. (1 − s)2

As a consequence, the Dol´eans-Dade exponent M = E(X), which is a local martingale, is not a martingale because   1 1 2 E(M(1)) = E e 0 X(s)dW(s)− 2 X(s) ds ≤ e−1 < M(0) = 1. The random variable M(1) satisfies 0 < M(1) ≤ e−1 and thus application of the martingale representation theorem to the square integrable martingale E[M(1) | Ft ] 1 yields (11.3.7) where E 0 γ 2 (s)ds < +∞. On the other hand, application of

260

Completeness

the martingale  1 representation theorem to the local martingale M provides (11.3.8), where P( 0 ψ 2 (s)ds < +∞) = 1. Since M is not a martingale, it follows that 1 E 0 ψ 2 (s)ds = +∞. Therefore ψ = γ .

11.4 Hedging Equation In this section we introduce the so-called hedging equation – a basic tool for examining the completeness problem. It allows us to construct the required representation  T∗ ˆ ·)), (ϕt , dP(t, P − a.s. (11.4.1) Xˆ = x + 0

for any Xˆ ∈ L1 (, FT ∗ , P) from the decomposition of its conditional expectation  t  t gXˆ (s, y)π˜ (ds, dy), t ∈ [0, T ∗ ]. E[Xˆ | Ft ] = EXˆ + fXˆ (s), dW(s) + 0

0

U

(11.4.2) Recall that the existence of a unique pair fXˆ ∈ (U), gXˆ ∈ 1,2 in (11.4.2) follows from Theorem 6.1.1. Theorem 11.4.1 Let Xˆ ∈ L1 (, FT ∗ , P) and fXˆ , gXˆ be given by (11.4.2). Then an ˆ i.e. (11.4.1) holds, if admissible strategy (ϕt ) with initial capital x ∈ R replicates X, ˆ and only if x = E[X] and ϕ solves the hedging equation ⎧ ⎪ dP × ds − a.s., ⎨ fXˆ (s) = (ϕs , 1 (s, ·)), (11.4.3) ⎪ ⎩ g (s, y) = (ϕs , 2 (s, ·, y)), dP × ds × dν − a.s. Xˆ Proof 

Writing (11.4.1) in the form  T∗  ˆ ·)) = E(X) ˆ + (ϕs , dP(s, fXˆ (s), dW(s) +

T∗

x+ 0

0

T∗

0

 U

gXˆ (s, y)π˜ (ds, dy), (11.4.4)

ˆ The process and taking expectations of both sides, we obtain x = E(X).  t  t  t ˆ ·)) − f ˆ (s), dW(s) − (ϕs , dP(s, gXˆ (s, y)π˜ (ds, dy) Mt : = X 0

 =

0

+

0

t

0

U

((ϕs , 1 (s, ·)) − fXˆ (s)), dW(s)

 t 0

 (ϕs , 2 (s, ·, y)) − gXˆ (s, y) π˜ (ds, dy),

 U

t ∈ [0, T ∗ ]

is thus a martingale equal to zero. With the use of Theorem 6.1.1 we obtain (11.4.3).

11.5 Completeness for the HJM Model

261

In view of (11.3.3) and (11.4.2) it is easy to see that (11.4.3) is sufficient for ϕ to replicate X. Necessity is, however, not so obvious and requires the assumption ˆ that admissible strategies generate discounted wealth processes X(t), which are ˆ is a positive local martingale then (11.4.3) martingales. If we require instead that X(t) does not have to hold, see Proposition 11.3.4.

11.5 Completeness for the HJM Model This section is concerned with the completeness problem of the HJM model df (t, T) = α(t, T)dt + σ (t, T), dZ(t) ,

t ∈ [0, T ∗ ],

T > 0,

(11.5.1)

with a U = Rd -valued L´evy process Z. Recall that the model satisfies the (MP) condition. We consider two cases when the support of the L´evy measure ν of Z is finite (finite activity noise) and when the support of ν has a concentration point. The results involve the following random field  T (t, T) := σ (t, s)ds, t ∈ [0, T ∗ ], T > 0, t∧T

which appeared in the formulation of Theorem 8.1.1 on the HJM drift conditions. To be precise, we should write (t, T, ω), but the dependence on ω will be omitted to keep the notation simple. It is clear that (t, T) is a continuous function of T ≥ 0.

11.5.1 L´evy Measure with Finite Support First, we treat the completeness problem in the case in which support of the L´evy measure of Z is finite: supp{ν} = {y1 , y2 , . . . , yn },

yi ∈ U, i = 1, 2, . . . , n.

(11.5.2)

Then Z is a finite activity process with a finite number of jump sizes. We will often replicate contingent claims with the use of a finite number of bonds with maturities, say T1 , T2 , . . . , TK , with some K. Then ϕt =

K 

ϕ(t, Tk )δ{Tk } ,

t ∈ [0, T ∗ ],

k=1

with some predictable processes ϕ(t, T1 ), . . . , ϕ(t, TK ), t ∈ [0, T ∗ ]. Now we formulate conditions for ϕ to replicate a given contingent claim X. Solution of the completeness problem in the one-dimensional case is given by the following result. Theorem 11.5.1 Let Z be a real-valued L´evy process with jumps satisfying (11.5.2). Let us assume that there exist maturities T1 , T2 , . . . , Tn+2 in [T ∗ , +∞) such that dP × dt-a.s.: (t, Ti ) = (t, Tj ),

i = j,

t ∈ [0, T ∗ ].

262

Completeness

Then each Xˆ ∈ L1 (, FT ∗ , P) can be replicated by strategies involving Ti -bonds only with i = 1, 2, . . . , n + 2. In particular, the market is complete in the class Xˆ ∈ L∞ (, FT ∗ , P). If d ≥ 1 then the conditions for completeness involve the following functions (t, T) := ( 1 (t, T), . . . ,  d (t, T)), e− (t,T),y1 − 1, . . . , e− (t,T),yn − 1. (11.5.3) For fixed (ω, t), the functions in (11.5.3) depend continuously on the argument T > 0. Theorem 11.5.2 (a) If the functions (11.5.3) of argument T are linearly dependent with positive dP × dt-measure, then the market is not complete in the class Xˆ ∈ L∞ (, FT ∗ , P) and M-valued admissible strategies. (b) If the functions (11.5.3) are linearly independent dP × dt-a.s., then each Xˆ ∈ L∞ (, FT ∗ , P) can be replicated by a strategy consisting, at any time, of d + n bonds with different maturities. In particular, the market is complete in the class Xˆ ∈ L∞ (, FT ∗ , P). Replicating strategies in Theorem 11.5.2 (b) consist of n + d bonds, but the choice of bonds is time and ω-dependent. The result is thus weaker than in Theorem 11.5.1 where maturities of the involved bonds are fixed. In the multidimensional case we can, however, impose stronger assumptions that also guarantee that maturities of replicating bonds can be fixed in advance. Theorem 11.5.3 Let us assume functions (11.5.3) of argument T ≥ T ∗ are linearly independent dP × dt- a.s. Moreover, assume that the functions t −→ (t, T),

t −→ e− (t,T),yi − 1,

i = 1, 2, . . . , n; T ∈ [T ∗ , +∞)

are analytic on the interval [0, T ∗ ]. Then there exists a set of dates T1 , T2 , . . . , Td+n ∈ [T ∗ , +∞), such that each Xˆ ∈ L1 (, FT ∗ , P) can be replicated with the use of bonds with maturities T1 , T2 , . . . , Td+n . Specifically, the market is complete. Before we present proofs of the preceding results, we present first some of their implications. In view of Theorem 11.5.2 and Theorem 8.2.11 we obtain the following result connecting completeness and the uniqueness of the martingale measure. Theorem 11.5.4 Let Z be a finite activity L´evy process in Rd with a finite number of jump sizes. Then the HJM model satisfying (MP) is complete in the class Xˆ ∈ L∞ (, FT ∗ , P) if and only if the martingale measure is unique. Theorem 11.5.2 is ilustrated by a model with constant volatility.

11.5 Completeness for the HJM Model

263

Proposition 11.5.5 Let Z be a L´evy process in Rd without Wiener part and jumps satisfying (11.5.2). If the volatility is constant: σ (t, T) ≡ σ and y1 + y2 + · · · + yn = 0, then each Xˆ ∈ L1 (, FT ∗ , P) is attainable. Hence, the market is complete in the class Xˆ ∈ L∞ (, FT ∗ , P). Proof In the proof we use Theorem 11.5.2. Since (t, T) = (T − t)σ the condition for completeness amounts to the linear independence of the functions h1 (T) := e−(T−t) σ ,y1 − 1, . . . , hn (T) := e−(T−t) σ ,yn − 1. It is clear that the linear independence of h1 , . . . , hn is implied by the linear independence of h˜ 1 , . . . , h˜ n , 1 where h˜ i (T) = hi (T) + 1, i = 1, 2, . . . , n. Now we examine linear independence of the latter set. Its Wro´nskian W(T) (see Pontriagin [104], Section 3.18) equals ⎡ ⎤ h˜ 2 (T) . . . h˜ n (T) 1 h˜ 1 (T) ⎢ ⎥ ⎢ h˜  (T) h˜ 2 (T) . . . h˜ n (T) 0 ⎥ ⎢ 1 ⎥ ⎢ ⎥ W(T) = DET ⎢ .. .. .. .. .. ⎥ ⎢ . . . . . ⎥ ⎢ ⎥ ⎣ (n) ⎦ ˜h (T) h˜ (n) (T) . . . h˜ (n) (T) 0 n 1 2 ⎡ = (−1)

n+2

⎢ ⎢ DET ⎢ ⎢ ⎣

h˜ 1 (T)

h˜ 2 (T)

...

h˜ n (T)

.. .

.. .

.. .

.. .

h˜ (n) 1 (T)

(n) h˜ 2 (T)

...

(n) h˜ n (T)

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

Since (j) h˜ i (T) = (− yi , σ )j · h˜ i (T),

i = 1, 2, . . . , n,

j = 0, 1, . . . , n,

we have W(T)

⎡

(− y1 , σ )h˜ 1 (T)

⎢ ⎢(− y1 , σ )2 h˜ 1 (T) ⎢ = (−1)n+2 DET ⎢ .. ⎢ ⎢ . ⎣ (− y1 , σ )n h˜ 1 (T)

(− y2 , σ )h˜ 2 (T)

...

(− y2 , σ )2 h˜ 2 (T)

...

.. .

.. .

(− y2 , σ )n h˜ 2 (T)

...

⎤ (− yn , σ )h˜ n (T) ⎥ (− yn , σ )2 h˜ n (T)⎥ ⎥ ⎥. .. ⎥ ⎥ . ⎦ (− yn , σ )n h˜ n (T)

264

Completeness

This yields W(T) = (−1)n+1 σ , y1 + y2 + · · · + yn e−(T−t) σ ,y1 +y2 +···+yn ⎡ ⎢ ⎢ ⎢ · DET ⎢ ⎢ ⎢ ⎣

⎤

1

1

...

1

(− y1 , σ )

(− y2 , σ )

...

(− yn , σ )

.. .

.. .

.. .

.. .

(− y1 , σ )n−1

(− y2 , σ )n−1

...

(− yn , σ )n−1

⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

However, the matrix is of the Vandermonde type (see Knapp [83, p. 215]) and thus $ σ · (yi −yj ) = 0. W(T) = (−1)n+1 σ , y1 +y2 + · · · +yn e−(T−t) σ ,y1 +y2 +···+yn 1≤i 0, k = 1, 2, . . . , so we can choose a sequence {ak }∞ k=1 s.t. ak ∈ Aν (ω, t) ∩ Dk

∀ k = 1, 2, . . . .

Let us examine now (11.5.19) with n = 2, β1 = 1, β2 = −1 and y1 = a2k+1 , y2 = a2k+2 for k = 0, 1, . . .. After dividing by γ , the left side of (11.5.19) has the form   1  1  (|a2k+1 | ∧ 1) + (|a2k+2 | | ∧ 1) β1 g(a2k+1 ) + β2 g(a2k+2 ) = γ γ and thus satisfies

 2(|y | ∧ 1) 1  0  = 0. β1 g(a2k+1 ) + β2 g(a2k+2 ) = k−→∞ γ γ lim

(11.5.20)

272

Completeness

For estimating the right side of (11.5.19) we use the following     2 (t, a2k+1 ) − 2 (t, a2k+1 ) Cb (R+ )

=

sup T∈[0,+∞)

≤

  ˆ  ˆ T)(e− (t,T),a2k+1 − 1) P(t−, T)(e− (t,T),a2k+1 − 1) − P(t−, ˆ |P(t−, T)| ·

sup T∈[0,+∞)

sup T∈[0,+∞)

   − (t,T),a2k+1  − e− (t,T),a2k+2 . e

The first supremum is clearly finite. To deal with the second supremum let us fix δ > 0 such that | a2k+1 − a2k+2 |< δ for k = 0, 1, 2, . . .. Then     sup e− (t,T),a2k+1 − e− (t,T),a2k+2  T∈[0,+∞)

≤ ≤

sup

    sup De− (t,T),y 

sup

sup

T∈[0,+∞) |y−y0 | 0 such that the set Assume (11.5.11) and that there exists  ' & ˜ A = (ω, t) ∈  × [0, T ∗ ] : sup | (t, T) |≤ 

Theorem 11.5.15

T>0

is of positive dP × dt measure and  ∗ +∞  T  ˜ 2(+ε)|y | i E e ν({yi })ds < +∞ 0

i=1

for some ε > 0. Then there exists Xˆ ∈ L2 (, FT ∗ , P) which cannot be replicated. Proof of Theorem 11.5.14

Let us define #√ gXˆ (t, yi ) :=

& where ik := inf i : ν({yi }) ≤ and, by (11.5.23),

1 k3

'

0

k

for i = ik , for

i = ik ,

(11.5.25)

. Due to (11.5.24) the process gXˆ is well defined

lim sup | gXˆ (t, yi ) |= +∞. i→∞

(11.5.26)

274

Completeness

It follows from  T∗  ∞ 0

| gXˆ (t, yi ) |2 ν({yi })dt ≤ T ∗

i=1

∞  1 < +∞ k2

(11.5.27)

k=1

· that gXˆ ∈ 1,2 and that 0 U gXˆ (s, y)π˜ (ds, dy) is a square integrable martingale. We will show that the square integrable random variable,  T∗  gXˆ (s, y)π˜ (ds, dy), (11.5.28) Xˆ := 0

U

cannot be replicated. Let us fix (ω, t) ∈ A and assume that (11.5.17) is satisfied by some ϕ. Then, by Lemma 11.5.12, there exists γ = γ (ω, t) > 0 such that ∀n∈N

∀ {βk }nk=1 , βk ∈ R

∀ {yik }nk=1

n n         βk gXˆ (t, yik ) ≤ γ  βk 2 (t, yi )  k=1

k=1

Cb (R+ )

.

(11.5.29)

Let us check (11.5.29) with n = 1, β1 = 1 and for i1 = 1, 2, . . . successively, that is,   ˆ  ∀i = 1, 2, . . . . (11.5.30) |gXˆ (t, yi )| ≤ γ sup P(t−, T)(e− (t,T),yi − 1) T∈[0,+∞)

By definition of the set A, for any i = 1, 2, . . ., we have  ⏐  ⏐ ˆ ⏐ˆ  ⏐ lim sup sup P(t−, T)(e− (t,T),yi − 1) ≤ sup ⏐P(t−, T)⏐ T∈[0,T ∗ ]

i→∞ T∈[0,+∞)

· lim sup

sup

| e− (t,T),yi − 1 |< +∞.

i→∞ T∈[0,+∞)

However, recall that the left side of (11.5.30) satisfies (11.5.26), so the required constant γ does not exist. Hence (11.5.17) is no satisfied for any (ω, t) ∈ A. But the set A is of positive dP × dt-measure, so in view of Lemma 11.5.13 (11.5.17) does not hold dP × ds × dν-a.s. for any admissible strategy ϕ. Proof of Theorem 11.5.15: The arguments are similar as in the proof of Theorem 11.5.14. Let us define ˜

i = 1, 2, . . . gXˆ (t, yi ) := e(+ε)|yi | , t Then gXˆ ∈ 1,2 and 0 U gXˆ (s, y)π˜ (ds, dy) is a square integrable martingale. The square integrable random variable  T∗  gXˆ (y)π˜ (ds, dy), Xˆ := 0

U

11.6 Completeness for Affine Models

275

cannot be replicated because for any (ω, t) ∈ A we have ˜ | gXˆ (t, yi ) | e(+ε)|yi | = +∞, ≥ lim sup − (t,T),yi − 1 | ˜ Cb (R+ ) i→∞ | e i→∞ e|yi | + 1

lim sup

and, consequently, (11.5.17) does not have a solution. Proof of Lemma 11.5.12 The result is an extension of the moment problem solution (see Yosida [122]). Necessity is obvious, (11.5.13) holds with γ = |e∗ |E∗ . To prove sufficiency let us define a linear subspace M of E by n ' &  βi h(ai ); n ∈ N, βi ∈ R, ai ∈ A M= e∈E:e= i=1

e˜ ∗ :

M −→ R by the formula n n    βi h(ai ) = βi g(ai ). e˜ ∗

and a linear transformation

i=1

n

i=1

Notice, that for e1 , e2 ∈ M with e1 = i=1 βi h(ai ) and e2 = (11.5.13), we obtain n m        ∗    ∗ βi g(ai ) − βj g(aj ) ˜e (e1 ) − e˜ (e2 ) =  i=1 n 

 ≤ γ

j=1 m 

βi h(ai ) −

i=1

m



j=1 βj h(uj ),

by

   βj h(aj ) = γ |e1 − e2 |E .

j=1

E

If e1 = e2 then e˜ ∗ (e1 ) = e˜ ∗ (e2 ), so this transformation is well defined because its value does not depend on the representation. It is also continuous and thus by the Hahn–Banach theorem it can be extended to the functional e∗ ∈ E∗ , which clearly satisfies (11.5.12). Proof of Lemma 11.5.13 The assertion follows from the Fubini theorem applied to the function h = 1A where A := {(x, y) ∈ E1 × E2 : f1 (x, y) = f2 (x, y)}.

11.6 Completeness for Affine Models Let us consider an affine model P(t, T) = e−C(T−t)−D(T−t)R(t) ,

t ∈ [0, T ∗ ],

T > 0,

with short rate satisfying dR(t) = F(R(t))dt + G(R(t−))dZ(t),

t ≥ 0.

(11.6.1)

Here, Z is a one-dimensional L´evy process and the functions C(·), D(·), F(·), G(·) are deterministic. The model is assumed to satisfy the (MP) condition.

276

Completeness

Since R(t) = −

  1 ln P(t, T) + C(T − t) , D(T − t)

it follows that, for each T > 0, the filtration (FtR ) generated by R is identical with T (FtP ) generated by P(·, T), i.e. T

FtR = FtP ,

t ∈ [0, T].

From (11.6.1) we have that dZ(t) =

dR(t) F(R(t)) − dt, G(R(t−)) G(R(t−))

which implies that (FtR ) is identical with the filtration (Ft ) generated by Z. Hence, for each T > 0, T

FtP = FtR = Ft ,

t ∈ [0, T].

(11.6.2)

Recall that by Proposition 7.5.1 the model can be written as the HJM model df (t, T) = α(t, T)dt + σ (t, T)dZ(t),

(11.6.3)

with α(t, T) := F(R(t))D (T − t) − C (T − t) − D (T − t)R(t),

(11.6.4)

σ (t, T) := D (T − t)G(R(t−)).

(11.6.5)

This enables us to examine the completeness problem for affine models by using the results proven in previous sections concerned with the HJM model. Theorem 11.6.1 (a)

Let the affine model satisfying (MP) have short rate of the form

 dR(t) = (βR(t) + c)dt + R(t)dW(t), R(0) > 0, β ∈ R, c ≥ 0,

t ∈ [0, T ∗ ], (11.6.6)

where W is a one-dimensional Wiener process. Then any claim Xˆ ∈ L1 (, FT ∗ , P) = L1 (, FTR∗ , P) is attainable and can be replicated by a strategy involving one T-bond only, where T is any number such that T ≥ T ∗ . In particular, the model is complete. (b) 1

dR(t) = (βR(t) + c)dt + (R(t−)) α dZ α (t),

β ∈ R, c ≥ 0,

t ∈ [0, T ∗ ], (11.6.7)

where Z α is an α-stable martingale with index α ∈ (1, 2) and positive jumps only. Then the market is not complete in the class Xˆ ∈ L∞ (, FT ∗ , P). Moreover,

11.7 Completeness for Factor Models

277

for any T ≥ T ∗ , there exists a bounded claim of the form Xˆ = h(P(t, T), t ∈ [0, T ∗ ]), where h is a deterministic function, which is not attainable in the class of admissible M-valued strategies. Proof It follows from Theorem 10.2.7 that both equations generate affine models which satisfy (MP) with some functions C(·), D(·). Writing the model in the HJM parametrization, we obtain, by (11.6.3)–(11.6.5), that the volatility equals  σ (t, T) = D (T − t) R(t), for (11.6.6) and 1

σ (t, T) = D (T − t)(R(t)) α , for (11.6.7). (a) The assertion follows from Theorem 11.5.2 and the relation T

FtP = Ft ,

t ∈ [0, T ∗ ],

T ≥ T ∗,

(11.6.8)

which follows from (11.6.2). (b) We can apply Theorem 11.5.11. Since D is bounded (see Remark 10.2.10) we have  +∞  +∞ 1 | σ (t, T) | dT =| (R(t−)) α | · | D (T − t) | dT < +∞, 0

0

and thus (11.5.11) is satisfied. Since the L´evy measure of Z α has a concentration point, the market is not complete in the class Xˆ ∈ L∞ (, FT ∗ , P). The rest of the assertion follows from (11.6.8).

11.7 Completeness for Factor Models We present now a partial solution of the completeness problem for a factor model f (t, T) = G(T − t, X(t)),

(11.7.1)

with X(t) solving dX(t) = a(t)dt + b(t), dZ(t) ,

t ≥ 0.

(11.7.2)

Here Z is a d-dimensional L´evy process, (a(t), t ≥ 0)- an adapted process and (b(t), t ≥ 0)- a predictable process. Theorem 11.7.1 Let us assume that for the model given by (11.7.1)–(11.7.2) the (MP) condition is satisfied. (a) Let Z be a Wiener process with a drift. If d = 1 then the market is complete and any Xˆ ∈ L1 (, FT ∗ , P) is attainable. If d > 1 then the market is not complete.

278

Completeness

(b) Let the L´evy measure of Z have a concentration point y0 = 0. If G is linear in x, i.e. G(t, x) = G1 (t)x + G2 (t) and  +∞ | G1 (s) | ds < +∞, 0

then the market is not complete. Proof

By the Itˆo formula

df (t, T) = dG(T − t, X(t)) 1 = −Gt (T − t, X(t))dt + Gx (T − t, X(t−))dX(t) + Gxx (T − t, X(t))b2 (t)dt 2 & ' + G(T − t, X(t)) − G(T − t, X(t−)) − Gx (T − t, X(t−))X(t) . (11.7.3) (a) If Z is continuous then the last line in (11.7.3) disappears and, taking into account (11.7.2), we obtain ! 1 df (t, T) = − Gt (T − t, X(t)) + Gx (T − t, X(t))a(t)+ Gxx (T − t, X(t))b2 (t) dt 2 + Gx (T − t, X(t))b(t), dZ(t) .

(11.7.4)

Coming back to the HJM notation, it follows that (t, T) =  d (t, T)) is given by  T  T  T−t i i i  i σ (t, u)du = b (t)Gx (u − t, X(t))du = b (t) Gx (s, X(t))ds,  (t, T) = ( 1 (t, T), . . . ,

t

t

0

i = 1, 2, . . . , d, so, for fixed t, vectors  1 (t, T), . . . ,  d (t, T) are linearly dependent functions of T in the case d > 1. The assertion follows from Theorem 11.5.2. (b) If G(t, x) = G1 (t)x + G2 (t) then Gx = G1 (t) and consequently G(T − t, X(t)) − G(T − t, X(t−)) − Gx (T − t, X(t−))X(t)   = G1 (T − t) X(t) − X(t−) − X(t) = 0. From (11.7.3) we obtain   df (t, T) = G1 (T − t)a(t) − G1 (T − t)X(t) − G2 (T − t) dt + G1 (T − t)b(t), dZ(t) and can apply Theorem 11.5.11 with σ (t, T) = G1 (T − t)b(t). Since  +∞  +∞  +∞ | σ (t, T) | dT =| b(t) | | G1 (u − t) | du ≤| b(t) | | G1 (s) | ds < +∞, 0

t

0

11.7 Completeness for Factor Models

279

so (11.5.11) is satisfied, and the L´evy measure has a concentration point, the assertion follows. As an application of Theorem 11.7.1 we solve the completeness problem for models with Ornstein–Uhlenbeck factors (see Chapter 9). Proposition 11.7.2 Let the factor process be given by dX x (t) = (a + AX x (t))dt + dW(t),

X x (0) = x ∈ Rd ,

where a ∈ Rd , A is a d × d-matrix and W the Wiener process in Rd with identity covariance matrix. Let the factor model with factor X x and initial curve G(0, x) := Kx, x , where K is a positive d × d-matrix, satisfy (MP). Then the following statements are true. (a) If d = 1 then the market is complete. Moreover, any claim Xˆ ∈ L1 (, FT ∗ , P) of ˆ the form Xˆ = X(P(t, T), t ∈ [0, T ∗ ]), for some T ≥ T ∗ , can be replicated by a strategy involving the T-bond only. (b) For d > 1 the model is not complete. Moreover, for any T ≥ T ∗ , there exists a ˆ bounded claim of the form Xˆ = X(P(t, T), t ∈ [0, T ∗ ]), which is not attainable. Proof Note that since P(t, T) = F(T − t, X(t)), where the function F is described in Proposition 9.1.8, we have for each T T

FtP ⊆ Ft ,

t ∈ [0, T].

ˆ Therefore the assertion for claims of the form Xˆ = X(P(t, T), t ∈ [0, T ∗ ]) follows from Theorem 11.7.1. Proposition 11.7.3 Let us assume that the factor is the short rate given by dRx (t) = (a + bRx (t))dt + dZ(t),

Rx (0) = x,

t ≥ 0,

where a ≥ 0, b < 0 and (MP) is satisfied. If the L´evy measure of Z has a nonzero concentration point then the model is not complete. Moreover, for any T ≥ T ∗ ˆ there exists a bounded claim of the form Xˆ = X(P(t, T), t ∈ [0, T ∗ ]), which is not attainable. Proof We have shown in Proposition 9.1.7 that if the model satisfies (MP) then the bond curve is given by     bt bt 2 bt 2 F(t, x) = e− x(e −1)/b+a(e −1)/b −at/b eJ (e −1)/b −t/b . (11.7.5) Consequently, G(t, x) = G1 (t)x + G2 (t),

280

Completeness

where G1 (t) := ebt ,

G2 (t) :=

Since b < 0, we have



  d  −a(ebt − 1)/b2 + at/b − J (ebt − 1)/b2 − t/b . dt

+∞

 G1 (s)ds =

0

+∞

ebs ds < +∞,

0

and, by Theorem 11.7.1 (b), the market is not complete providing that the L´evy measure has a nonzero concentration point. Since, for any T > 0, P(t, T) = F(T − t, R(t)), t ∈ [0, T] we obtain, by (11.7.5), that  bt  at t e −1 a bt b − , t ∈ [0, T]. ln P(t, T) + 2 (e − 1) − − J R(t) = b b 1 − ebt b b2 T

This implies that FtP = FtR for t ∈ [0, T]. Since the filtrations generated by Z and R are also equal, we obtain T

FtP = FtR = Ft ,

t ∈ [0, T ∗ ],

for T ≥ T ∗ and the assertion follows.

11.8 Approximate Completeness Here we work, as before, with a bond market defined on a probability space (, F, P) with filtration (Ft ) generated by a L´evy process Z and assume that (MP) is satisfied. Recall that the discounted bond prices have the form  ˆ T) = 1 (t, T), dW(t) + 2 (t, T, y)π˜ (dt, dy), t ∈ [0, T ∗ ], T > 0, dP(t, U

(11.8.1) where (1 (t, T)) ∈ (U) and (2 (t, T, y)) ∈ 1,2 and the discounted wealth process of an M([0, +∞))-valued strategy (ϕt ) is given by  t ˆ ·)) ˆ = X(0) + (ϕs , dP(s, X(t) 0

 t  t (ϕs , 2 (s, ·, y))π˜ (ds, dy), t ∈ [0, T ∗ ]. := X(0) + (ϕs , 1 (s, ·)), dW(s) + 0 U

0

We use a subclass A of admissible strategies defined by 8 #  T∗  T ∗ ∗ 2 2 (ϕs , 1 (s, ·)) ds+E (ϕs ,2 (s, ·, y)) dsν(dy) < +∞ . A := ϕs , s ∈ [0, T ] : E 0

0

U

(11.8.2)

11.8 Approximate Completeness

281

ˆ For (ϕt ) ∈ A the process (X(t)) is a square integrable martingale. 2 ˆ If a claim X ∈ L (, FT ∗ , P) is not replicable at time T ∗ by strategies from A, one may try to find a sequence (xn , ϕ n ), n = 1, 2, . . . where xn ∈ R and ϕ n ∈ A such that lim E





xn +

n→+∞

T∗

0

2 !  ˆ ·)) − Xˆ (ϕtn , dP(t, = 0.

(11.8.3)

ˆ If each Xˆ ∈ Such a sequence will be called an approximating sequence for X. 2 L (, FT ∗ , P) admits an approximating sequence, the market is called approximately complete in the class A. First we characterize approximating sequences for Xˆ ∈ L2 (, FT ∗ , P) with the use of the decomposition  t  t E[Xˆ | Ft ] = EXˆ + fXˆ (s), dW(s) + gXˆ (s, y)π˜ (ds, dy), t ∈ [0, T ∗ ], 0

U

0

(11.8.4) where fXˆ , gXˆ are uniquely determined and satisfy 

T∗

E 0

 | fXˆ (t) | dt < +∞,

Proposition 11.8.1 if and only if

T∗

E

2



0

U

| gXˆ (t, y) |2 dtν(dy) < +∞.

(xn , ϕ n ) is an approximating sequence for Xˆ ∈ L2 (, FT ∗ , P) xn − EXˆ −→ 0, 

T∗

E 

T∗

E 0

 U

0

| fXˆ (s) − (ϕsn , 1 (s, ·)) |2 ds −→ 0,

| gXˆ (s, y) − (ϕsn , 2 (s, ·, y)) |2 dsν(dy) −→ 0.

(11.8.5) (11.8.6) (11.8.7)

Proof Since the discounted wealth process of an approximating sequence (xn , ϕ n ) equals  t  t ˆX n (t) = xn + (ϕsn , 1 (s, ·)), dW(s) + (ϕsn , 2 (s, ·, y))π˜ (ds, dy), t ∈ [0, T ∗ ], 0

0

U

(11.8.8) the hedging error equals ˆ 2] = E E[(Xˆ n (T ∗ ) − X)



ˆ + xn − E(X)



T∗ 0



T∗

+ 0



;

(ϕsn , 1 (s, ·)) − fXˆ (s), dW(s)

2 !  (ϕsn , 2 (s, ·, y)) − gXˆ (s, y) π˜ (ds, dy) .

 U

:

282

Completeness

Using the independence and the zero mean property of the stochastic integrals in (11.8.8) we obtain ˆ 2 ] = (xn − E(X)) ˆ 2+E E[(Xˆ n (T ∗ ) − X)



T∗

:

0



+E



T∗

0

2 !  (ϕsn , 2 (s, ·, y)) − gXˆ (s, y) π˜ (ds, dy)

U



T∗

| fXˆ (s) − (ϕsn , 1 (s, ·)) |2 ds

0



T∗

0

; 2 !



ˆ 2+E = (xn − E(X)) +E

(ϕsn , 1 (s, ·)) − fXˆ (s), dW(s)

 U

| gXˆ (s, y) − (ϕsn , 2 (s, ·, y)) |2 dsν(dy),

so the assertion follows. Our method of examining approximate completeness in the sequel involves decompositions (11.8.4). Since E[Xˆ 2 ] = E



T∗

0

 | fXˆ (t) |2 dt + E

T∗



0

U

| gXˆ (t, y) |2 dtν(dy).

Let us introduce the product Hilbert space H of predictable processes (f (t), g(t, y), t ∈ [0, T ∗ ], y ∈ U) such that 

T∗

E



T∗

f 2 (t)dt + E

0

0

 g2 (t, y)dtν(dy) < +∞, U

with scalar product  ((f1 , g1 ), (f2 , g2 ))H = E

T∗



T∗

f1 (t)f2 (t)dt + E

0

 g1 (t, y)g2 (t, y)dtν(dy).

0

U

The following result will be our basic tool for examining approximate completeness in the sequel. Proposition 11.8.2 Let A˜ ⊆ A be a Banach space such that the operator Kϕ(t, y) = ((ϕt , 1 (t, ·)); (ϕt , 2 (t, ·, y))),

˜ ϕ ∈ A,

t ∈ [0, T ∗ ], y ∈ U, (11.8.9)

is a linear bounded operator from A˜ into H. Then the market with discounted prices (11.8.1) is approximately complete in the class A˜ if and only if Ker K∗ = {0}, where K∗ stands for the adjoint operator of K.

11.8 Approximate Completeness

283

Proof In view of Proposition 11.8.1, approximate completeness is equivalent to the statement that the image of K is dense in H. However, for (f , g) ∈ H: (Kϕ, (f , g))H = (ϕ, K∗ (f , g))A˜ ,

ϕ ∈ H,

where the right side denotes the value of the functional K∗ (f , g) on the element ϕ. Therefore ImK is dense in H if and only if the implication K∗ (f , g) = 0

⇒

(f , g) = 0

holds.

11.8.1 HJM Model Now we examine approximate completeness in the HJM model df (t, T) = α(t, T)dt + σ (t, T), dZ(t) ,

(11.8.10)

with a d-dimensional L´evy process Z. We begin with a result that approximate completeness may appear if the support of the L´evy measure of Z is infinite. In the opposite case the concepts of approximate completeness and completeness coincide. Theorem 11.8.3 (a) If the market (11.8.10) is approximately complete then, for each t ∈ [0, T ∗ ], P-almost surely, the vectors  1 (t, T),  2 (t, T), . . . ,  d (t, T) are linearly independent as functions of T ∈ (0, +∞). (b) Let the jumps of Z take values in a finite set. Then the market (11.8.10) is approximately complete if and only if it is complete. Proof (a) If the vectors  1 (t, T),  2 (t, T), . . . ,  d (t, T) are linearly dependent with positive dP × dt measure then one can find fXˆ such that for some ε > 0 and any ϕ ∈ A,  T∗ E | fXˆ (s) − (ϕs , 1 (s, ·)) |2 ds > ε. 0

A precise construction of fXˆ can be deduced from the proof of Theorem 11.5.2 (a). (b) In view of Theorem 11.5.2 the market is complete if and only if the vectors (t, T) := ( 1 (t, T), . . . ,  d (t, T)), e− (t,T),y1 − 1, . . . , e− (t,T),yn − 1 are linearly independent. Above y1 , . . . , yn denote all possible values of jumps of Z. If the linear independence of the preceding set fails, then one can construct fXˆ and gXˆ such that for some ε > 0 and any ϕ ∈ A,

284

Completeness 

T∗

E 0

 | fXˆ (s) − (ϕs , 1 (s), ·) |2 ds + E

T∗



0

U

| gXˆ (s, y)

− (ϕs , 2 (s, ·, y)) |2 dsν(dy) > ε. In the construction one can follow the proof of Theorem 11.5.2 (b). We assume in the sequel that Z does not have the Wiener part. We will also restrict admissible strategies to measure valued strategies requiring that measures have densities with respect to some finite measure μ on [0, +∞) with infinite support, i.e. ϕ(t, dT) = ϕ(t, T)μ(dT),

t ∈ [T ∗ ].

The case in which the support of μ equals [a, b], with some 0 < a < b < +∞, corresponds to trading with bonds with maturities from [a, b] only. In this framework the operator K : A −→ H in (11.8.9) takes the reduced form  +∞ 2 (t, T, y)ϕ(t, T)μ(dT), (11.8.11) (Kϕ)(t, y) := 0

and the space H is given by 

T∗

H := {g = g(t, y) : E

 g2 (t, y)dtν(dy) < +∞}.

0

U

More formally, H = L2 ( × [0, T ∗ ] × U, P × B(U), P × dt × ν). Now we formulate and prove the main result of this section. Theorem 11.8.4

Let in the HJM model (11.8.10) forward rates be positive,

supp{ν} ⊆ Rd+ ,

(t, T) ∈ Rd+ ,

t ∈ [0, T ∗ ],

T>0

(11.8.12)

and ess

| (s, T) |< +∞

sup

s∈[0,T ∗ ], T≥s, ω∈

(11.8.13)

hold. Let A˜ be the space of M-valued strategies with densities, i.e. ϕt (dT) = ϕ(t, T)μ(dT), equipped with the norm #  2 ˜ A := ϕ = ϕ(t, T) :| ϕ | ˜ := E A

T∗ 0

 0

Then the following statements are true.

+∞

8 ϕ (t, T)dt μ(dT) < +∞ . (11.8.14) 2

11.8 Approximate Completeness

285

(a) The market is approximately complete in the class A˜ if and only if the following implication holds: for almost all (ω, t)  (e− (t,T),y − 1)h(t, y)ν(dy) = 0, for μ-almost all T ∈ [t, +∞) U

⇒

h(t, y) = 0

ν − a.s..

(11.8.15)

(b) If d = 1 and for almost all (ω, t), there exists an interval [a(ω), b(ω)], contained in the support of μ and such that {(t, T, ω) : T > t} ⊇ [a(ω), b(ω)],

P − a.s.,

˜ then the market is approximately complete in the class A. Proof The proof is divided into two steps. In Step 1 we show that the operator K is bounded. In Step 2 we determine K∗ and prove (a) and (b). Step 1: To prove that K is bounded note that for ϕ ∈ A˜ we have  T∗  2 (ϕs , 2 (s, ·, y))2 dsν(dy) | Kϕ |H = E 0

 =E

U

T∗

 

T∗

 

0

 ≤E 0

U

U

+∞

2 ϕ(s, T)2 (s, T, y)μ(dT) dsν(dy)

0 +∞

  ϕ 2 (s, T)μ(dT)

0

+∞ 0

 22 (s, T, y)μ(dT) dsν(dy).

ˆ T)[e− (t,T),y − 1] (see Proposition 11.2.1) and forward Since 2 (t, T, y) = P(t−, rates are positive, we continue the estimation as follows  T ∗   +∞   +∞  ϕ 2 (s, T)μ(dT) | Kϕ |2H ≤ E Pˆ 2 (s−, T)ψ((s, T))μ(dT) ds, 0

0

0

where ψ is the following function  ψ(x) := | e− x,y − 1 |2 ν(dy), U

Since 1 − e−z ≤ z; z ≥ 0 we obtain   | e− x,y − 1 |2 ν(dy) = |y|≤1

 |y|≤1



 |y|>1

e− x,y − 1 x, y

2 | x, y |2 ν(dy)

| y |2 ν(dy),

x ∈ Rd+ .

| e− x,y − 1 |2 ν(dy) ≤ ν({| y |> 1}) < +∞,

x ∈ Rd+ ,

≤| x |2 Clearly,

x ∈ Rd+ .

|y|≤1

286

Completeness

so ψ(x) ≤ A + B | x |2 , x ∈ Rd+ , (11.8.16)  where A := ν({| y |> 1}) and B := |y|≤1 | y |2 ν(dy). Therefore we obtain  T ∗   +∞   +∞  2 2 | Kϕ |H ≤ E ϕ (s, T)μ(dT) (A + B | (s, T) |2 )μ(dT) ds 0

0

0

≤ μ([0, +∞))(A + B

ess sup

ω∈,0≤s≤T ∗ ,s≤T

| (s, T) |2 ) | ϕ |2˜ , A

and the assertion follows. Step 2: We determine K∗ . For ϕ ∈ A˜ and g ∈ H, we have  ∗   T Kϕ(t, y) · g(t, y)dtν(dy) (K(ϕ), g)H = E 

0

=E 

U

T∗

 

T∗



0

=E

U

0

+∞

0

 2 (t, T, y)ϕ(t, T)μ(dT) · g(t, y)dtν(dy)  2 (t, T, y) · g(t, y)ν(dy) ϕ(t, T)dtμ(dT)

+∞  U

0

= (ϕ, K∗ (g))A˜ , so it follows that K∗ g can be identified with the random field  2 (t, T, y) · g(t, y)ν(dy). (K∗ g)(t, T) = U

To prove (a), note that the condition K∗ g = 0 means that dt × μ(dT) – almost surely,  ! ˆ P(t−, T) e− (t,T),y − 1 g(t, y)ν(dy) = 0, U

or, equivalently, that



! e− (t,T),y − 1 g(t, y)ν(dy) = 0. U

The result follows. Part (b) follows from Proposition 11.8.5. 11.8.5 Let h be a function from L2 ((0, +∞), ν), where ν satisfies Proposition +∞ 2 (y ∧ 1)ν(dy) < +∞. If there exist 0 < λ1 < λ2 < +∞ such that 0  +∞ (e−λy − 1)h(y)ν(dy) = 0, H(λ) := 0

on a dense subset of (λ1 , λ2 ), then h = 0, as an element of L2 ((0, +∞), ν).

11.8 Approximate Completeness

287

We establish first the following technical lemma. Lemma 11.8.6 Let us assume that h ∈ L2 ((0, +∞), ν), with ν satisfying 1)ν(dy) < +∞. Then the function  +∞ (e−λy − 1)h(y)ν(dy), λ > 0 H(λ) :=

 +∞ 0

(y2 ∧

0

is differentiable and H  (λ) = −



+∞

e−λy y h(y)ν(dy),

λ > 0.

(11.8.17)

0

Proof

We split H into two parts H(λ) = H0 (λ) + H∞ (λ),

where



1

H0 (λ) :=

(e−λy − 1)h(y)ν(dy),

λ > 0,

0

 H∞ (λ) :=

+∞

(e−λy − 1)h(y)ν(dy),

λ > 0.

1

We show the result for H0 . The argument for H∞ is similar and easier. Let us denote g(λ, y) := (e−λy − 1)h(y). Then ∂g(λ, y) = −ye−λy h(y) ∂λ and

so

   ∂g(λ, y)     ∂λ  ≤ y | h(y) |,  0

  

   ν(dy) ≤ 

1  ∂g(λ, y) 

∂λ

1

y | h(y) | ν(dy)

0



1

≤ 0

 1  2 y ν(dy) 2

1

1 2

2

h (y)ν(dy)

< +∞.

0

By the classical result on differentiation of integrals with respect to a parameter the formula (11.8.17) follows. Proof of Proposition 11.8.5 H  (λ) = −

 0

It follows from Lemma 11.8.6 that +∞

e−λy y h(y)ν(dy),

λ ∈ (λ1 , λ2 ).

288

Completeness

Thus the Laplace transform of yh(y)ν(dy) vanishes on a dense subset of the interval (λ1 , λ2 ), so it vanishes for all λ > 0 and thus yh(y) = 0, ν-almost surely. Remark 11.8.7 The severe assumptions required in Theorem 11.8.4 were used to show that K is a bounded operator.

11.8.2 Factor Model Theorem 11.8.4 allows us to solve the approximate completeness problem for factor models f (t, T) = G(T − t, X(t)),

(11.8.18)

G(t, x) = G1 (t)x + G2 (t),

(11.8.19)

dX(t) = a(t)dt + b(t)dZ(t),

(11.8.20)

where

and Z is a real-valued L´evy process with no Wiener part. Proposition 11.8.8 Let G(t, x) have the decomposition (11.8.19) satisfying  +∞ G1 (t) ≥ 0, G2 (t) ≥ 0, t ≥ 0, | G1 (t) | dt < +∞. 0

Let the factor (11.8.20) be a positive process, b(t) > 0, t ≥ 0, and the L´evy measure of Z has a nonzero concentration point and supported by (0, +∞). Then the model (11.8.18) is not complete and is approximately complete in the class of admissible strategies A˜ defined by (11.8.14). Specifically, the result is true for the short-rate factor dRx (t) = (a + bRx (t))dt + dZ(t),

Rx (0) = x,

t ≥ 0,

(11.8.21)

where b < 0. Proof The fact that the model is not complete follows from Theorem 11.7.1 (b). Passing to the HJM parametrization yields σ (t, T) = G1 (T − t)b(t), so  T  T−t (t, T) = G1 (u − t)b(t)du = b(t) G1 (s)ds, t < T. t

It follows that for each t ∈

0

[0, T ∗ ]

there exists an interval contained in the set {(t, T); T > t},

so, by Theorem 11.8.4 (b), the model is approximately complete. For the process satisfying (11.8.21) we have shown in Proposition 11.7.3 that G1 (t) = ebt . So, G1 with negative b is integrable and the reasoning above applies.

11.8 Approximate Completeness

289

11.8.3 Affine Model Let us consider the affine model P(t, T) = e−C(T−t)−D(T−t)R(t) ,

t ∈ [0, T ∗ ], T ≥ t,

with short rate of the form 1

dR(t) = (aR(t) + b)dt + R(t−) α dZ α (t),

R(0) = x ≥ 0,

a ∈ R, b ≥ 0, (11.8.22)

where Z α is a stable martingale with index α ∈ (1, 2). Let us recall that in this case s

ˆ T) = e−C(T−s)−D(T−s)R(s)− P(s,

0

R(u)du

,

(11.8.23)

and that ˆ T)(e−D(T−t)R(t−) 2 (t, T, y) = P(t−,

1 α

y

− 1),

t ∈ [0, T ∗ ], T ≥ 0, y ∈ (0, +∞) (11.8.24)

(see Proposition 11.2.1). In Section 11.6 we have shown that this model is not complete (see Theorem 11.6.1). Theorem 11.8.9 dense in the set

Let μ be a finite measure on [0, +∞) such that its support is {D(v), v ≥ 0}.

Let A˜ be the space of M-valued strategies ϕt (·) such that ϕt (dT) = ϕ(t, T)μ(dT), equipped with the norm  | ϕ |2˜ := A

+∞

ess sup

ω∈,t∈[0,T ∗ ]

ϕ 2 (t, T)μ(dT) < +∞.

(11.8.25)

0

Then the affine model with short rate (11.8.22) is approximately complete in the ˜ class A. Proof

We show first that the operator K given by  +∞ 1 α ˆ (Kϕ)(t, y) := P(t−, T)(e−D(T−t)R(t−) y − 1)ϕ(t, T)μ(dT), 0

t ∈ [0, T ∗ ], T ≥ 0 is a bounded operator from A˜ into H.

290

Completeness

˜ Using (11.8.23) and (11.8.24) we obtain for ϕ ∈ A,  | Kϕ |2H = E

T∗

0

 ≤E ≤E 0

(ϕs , 2 (s, ·, y))2 dsν(dy) U

T∗

 

T∗



0





+∞

  ϕ 2 (s, T)μ(dT)

0

U

+∞

0

+∞

  ϕ 2 (s, T)μ(dT)

0

+∞

22 (s, T, y)μ(dT) dsν(dy)

e−2

s 0

R(u)du

1 ψ(R α (s−)D(T − s))μ(dT) ds,

0

where



| e−xy − 1 |2 ν(dy),

ψ(x) :=

x ≥ 0.

U

However, for arbitrary x > 0,



ψ(x) =

+∞

(1 − e−xy )2

0

where



+∞

c :=

(1 − e−u )2

0

1 y1+α

dy = xα c,

1 du < +∞. u1+α

Recall that in Theorem 10.2.7 we showed that D(·) solves D (v) = −cα Dα (v) + aD(v) + 1,

D(0) = 0,

1 and C (v) = bD(v), C(0) = 0, where cα = α(α−1) (2−α) > 0. Note that D(v) ↑ xα as v ↑ +∞, where xα is a solution of the equation K(x) = 0, x ≥ 0, where

K(x) := −cα xα + ax + 1. Therefore | Kϕ |2H ≤ cE

 T ∗   +∞ 0

  +∞

ϕ 2 (s, T)μ(dT)

0

e−2

s 0

0

≤ c xα μ([0, +∞)) E

 T ∗   +∞

 s ϕ 2 (s, T)μ(dT) R(s−)e−2 0 R(u)du ds

0

0

  T∗

≤ c xα μ([0, +∞)) | ϕ |2˜ E A

R(s−)e−2

s 0



R(u)du ds .

0

However,     T∗ s −2 0 R(u)du R(s−)e ds = E E 0

R(u)du Dα (T − s)R(s−)μ(dT) ds

T∗

R(s)e−2

s 0

R(u)du

 ds

0

 =E 0

T∗

!  1 s  T∗ d 1 1  − e−2 0 R(u)du ds = E 1 − e−2 0 R(u)du ≤ . ds 2 2 2

11.8 Approximate Completeness

291

Finally, 1 c xα μ([0, +∞)) | ϕ |2˜ A 2 and the boundedness of K follows. In the present situation  +∞ 1 α ˆ K∗ g(t, T) = P(t−, T)(e−D(T−t)R(t−) y − 1)g(t, y)ν(dy), | Kϕ |2H ≤

0

so the condition K∗ g = 0 means that P − a.s. for almost all t ∈ [0, T ∗ ] and for μ-almost all T ≥ t  +∞ 1 α (e−D(T−t)R(t−) y − 1)g(t, y)ν(dy) = 0. 0

However, for t > 0, R(t−) > 0 and on the interval (t, +∞) the function D(T − t) of the T argument is increasing. Since the support of μ is dense in {D(v), v ≥ 0}, by Proposition 11.8.5 we obtain that g = 0.

Part IV Stochastic Equations in the Bond Market

12 Stochastic Equations for Forward Rates

Four types of stochastic equations describing the movement of forward rates are introduced: the Heath–Jarrow–Morton equation, Morton’s equation, the Heath–Jarrow–Morton–Musiela equation and the Morton–Musiela equation.

12.1 Heath–Jarrow–Morton Equation In the Heath–Jarrow–Morton approach to the bond market the forward rates are represented in the form 

t

f (t, T) = f (0, T) +

 α(s, T)ds +

0

 = f (0, T) +

t

α(s, T)ds +

0

t

σ (s, T), dZ(s) ,

0 d  t j=1

σ j (s, T)dZ j (s),

(12.1.1) t ∈ [0, T ∗ ], T > 0,

0

with 0 < T ∗ < +∞ or, equivalently df (t, T) = α(t, T)dt + σ (t, T), dZ(t) ,

t ∈ [0, T ∗ ], T > 0.

Here Z(s) = (Z 1 (s), . . . , Z d (s)) is a d-dimensional L´evy process, σ (s, T) = (σ 1 (s, T), . . . , σ d (s, T)) is the d-dimensional volatility field and α(s, T) is the drift term. The representation does require the specification of α, σ and Z. To ensure that the forward rates have such basic properties like mean reversion and positivity, it is convenient to represent f as a solution of stochastic equations, rather than as a sum of two integrals. It is natural to assume that the volatility is a function of forward rates, say σ (t, T) = g(t, T, f (t−, T)),

t ∈ [0, T ∗ ], T > 0,

296

Stochastic Equations for Forward Rates

where g is a fixed deterministic function g(t, T, z), 0 ≤ t ≤ T < +∞, z ≥ 0. Recall that the HJM model given by (12.1.1) satisfies (MP) if and only if  T  α(t, T) = DJ σ (t, s)ds , σ (t, T) , (12.1.2) t

where J : Rd → R stands for the Laplace transform of Z and DJ for its derivative. This leads to the equation 1  T  2 df (t, T) = DJ g(t, s, f (t−, s))ds , g(t, T, f (t−, T)) dt t

+ g(t, T, f (t−, T)), dZ(t) ,

t ∈ [0, T ∗ ], T > 0,

(12.1.3)

which will be called the Heath–Jarrow–Morton equation (HJM equation).

12.2 Morton’s Equation Morton’s equation is a particular case of the HJM equation (12.1.3) with onedimensional L´evy process and linear function G, i.e. volatility has the form σ (t, T) = λ(t, T)f (t−, T),

t ∈ [0, T ∗ ], T ≥ 0,

(12.2.1)

where λ is a deterministic, positive and continuous function. Although (12.2.1) describes a simple form of dependence of volatility on the forward rate, the drift becomes nonlinear and nonlocal, i.e.  T α(t, T) = J  λ(t, u)f (t−, u)du λ(t, T)f (t−, T), t ∈ [0, T ∗ ], T ≥ 0. t

(12.2.2) The resulting equation,

df (t, T) = J





T

λ(t, u)f (t−, u)du λ(t, T)f (t−, T)dt + λ(t, T)f (t−, T)dZ(t),

t

(12.2.3) will be called Morton’s equation. The problem of solvability of (12.2.3), with λ(t, T) ≡ 1, has been first stated by Morton in [95] in the case when Z was a Wiener process and solved with negative answer: linear volatility implies that there is no solution to (12.2.3) in that case (see [95] Section 4.7 or Filipovi´c [52], Section 7.4). This fact was one of the main reasons why the well known Brace–Ga¸tarek–Musiela (BGM) model was formulated in terms of Libor rates and not in terms of forward rates (see Brace, Ga¸tarek and Musiela [15]).

12.3 The Equations in the Musiela Parametrization

297

12.3 The Equations in the Musiela Parametrization It is sometimes convenient to change the bond market notation by passing from the (t, T) parametrization used so far, which we call the natural frame, to the new one (t, x) introduced by Musiela and which will be called the moving frame. For a current time point t and maturity T we use time to maturity x = T − t. The forward rate in the moving frame is given by r(t, x) := f (t, t + x),

t ∈ [0, T ∗ ], x ≥ 0.

We will concentrate on the case in which σ (s, T) = (σ j (s, T), j = 1, . . . , d) = (g j (T − s, f (s−, T), j = 1, . . . , d). Assume that the function valued process r(t, ·), t ∈ [0, T ∗ ] takes values in a Hilbert space H of some functions defined on [0, +∞), such as    +∞ | h(x) |2 eγ x dx < +∞ , (12.3.1) L2,γ := h : [0, +∞) → R : 0

or

  H 1,γ h : [0, +∞) → R :

+∞

 (| h(x) |2 + | h (x) |2 )eγ x dx < +∞ .

(12.3.2)

0

We will show that the solution of the HJM equation regarded in the Musiela parametrization satisfies the so-called Heath–Jarrow–Morton–Musiela (HJMM) stochastic evolution equation dr(t, x) =

d  ∂  G j (r(s−))(x)dZ j (s), r(t, x) + F(r(t))(x) dt + ∂x j=1

(12.3.3)

r(0, ·) = r0 . The transformations F, G j , j = 1, . . . , d are of the form G (h)(x) = g (x, h(x)), j = 1, . . . d, j

j

 F(h)(x) = DJ(

x

G(h(v))dv), Gh(x) 0

for h ∈ H and x ≥ 0. Moreover, S(t):

∂ ∂x

denotes the generator A of the shift semigroup

S(t)h(x) = h(t + x), t ≥ 0, x ≥ 0. For basic definitions related to stochastic evolution equations, we refer to Appendix C where the concepts of strong, weak and mild solutions are defined. As weak and mild solutions usually coincide we often speak about solutions meaning weak and/or mild solutions. Strong solutions exist very rarely.

298

Stochastic Equations for Forward Rates

We prove now that r is a mild solution of the equation (12.3.3), that is,  t d  t  r(t) = S(t)r0 + S(t − s)F(r(s))ds + S(t − s)G j (r(s−))dZ j (s). (12.3.4) 0

j=1

0

Assume that r(t, x) = f (t, x), r(t−, x) = f (t−, t + x), t ≥ 0, x ≥ 0 and that the integral version of (12.1.1),  t d  t  f (t, T) = f (0, t) + α(s, T)ds + σ j (s, T)dZ j (s), (12.3.5) 0

j=1

0

where σ j (s, T) = g j (T − s, f (s−, T)),

α(s, T) = DJ



T

 σ (s, v)dv , σ (s, T) , s ≤ T,

s

(12.3.6) holds with full probability for all 0 ≤ s ≤ T < +∞. Thus inserting T = t + x into (12.3.5) yields  t d  t  α(s, t + x)ds + σ j (s, t + x)dZ j (s). r(t, x) = f (t, t + x) = f (0, t + x) + 0

j=1

0

Moreover, σ j (s, t + x) = g j (t + x − s, f (s−, t + x)) = g j (t + x − s, r(s−, t + x − s)). Consequently, d  t d  t   j σ (s, t + x)dZ (s) = g j (t + x − s, f (s−, t + x))dZ j (s) j=1

0

j=1

=

d   j=1

=

0

g j (t + x − s, r(s−, t + x − s))dZ j (s)

0

d   j=1

t

t

S(t − s)G j (r(s−))(x)dZ j (s).

0

Similarly, α(s, t + x) 4  = DJ 4



t+x

5 g(u − t, f (t−, u)du), g(t + x − s, f (s−, t + x)

s t−s+x

= DJ

5 g(v + s − t, r(t−, v + s − t))dv), g(t − s + x, r(s−, t − s + x)

0

= S(t − s)F(r(s))(x),

12.3 The Equations in the Musiela Parametrization and



t

 α(s, t + x)ds =

0

t

299

S(t − s)F(r(s))(x)ds.

0

Addition of the obtained formulae yields the required identity. In a similar way Morton’s equation in the Musiela parameterization becomes:    x ∂ r(s, u)λ(u)du) λ(x)r(t, x) dt r(t, x) + J  dr(t, x) = ∂x 0 + λ(x)r(t−, x)dZ(t),

t ∈ [0, T ∗ ],

x ≥ 0,

and will be called the Morton–Musiela equation. Here λ is a fixed function and the corresponding one-dimensional volatility σ (t, T, f (t−, T) = λ(T − t)f (t−, T). This corresponds to the function g(x, z) = λ(x)z. Its mild version is of the form   x  t  St−s J r(s, u)λ(u)du λ(x)r(s, x) ds r(t, x) = r(0, x) + 0

 +

0 t



 St−s λ(x)r(s−, x) dZ(s),

t ∈ [0, T ∗ ],

x ≥ 0. (12.3.7)

0

The main advantage of passing to the Musiela parametrization is that the forward rate process becomes Markovian and evolves in a fixed state space. General results on Markov processes and stochastic evolution equations can be used. In the present part of the book the existence and uniqueness questions for the equations introduced are settled. They should provide a starting point to investigate asymptotic behaviour, time-reversion of forward curves and more special questions such as consistency of the models. The equation (12.3.3) was intensively studied in the case in which Z is a Wiener process in Rd (see e.g. Filipovi´c [54], Peszat and Zabczyk [100] and references therein). Then the function DJ is linear, i.e. DJ(z) = Qz, z ∈ Rd , where Q is the covariance matrix of Z. There are also several results for the case of general, also infinite dimensional, L´evy process Z (see e.g. Peszat and Zabczyk [100], Filipovi´c and Tappe [56], Rusinek [109], Rusinek [111], Rusinek [112], Marinelli [90], Filipovi´c, Tappe and Teichmann [57], Peszat and Zabczyk [101]). In particular, in [90] the local solvability of (12.3.3) was studied for a L´evy process Z having exponential moments. An extensive study of the HJM equations in Banach spaces can be found in Brze´zniak and Tayfun [23].

13 Analysis of the HJMM Equation

In this chapter we study the solvability of the HJMM equation through the general theory of stochastic evolution equations. We establish the existence of local and global solutions. In particular, we deal with local solutions of the Morton–Musiela equation.

13.1 Existence of Solutions to the HJMM Equation In this section we examine the existence of weak solutions (see Appendix C) of the HJMM equation dr(t, x) =

∂  r(t, x)+F(r(t))(x) dt+G(r(t−))(x)dZ(t), r(0, ·) = r0 , t ∈ [0, T ∗ ], ∂x (13.1.1)

with a one-dimensional L´evy process Z and coefficients of the form G(r)(x) = g(x, r(x)), x ≥ 0, r ∈ H,  x  g(v, r(v))dv g(x, r(x)), F(r)(x) = J

(13.1.2) x ≥ 0, r ∈ H.

(13.1.3)

0

In the preceding g : R × R −→ R is some function and J  denotes the derivative of the Laplace exponent of the process Z. Recall that for a L´evy process Z with characteristic triplet (a, q, ν), J is given by  1 z ∈ R, J(z) = −az + qz2 + (e−zy − 1 + zy1(−1,1) (y)) ν(dy), 2 R and J  (z) = −a + qz +

 R

y(1(−1,1) (y) − e−zy ) ν(dy),

z ∈ R,

13.1 Existence of Solutions to the HJMM Equation

301

providing that the preceding integrals exist. We focus on local and global solutions of 2,γ 1,γ equation (13.1.1) living in the spaces H = L+ ([0, +∞)) and H+ ([0, +∞)), (see (12.3.1) and (12.3.2)). In view of Theorem C.1.4, in Appendix C, we need conditions for local Lipschitz property and linear growth of the transformations F : H −→ H,

G : H −→ H.

They will be formulated in terms of the function g and the characteristic triplet of the process Z. Our analysis uses the following decomposition of the Laplace exponent J 1 J(z) = −az + qz2 + J1 (z) + J2 (z) + J3 (z) + J4 (z), 2

z ∈ R,

(13.1.4)

where  J1 (z) :=

−1 −∞



1

J3 (z) :=



(e−zy − 1)ν(dy), −zy

(e

J2 (z) :=

0 −1

 − 1 + zy)ν(dy),

(e−zy − 1 + zy)ν(dy),

+∞

J4 (z) :=

0

(e−zy − 1)ν(dy).

1

It is natural to require that forward rates are positive processes. Therefore, before passing to the existence results, we establish a theorem providing necessary and sufficient conditions on the volatility function g implying the positivity of the solutions of the HJMM equation. Theorem 13.1.1 Assume that the coefficients G and F of the equation (13.1.1) are locally Lipschitz in H, where H = L2,γ or H = H 1,γ . Then (13.1.1) is positivity preserving if and only if r + g(x, r)u ≥ 0 g(x, 0) = 0

for all r ≥ 0, x ≥ 0, u ∈ supp ν,

(13.1.5)

for all x ≥ 0.

(13.1.6)

Proof We will use Theorem C.1.5 in a similar way to Peszat and Zabczyk [100]. Let us consider the L´evy–Itˆo decomposition of Z Z(t) = at + qW(t) + Z0 (t) + Z1 (t), where  t  t yπ˜ (ds, dy), Z1 (t) := Z0 (t) := 0

|y|≤1

0

|y|>1

yπ(ds, dy),

and a sequence of its approximations of the form Z n (t) = at + qW(t) − tmn + (Z0n (t) + Z1 (t)), with Z0n (t) :=

t

0 { 1n = 0 if and only if g(x, 0) = 0. The solution remains positive in the moment of jump of Z n if and only if  1 r + g(x, r)u ≥ 0, r ≥ 0, u ∈ supp{ν} ∪ , +∞ . n Passing to the limit n → +∞ we obtain (13.1.5).

13.1.1 Local Solutions 2,γ

For the solvability of (13.1.1) in L+ we will need the following conditions on g. ⎧ ⎪ (i) The function g is continuous on R2+ and ⎪ ⎪ ⎪ ⎪ ⎪ g(x, 0) = 0, g(x, y) ≥ 0, x, y ≥ 0. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨(ii) For all x, y ≥ 0 and u ∈ supp ν, (G1) ⎪ y + g(x, y)u ≥ 0. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (iii) There exists a constant C > 0 such that ⎪ ⎪ ⎪ ⎩ | g(x, u) − g(x, v) |≤ C | u − v |, x, u, v ≥ 0. Conditions (G1(i)) and (G1(ii)) ensure that the solution of (13.1.1) is positive (see Theorem 13.1.1) while (G1(iii)) is needed for the Lipschitz property of F and G. Theorem 13.1.2 some z0 > 0, (L1)

Assume that (G1) holds and either Z is a Wiener process or for 

−1

−∞

| y |2 ez0 |y| ν(dy) < +∞,

 and 1

+∞

y2 ν(dy) < +∞.

13.1 Existence of Solutions to the HJMM Equation

303

2,γ

Then for arbitrary initial condition r0 ∈ L+ there exists a unique local solution of 2,γ (13.1.1) in L+ . Condition (L1) in the theorem implies that  | y |2 ν(dy) < +∞, |y|>1

so Z satisfies E[Z 2 (t)] < +∞,

t ∈ [0, T ∗ ].

Consequently, Z is a square integrable martingale with drift. 1,γ

For the existence of local solutions in H+ we will need more stringent conditions on the function g. ⎧   2 ⎪ ⎪(i) The functions gx , gy are continuous on R+ and ⎪ ⎪ ⎪ ⎪ gx (x, 0) = 0, x ≥ 0. ⎪ ⎪ ⎨ (G2) (ii) supx,y≥0 | gy (x, y) |< +∞. ⎪ ⎪ ⎪ ⎪ ⎪ (iii) There exists a constant C > 0 such that ⎪ ⎪ ⎪ ⎩ | gx (x, u)−gx (x, v) | + | gy (x, u)−gy (x, v) |≤ C | u−v |, x, u, v ≥ 0. Theorem 13.1.3 (L2)

Assume that (G1) and (G2) hold and for some z0 > 0,  −1  +∞ | y |3 ez0 |y| ν(dy) < +∞, and y3 ν(dy) < +∞. −∞

1

Then for arbitrary initial condition r0 ∈ 1,γ (13.1.1) in H+ . Since (L2) implies that

1,γ H+

there exists a unique local solution of

 |y|>1

| y |3 ν(dy) < +∞,

we see that Z satisfies E[| Z(t) |3 ] < +∞,

t ≥ 0.

We pass to the proofs of the theorems. Proof of Theorem 13.1.2 We use the fact that locally Lipschitz coefficients of a general evolution equation imply the existence of its local solutions (see Peszat and Zabczyk [100]). We show thus that F and G given by (13.1.2) are locally Lipschitz in L2,γ . Step 1: We show that condition (L1) is satisfied if and only if the function J  is locally Lipschitz.

304

Analysis of the HJMM Equation

Differentiation of (13.1.4) yields J  (z) = −a + qz + J1 (z) + J2 (z) + J3 (z) + J4 (z), with



J1 (z) := − J3 (z) :=

−1 −∞



1

ye−zy ν(dy),

J2 (z) :=

y(1 − e−zy )ν(dy),



0 −1

0

y(1 − e−zy )ν(dy),



J4 (z) := −

z ∈ R,

+∞

ye−zy ν(dy),

1

providing that the preceding integrals exist (see for instance Lemma 8.1 and 8.2 in Rusinek [109]). Similarly, J  (z) = q + J1 (z) + J2 (z) + J3 (z) + J4 (z), z ∈ R, where  −1  0 y2 e−zy ν(dy), J2 (z) := y2 e−zy ν(dy), J1 (z) := −∞



J3 (z) :=

1

−1

y2 e−zy ν(dy),

J4 (z) :=

0



+∞

y2 e−zy ν(dy).

1

The function J  is Lipschitz on [0, z0 ) for some z0 > 0 if and only if J  is bounded on [0, z0 ). Since J2 and J3 are bounded on [0, z0 ) one needs conditions that yield that J1 and J4 are bounded. This leads to L(1). Step 2: By (G1(iii)) it is clear that G is Lipschitz in L2,γ . We prove that if J  is locally Lipschitz and (G1) holds then F is locally Lipschitz as well. For any r, r¯ ∈ L2,γ we have  +∞   x 2  g(y, r(y))dy g(x, r(x)) J | F(r) − F(¯r) |L2,γ = 0

0

−J 



x

2 g(y, r¯ (y))dy g(x, r¯ (x)) eγ x dx

0

≤ 2I1 + 2I2 , where



+∞ 



+∞ 



J

I1 := 0

 I2 :=

x

 g(y, r(y))dy − J 

x

2  2 g(y, r¯ (y))dy g(x, r¯ (x)) − g(y, r(x))dx eγ x dx.

0

J

0

x

2 g(y, r¯ (y))dy g2 (x, r(x))eγ x dx,

0

0

By (G1(iii)) we have  x  x   g(y, r(y))dy = g(y, r(y)) − g(y, 0) dy ≤ C 0

0



≤C

+∞

−γ y

e 0

x

r(y)dy

0

1/2  dy

+∞ 0

1/2 C | r(y) | e dy ≤ √ | r |L2,γ . γ 2

γy

13.1 Existence of Solutions to the HJMM Equation

305

Denoting by D = D(| r |L2,γ , | r¯ |L2,γ ) the local Lipschitz constant of J  , we obtain  I1 ≤ D

+∞  x 

0

 2 g(y, r(y)) − g(y, r¯ (y)) dy g2 (x, r(x))eγ x dx

0



≤ D | g(·, r(·)) − g(·, r¯ (·)) |2L2,γ

+∞

0

 ≤ D | g(·, r(·)) − g(·, r¯ (·)) |2L2,γ ·  ≤ DC

2

+∞

g2 (x, r(x))eγ x dx

+∞ 

0

(r(x) − r¯ (x)) e dx · C 2 γx

0



2 g(x, r(x)) − g(x, 0) eγ x dx

+∞

2

r2 (x)eγ x dx

0

≤ DC4 | r − r¯ |2L2,γ | r |2L2,γ . Similarly, for a local boundary B of J  we get I2 ≤ BC2 | r − r¯ |2L2,γ , and thus the local Lipschitz property of F follows. Proof of Theorem 13.1.3: We prove that F and G are locally Lipschitz in H 1,γ . Step 1: We show that (L2) is equivalent to the fact that J  is locally Lipschitz. Using (13.1.4) we obtain J  (z) = J1 (z) + J2 (z) + J3 (z) + J4 (z), where J1 (z) := − J3 (z)



−1 −∞



1

:= −

y3 e−zy ν(dy),

J2 (z) := −

3 −zy

J4 (z)

y e 0

ν(dy),



0 −1



y3 e−zy ν(dy),

+∞

:= −

y3 e−zy ν(dy).

1

For fixed z0 > 0 the functions J2 , J3 are bounded on [0, z0 ). It follows that J  is bounded on [0, z0 ) if and only if J1 and J4 are bounded on [0, z0 ), which leads to (L2). Let us notice that (L2) implies (L1) and, consequently, that J  is also locally Lipschitz. Step 2: We show that if J  and J  are locally Lipschitz and (G1), (G2) hold then 1,γ F and G are locally Lipschitz in H+ . To get the required estimation for G we need to estimate  I0 := 0

+∞

!2

gy (x, r(x))r (x) − gy (x, r¯ (x))¯r (x)

eγ x dx.

306

Analysis of the HJMM Equation

Using Lemma 15.2.3 we obtain the following inequalities  +∞  2  2    I0 ≤ 2 gy (x, r(x)) r (x) − r¯  (x)) eγ x dx 0  +∞  2  2    + gy (x, r(x)) − gy (x, r¯ (x)) r¯  (x)) eγ x dx 0

≤ 2 sup | gy (x, r) |2 x,r≥0

+ 2C



+∞

2



+∞

| r (x) − r¯  (x) |2 eγ x dx

0

| r(x) − r¯ (x) |2 | r¯  (x) |2 eγ x dx

0

≤ 2 sup | gy (x, r) |2 · | r − r¯ |2H 1,γ + 2C2 x,r≥0

4 | r − r¯ |2H 1,γ · | r¯ |2H 1,γ , γ

and thus the local Lipschitz property of G follows. To show the same for F it is sufficient to show the Lipschitz estimation for the formula  +∞    x d  g(y, r(y))dy g(x, r(x)) J I := dx 0 0 2  x g(y, r¯ (y))dy g(x, r¯ (x)) eγ x dx. −J  0

By explicit calculations we obtain I ≤ 3I1 + 3I2 + 3I3 , where 

+∞ 

J 

I1 :=



0

 I2 :=

 g(y, r(y))dy g2 (x, r(x)) − J 

0 +∞   J

0



x

J

I3 := 0

 0

x

2 g(y, r¯ (y))dy g2 (x, r¯ (x)) eγ x dx,

0

x

0 +∞ 

x



g(y, r(y))dy gx (x, r(x)) − J 

 0

x

2 g(y, r¯ (y))dy gx (x, r¯ (x)) eγ x dx,

2  x g(y, r(y))dy gy (x, r(x)) · r (x)−J  g(y, r¯ (y))dy gy (x, r¯ (x)) · r¯  (x) eγ x dx. 0

With the use of (G1) and (G2) we can estimate I1 and I2 in a similar way as we did in Step 2 of the proof of Theorem 13.1.2. In fact, by the estimation of I1 a new fourth-power term of r¯ appears, which can be estimated as follows  +∞  +∞ 4 γx 2 | r¯ (x) | e dx ≤ sup | r¯ (x) | | r¯ (x) |2 eγ x dx ≥ C | r¯ |4H 1,γ 0

x≥0

0

13.1 Existence of Solutions to the HJMM Equation

307

(see Lemma 15.2.3). To estimate I3 , we need additional inequalities for  +∞ !2 gy (x, r(x))r (x) − gy (x, r¯ (x))¯r (x) eγ x dx, 0

which is exactly I0 and is estimated in the preceding, and for  +∞ !2 gy (x, r¯ (x))¯r (x) eγ x dx. I4 := 0

Estimation for I4 follows from the bound on gy .

13.1.2 Global Solutions 2,γ

We pass now to the existence of global solutions of (13.1.1) in the spaces L+ and in 1,γ H+ . As one suspects, we need stronger conditions for the function g and the L´evy process Z than those formulated for local solutions in the previous section. Theorem 13.1.4 q = 0,

Assume that (G1) holds and that  +∞ supp{ν} ⊆ [0, +∞), max{y, y2 }ν(dy) < +∞.

(13.1.8)

0 2,γ

Then for arbitrary r0 ∈ L+ the equation (13.1.1) has a unique global solution 2,γ in L+ . The requirements for the function g are the same as in Theorem 13.1.2 but the process Z is a square integrable subordinator with drift. 1,γ

For global existence in H+ we need additional conditions for g. ⎧ ⎪ (i) Partial derivatives gy , gxy , gyy are bounded on R2+ . ⎪ ⎪ ⎨ √ (G3) (ii) 0 ≤ g(x, y) ≤ c y, x, y ≥ 0. ⎪ ⎪ ⎪ 2,γ ⎩ (iii) | gx (x, y) |≤ h(x), x, y ≥ 0, for some h ∈ L+ . Theorem 13.1.5 q = 0,

Assume that conditions (G1), (G2) and (G3) are satisfied and  +∞ supp{ν} ⊆ [0, +∞), max{y, y3 }ν(dy) < +∞. (13.1.9) 0

Then for arbitrary r0 ∈

1,γ H+

1,γ

there exists a unique global solution of (13.1.1) in H+ .

The class of L´evy processes in Theorem 13.1.5 is much narrower than that required for the existence of local solutions in Theorem 13.1.3. Now Z must not have the Wiener part and has positive jumps only. It is of finite variation and E[| Z(t) |3 ] < +∞,

t ≥ 0.

Thus Z is a subordinator with drift that is integrable in the third power. We pass to the proofs of the theorems.

308

Analysis of the HJMM Equation

Proof of Theorem 13.1.4 As local Lipschitz conditions for F and G in L2,γ were shown in the proof of Theorem 13.1.2, our goal now is to show the linear growth condition for the transformation F. The analogous result for G is trivial. Step 1: We show that, under (13.1.8), J  is bounded on [0, +∞). It follows from (13.1.4) that J  (0) is finite if and only if both J1 (0) and J4 (0) are finite, which is equivalent to  | y | ν(dy) < +∞. (13.1.10) |y|>1

Moreover, it follows that lim | J1 (z) |= +∞,

lim | J2 (z) |= +∞,

z→+∞

and that | J3 | is bounded ⇐⇒

z→+∞



| J4 (z) | is bounded ⇐⇒

1

0



yν(dy) < +∞, +∞

yν(dy) < +∞.

1

on [0, +∞) if and only if Z has neither Wiener part nor Consequently, J  is bounded  +∞ negative jumps and 0 yν(dy) < +∞. Step 2: We show that if J  is bounded on [0, +∞) then F has linear growth in L+ . Let C1 := supz≥0 J  (z) < +∞. The assertion follows from the estimation 2,γ

| F(r) |L2,γ  +∞   = J 0



≤ C1

x

2  γx g(y, r(y))dy g(x, r(x)) e dx ≤ C1

0 +∞



[g(x, r(x))−g(x, 0)]2 eγ x dx ≤ C1 C2

0

0

+∞

[g(x, r(x))]2 eγ x dx

0 +∞

r2 (x)eγ x dx ≤ C1 C2 | r |2L2,γ .

Proof of Theorem 13.1.5 Since the local Lipschitz property of F and G in H 1,γ were shown in the proof of Theorem 13.1.3, we prove now the linear growth of F and G. Step 1: We show that (13.1.9) implies that J  is bounded on [0, +∞). By similar analysis as in the proof of Theorem 13.1.3, one obtains that lim | J1 (z) |= +∞,

z→+∞

lim | J2 (z) |= +∞,

z→+∞

13.1 Existence of Solutions to the HJMM Equation and |

J3

| is bounded,

|

J4 (z)



+∞

| is bounded ⇐⇒

309

y2 ν(dy) < +∞.

1

So, J  is bounded on [0, +∞) if and only if the Wiener part of Z disappears, there  +∞ are no negative jumps and 1 y2 ν(dy) < +∞. From Step 1 of the proof of Theorem 13.1.4 we see that, under (13.1.9), also J  is bounded on [0, +∞). Step 2: We prove that if J  and J  are bounded on [0, +∞) and (G1), (G2), (G3) hold then G and F have linear growth. The linear growth of G follows from the estimation  +∞ d | g(x, r(x)) |2 eγ x dx dx 0  +∞ !2 = gx (x, r(x)) + gy (x, r(x))r (x) eγ x dx 0



+∞

≤2 0

[h(x)]2 eγ x dx + 2 sup | gy (x, r) |2 x,r≥0



+∞

| r (x) |2 eγ x dx

0

≤ 2 | h |2L2,γ + 2 sup | gy (x, r) |2 · | r |2H 1,γ .

(13.1.11)

x,r≥0

To show the linear growth of F let us start with the inequality  +∞     x  2 d   g(v, r(v))dv g(x, r(x))  eγ x dx J  dx 0 0  +∞    x 2     ≤2 g(v, r(v))dv g2 (x, r(x)) eγ x dx J 0

 +

0

0

+∞      J

0

+∞

2   g(v, r(v))dv [gx (x, r(x)) + gy (x, r(x))r (x)] eγ x dx.

The second integral can be estimated in the same way as (13.1.11). The linear growth of the first integral follows from the inequality  +∞  g(x, r) 4  +∞   | g(x, r(x)) |4 eγ x dx ≤ sup  √  | r(x) |2 eγ x dx. r 0 0 x,r≥0

13.1.3 Applications to the Morton–Musiela Equation In this section we use general results from the previous sections to examine the existence of solutions of the HJMM equation in the specific case in which g(x, y) = λ(x)y,

x, y ≥ 0,

310

Analysis of the HJMM Equation

where λ(·) is a continuous nonnegative bounded function on [0, +∞). As we already know we arrive then at Morton–Musiela equation ∂  dr(t, x) = r(t, x) + F(r(t))(x) dt+G(r(t−))(x)dZ(t), r(0, ·) = r0 , t ∈ [0, T ∗ ], ∂x (13.1.12) with G(r)(x) = λ(x)r(x),



F(r)(x) = J 

x

 λ(v)r(v) λ(x)r(x),

x ≥ 0.

0

(13.1.13) 2,γ

1,γ

As before, the state spaces are L+ and H+ . It turns out that the applicability of general results on the existence of global solutions is of limited use because F hardly ever satisfies the linear growth condition. This is why we have a special chapter on Morton–Musiela equations in which we develop the methodology introduced by Morton leading to better results. Proposition 13.1.6 If the drift transformation F given by (13.1.13) is of linear growth in L2,γ , then J  is bounded on [0, +∞). In particular, the characteristic triplet of Z satisfies  +∞ yν(dy) < +∞. (13.1.14) q = 0, supp{ν} ⊆ [0, +∞) and 0

It follows that only subordinators with drift make F of linear growth in L2,γ . However, in Section 15.1 we show, with the use of alternative methods, that solutions of (13.1.12)–(13.1.13) exist also for more general L´evy processes. Proof

Assume, to the contrary, that J  is unbounded and define rn (x) = n1[1,3] (x),

n = 1, 2, . . . .

Since for sufficiently large z ≥ 0 the function (J  (z))2 is increasing, we have, for large n 2 2 3   3    x λ2 (x)n2 eγ x dx λ2 1 J  λn(x − 1) eγ x dx 1 J 1 λ(y)ndy | F(rn ) |2 = ≥ ,   3 3 γx | rn |2 n2 eγ x dx e dx 1

1

where λ := infx≥0 λ(x). Since  3   2 eγ x dx ≥ J  λn(x − 1) 1

2  eγ x dx J  λn(x − 1)

3

2



2 

≥ J  (λn)

2

3

eγ x dx −→ +∞, n

13.1 Existence of Solutions to the HJMM Equation

311

the main claim holds. Splitting J  (z) as in the proof of Theorem 13.1.2 one can show that J  is bounded on [0, +∞) if and only if (13.1.14) holds. The existence of local solutions can be deduced from local Lipschitz properties studied for a general function g in Section 13.1.1. This requires assumptions on the function λ as well as on the jumps of the L´evy process. The following theorem is a direct consequence of Theorem 13.1.2. Theorem 13.1.7 (0) (1) (L1)

Assume that:

λ is continuous and infx≥0 λ(x) = λ > 0, supx≥0 λ(x) = λ¯ < +∞, supp ν ⊆ [− λ1¯ , +∞),  +∞ 2 y ν(dy) < +∞, 1

hold. Then there exists a unique local weak solution to the equation (13.1.12)– 2,γ (13.1.13) taking values in the space L+ . The positivity assumptions (G1)(i), (G1)(ii) required in Theorem 13.1.2 follow from (0), (1) while (G1)(iii) follows from (0). In the formulation of the theorem a simplified, but under (0), (1), equivalent version of the condition (L1) from Theorem 13.1.2 is used. Notice that the process Z now is required to be a square integrable martingale with drift but without negative large jumps. Similarly, as a consequence of Theorem 13.1.3 we obtain the following local 1,γ existence result in H+ . Theorem 13.1.8 (2) and (L2)

Assume that conditions (0), (1),

λ, λ are bounded and continuous on R+ ,  +∞ 1

y3 ν(dy) < +∞,

are satisfied. Then there exists a unique local weak solution to the equation (13.1.12) 1,γ taking values in the space H+ . Comparing with Theorem 13.1.7 we additionally require for the process Z that E[| Z(t) |3 ] < +∞,

t > 0.

14 Analysis of Morton’s Equation

This chapter is concerned with the solvability of the Morton equation. Necessary conditions and sufficient conditions on the L´evy process and the volatility function are specified for the equation to have a solution. An important role in the proofs is played by an operator version of the equation.

14.1 Results Here we study the solvability of the Morton equation,  T  df (t, T) = J λ(t, u)f (t−, u)du λ(t, T)f (t−, T)dt + λ(t, T)f (t−, T)dZ(t), t

(14.1.1) on the bounded domain

& ' (t, T) ∈ T := (t, T) ∈ R2 : 0 ≤ t ≤ T ≤ T ∗ ,

(14.1.2)

with an initial condition f (0, T) = f0 (T), T ∈ [0, T ∗ ].

(14.1.3)

Like Morton [95] we treat (14.1.1) as a family of stochastic equations indexed by T. This approach arises from the classical formulation of the HJM model. In Chapter 15 the solution will be treated as a function-valued process. Solutions of (14.1.1)–(14.1.3) will be searched in the class of random fields f (t, T), (t, T) ∈ T such that f (·, T) is adapted and c`adl`ag on [0, T] for all T ∈ [0, T ∗ ],

(14.1.4)

f (t, ·) is continuous on [t, T ∗ ] for all t ∈ [0, T ∗ ],

(14.1.5)

P( sup f (t, T) < +∞) = 1.

(14.1.6)

(t,T)∈T

Random fields satisfying (14.1.4)–(14.1.6) will be called regular.

14.1 Results

313

Results on the existence and nonexistence of solutions will be established under the following standing assumptions. The initial curve f0 is positive and continuous on [0, T ∗ ]. ¯ +∞) The support of the L´evy measure is contained in the interval (−1/λ,

(A1) (A2) and



∞

yν(dy) < +∞,

1

where λ :=

(A3)

λ¯ := sup λ(t, T) < +∞.

inf λ(t, T) > 0,

(t,T)∈T

(t,T)∈T

(14.1.7)

⎧ t ⎪ ⎨ For each 0 < t ≤ T ∗ the process 0 λ(s, T)dZ(s); T ∈ [t, T ∗ ] is continuous. ⎪ ⎩ The field |  t λ(s, T)dZ(s) | is bounded on T . 0

Under (A1)–(A3) the solvability of (14.1.1) depends on the growth of the function J  . Theorem 14.1.1

Assume that for some a > 0, b ∈ R, J  (z) ≥ a ln3 z + b,

(J1)

∀z > 0.

For arbitrary κ ∈ (0, 1), there exists a positive constant K such that if f0 (T) > K,

∀T ∈ [0, T ∗ ],

(14.1.8)

then, with probability greater or equal to κ, there are no regular solutions f : T −→ R+ of equation (14.1.1). Theorem 14.1.2

Assume that (J2)

  ¯ ∗ J  (z) = +∞. lim sup ln z − λT z→∞

(i) Then there exists a regular field f : T −→ R+ which solves (14.1.1). ∞ (ii) If, in addition, 1 y2 ν(dy) < +∞ holds, then the solution f is unique in the class of regular fields. The key step in the investigation of equation (14.1.1) is to reformulate it into an alternative fixed point problem: f (t, T) = Af (t, T), where Af (t, T) := a(t, T)e

t 0

J



T s

(t, T) ∈ T ,

 λ(s,u)f (s,u)du λ(s,T)ds

,

(14.1.9)

(t, T) ∈ T ,

314

Analysis of Morton’s Equation

and t

a(t, T) := f0 (T)e

0

2

λ(s,T)dZ(s)− q2

 t  +∞

t

λ2 (s,T)ds+

0

0 −1/λ¯





ln(1+λ(s,T)y)−λ(s,T)y π(ds,dy)

. (14.1.10)

The operator A will be called Morton’s operator and the equation (14.1.9) Morton’s operator equation. It was originally introduced by Morton in [95] for the case when Z was a Wiener process. Solutions of (14.1.9) will be searched also in the class of fields satisfying (14.1.4)–(14.1.6). The rest of the chapter is organized as follows. In the next subsection we comment on the assumptions (A1)–(A3). Then, in Section 14.2, applications are presented and the theorems are reformulated in terms of the characteristics (a, q, ν) of the noise process Z. The proofs of the theorems will be presented in the final two sections. In particular, in Section 14.3 we prove the equivalence of the equations (14.1.1) and (14.1.9), the result that is also used in Section 14.4.

14.1.1 Comments on Assumptions (A1)–(A3) Let us notice that in the equation (14.1.1) or equivalently in (14.1.9) intervene the values of J  (z) for all nonnegative z. To make the equations well posed one needs therefore to know that J  (z) exists for z ≥ 0. We have the following result. Proposition 14.1.3 Under (A2) the function J  is well defined on [0, +∞). Proof

By (A2) the function J can be written in the form 1 J(z) = −az + qz2 + J1 (z) + J2 (z) + J3 (z), 2

where  J1 (z) :=

0

−zy

−1/λ¯

(e

 − 1 + zy) ν(dy),

J2 (z) :=

1

(e−zy − 1 + zy) ν(dy),

0

 J3 (z) :=

∞

(e−zy − 1) ν(dy).

1

The fact that ν is a L´evy measure implies that J(z) is well defined for z ≥ 0. Moreover, it also implies that for z > 0 the functions J1 , J2 , J3 have derivatives of any order (see Lemma 8.1 and 8.2 in Rusinek [109]) and  0  1 J1 (z) = y(1 − e−zy )ν(dy), J2 (z) = y(1 − e−zy )ν(dy), J3 (z) −1/λ¯  ∞

=−

1

0

ye−zy ν(dy).

(14.1.11)

14.2 Applications of the Main Theorems It is clear that the condition



∞

315

yν(dy) < +∞

(14.1.12)

1

implies that J  (0) exists and then 

J (0) =

−a + J3 (0)



∞

= −a −

yν(dy). 1

Taking into account Morton’s operator equation and writing (14.1.10) in the form t

q2

t

2

a(t, T) = f0 (T) e 0 λ(s,T)dZ(s)− 2 0 λ (s,T)ds $ · (1 + λ(s, T)Z(s))e−λ(s,T)Z(s) ,

(t, T) ∈ T ,

s≤t

we see that (A1) and (A2) are necessary for (14.1.9) to have positive solutions. Condition (A3) together with (A1) and (A2) is needed to show the regularity of the field a in (14.1.10) (see Proposition 14.3.7 for details). Condition (A3) is satisfied if, for instance, λ(·, ·) is constant or, more generally, if it is of the form λ(t, T) =

N 

an (t)bn (T),

n=1

where {an (·)}, {bn (·)} are continuous functions. The assumption that λ(·, ·) is continuous does not imply, in general, (A3) (see Brze´zniak, Peszat and Zabczyk [22], Kwapie´n, Marcus and Rosi´nski [86] for counterexamples).

14.2 Applications of the Main Theorems Here we present Theorem 14.1.1 and Theorem 14.1.2 in terms of the characteristic triplet (a, q, ν) of the driving L´evy process Z. In the formulation of all results we implicitly assume that (A1), (A2) and (A3) are satisfied. As in the previous sections, the Laplace exponent J of Z is examined starting from the decomposition 1 J(z) = −az + qz2 + J1 (z) + J2 (z) + J3 (z), 2

z ≥ 0,

where  J1 (z) :=

0

−1/λ¯

−zy

(e

 − 1 + zy) ν(dy),

1

J2 (z) :=

(e−zy − 1 + zy) ν(dy),

0

 J3 (z) := 1

∞

(e−zy − 1) ν(dy).

316

Analysis of Morton’s Equation

The first theorem, which generalizes the result of Morton (see [95]) shows that for the existence of regular solutions to (14.1.9) the Gaussian part of Z must disappear and, what is rather surprising, Z must not have negative jumps. Theorem 14.2.1 If the L´evy exponent J of Z is such that q > 0 or ν{(− λ1¯ , 0)} > 0 then (J1) holds. Consequently, there are no regular solutions to (14.1.1). Proof

If q > 0 then J  satisfies J  (z) ≥ −a + qz + J3 (0),

z ≥ 0.

If ν{(− λ1¯ , 0)} > 0 then J  (z) ≥ −a + J1 (z) + J3 (0), and since J1 (z)

 =−

0 −1/λ¯

z ≥ 0,

y3 e−zy ν(dy) ≥ 0,

J1 is convex and as such it satisfies the inequality J1 (z) ≥ J1 (0)z + J1 (0). In both cases (J1) holds and thus the assertion follows from Theorem 14.1.1. Subordinators with drift provide a class of L´evy processes in which (14.1.1) has a solution. Proposition 14.2.2 If the process Z is a sum of a subordinator and a linear function then (J2) holds and (14.1.1) has a regular solution. In particular, this is true if Z is a compound Poisson process with drift and positive jumps only. In this case condition (J2) is trivially satisfied because J  is bounded. This follows from the calculation  1  ∞  1 J  (z) = −a + y(1 − e−zy )ν(dy) − ye−zy ν(dy) ≤ −a + yν(dy). 0

1

0

The converse implication is not true. If (14.1.1) has a regular solution, then Z does not have to be a subordinator with drift. In the following we present an example. Example 14.2.3

Let Z be a purely jump process with the L´evy measure ν(dy) =

1 1 1 (y)dy, y2 | ln y |γ (0, 2 )

where γ > 0. We show the following. (a) There exists a regular solution of (14.1.9) and (J2) holds for any γ > 0. (b) Z is a subordinator ⇐⇒ γ > 1.

14.2 Applications of the Main Theorems

317

To see this, we find J2 explicitly. Calculations yield 

1 2

J2 (z) =

y(1 − e−zy )

0

1 dy = 2 y | ln y |γ



z 2

0

1 − e−u 1 du. · u | ln uz |γ

For large z, we have 

1 2

J2 (z) ≤ c 0

1 | ln uz |

 du + γ

z 2 1 2

1 1 du. · u | ln uz |γ

The first integral tends to 0 with z → +∞. The second can be written in the form  2z  z 2 1 1 1 du = dv. · z γ 1 u | ln | v | ln v |γ 4 u 2 As a consequence, J2 (z) =0 z→+∞ ln z lim

and thus lim

z→+∞



 ln z − aJ2 (z) = +∞

for any a > 0. Thus (J2) holds and a solution exists. 1 To prove (b) we have to check that 0 yν(dy) < +∞, which is straightforward. In the sequel we consider Morton’s equation with L´evy processes without Gaussian part and without negative jumps. Criteria for the existence of solutions of (14.1.1) may will be formulated in terms of the L´evy measure of Z. In this case, the problem is related to the behaviour of the distribution function  x y2 ν(dy), x≥0 Uν (x) := 0

of the modified L´evy measure y2 ν(dy) near the origin and will be examined using a Tauberian theorem. For the formulation we need the concept of slowly varying functions. A positive function M varies slowly at 0 if for any fixed x > 0, M(tx) −→ 1, M(t)

as t −→ 0.

(14.2.1)

Typical examples are constants or, for arbitrary γ and small positive t, functions  1 γ M(t) = ln . t

318

Analysis of Morton’s Equation

If M varies slowly at zero, then for any ε > 0 the following estimation holds (see Lemma 2 p. 277 in Feller [51]) tε < M(t) < t−ε

(14.2.2)

for all positive t sufficiently small. If f (x) −→ 1, g(x)

as x −→ 0,

then we write f (x) ∼ g(x). Theorem 14.2.4

Assume that for some ρ ∈ (0, +∞), Uν (x) ∼ xρ · M(x),

as x → 0,

(14.2.3)

where M is a slowly varying function at 0. (i) If ρ > 1 then (J2) holds and there exists a regular solution of (14.1.1). (ii) If ρ < 1 then (J1) holds and there is no regular solution of (14.1.1). (iii) If ρ = 1, the measure ν has a density and  M(x) −→ 0

as x → 0,

and 0

1

M(x) dx = +∞, x

(14.2.4)

then (J2) holds and there exists a regular solution of (14.1.1). For the proof of Theorem 14.2.4 we need an auxiliary result. Proposition 14.2.5 The following conditions are equivalent 1 (i) 0 yν(dy) = +∞, (ii) limz→∞ J2 (z) = +∞. Moreover, if the measure ν has a density and Uν (x) ∼ x · M(x),

as x → 0,

(14.2.5)

where M is such that M(x)→c > 0 as x → 0, then each of the preceding conditions is equivalent to (iii)

1 0

M(x) x dx

= +∞.

Proof The equivalence of (i) and (ii) follows directly from the dominated convergence theorem. We show the equivalence of (i) and (iii). In virtue of (14.2.5) we have 

1

c 0

Uν (x) dx ≤ x2

 0

1

M(x) dx ≤ C x

 0

1

Uν (x) dx x2

14.2 Applications of the Main Theorems

319

for some constants 0 < c < C. However, (i) holds if and only if the last integral diverges. In fact, integration by parts yields  ⏐1  1   1  1  x ⏐ Uν (x) 1 1 x 2 2 ⏐ + dx = y g(y)dy · dx = − y g(y)dy yg(y)dy ⏐ 2 2 x 0 x x 0 0 0 0 0 

1

= lim M(x) − x→0

 y2 g(y)dy +

0

1

 yg(y)dy = c +

0

1



1

y2 ν(dy) +

0

yν(dy), 0

where g is a density of ν. Fix ρ ∈ (0, +∞). Let us notice that  1  1  2 −zy J2 (z) = y e ν(dy) = e−zy μ(dy)

Proof of Theorem 14.2.4

0

0

is a Laplace transform of the measure μ(dy) := y2 ν(dy). Thus it follows from Tauberian theorem (see Theorem 2, p. 445 in Feller [51]) that the condition (14.2.3) is equivalent to lim

z→∞

J2 (z) (ρ + 1)z−ρ · M( 1z )

= 1,

(14.2.6)

where  stands for the gamma function. (i) Assume that ρ > 1 and fix ε > 0 such that ρ − ε > 1. Using(14.2.2) we can find z0 > 0 such that M( 1z ) < zε for all z > z0 . In virtue of (14.2.6) for any z > z0 we have the following estimation   z  +∞ 1   −ρ  J2 (z) ≤ J2 (z0 ) + 2 v M vε−ρ dv < +∞. dv ≤ J2 (z0 ) + 2 v z0 z0 Thus condition (J2) holds because J  (z) = −a + J2 (z) + J3 (z),

z ≥ 0,

and J2 is a bounded function. The assertion follows from Theorem 14.1.2. To prove (ii) and (iii) notice first that in view of Proposition 14.2.5: lim J  (z) z→∞ 2

= +∞.

As a consequence of (14.2.2) when ρ ∈ (0, 1) and the assumption (14.2.4) when ρ = 1, we have   z 1 −ρ v ·M dv = +∞, ρ ∈ (0, 1] lim z→∞ a v for any a > 0. By the d’Hospital formula, for any a > 0, J  (z) J2 (z) = lim = 1,  z2 1 −ρ · M( )dv z→∞ (ρ + 1) z→∞ (ρ + 1) z−ρ · M( 1 ) a v v z lim

ρ ∈ (0, 1].

320

Analysis of Morton’s Equation

(ii) ρ ∈ (0, 1); Fix any ε > 0 such that ρ + ε < 1. By (14.2.2) we can find a constant a > 0 such that for any v > a we have M( 1v ) > v−ε . Then for z sufficiently large the following estimation holds  z  z 1 J2 (z) ≥ (1 − ε)(ρ + 1) v−ρ M( )dv ≥ (1 − ε)(ρ + 1) v−ρ v−ε dv v a a =

 (1 − ε)(ρ + 1)  1−(ρ+ε) − a1−(ρ+ε) . z 1 − (ρ + ε)

Consequently, J  satisfies (J1) and the assertion follows from Theorem 14.1.1. (iii) ρ = 1; Let c > 0 be any positive constant. Using (14.2.4) we can fix a constant a > 0 such that  1 < 1. 0 < 1 − 2c max M v∈[a,∞] v For large z satisfying z

J2 (z)

1 a v

· M( 1v )dv

≤ 2,

we have the estimates ln z − cJ2 (z)

  z J2 (z) 1 1   = ln z − c  ·M dv z 1 1 v a v a v · M v dv   z 1 1 ≥ ln z − 2c ·M dv v a v  1 ≥ ln z − 2c[ln z − ln a] max M v∈[a,z] v

   1 1 ≥ 1 − 2c max M ln z + 2c ln a · max M −→ ∞. v∈[a,∞] v∈[a,∞] v v z→∞ Thus condition (J2) holds because J  (z) = −a + J2 (z) + J3 (z),

z≥0

and J2 is a bounded function. The assertion follows from Theorem 14.1.2. We present now two examples for which the conditions M(x) −→ 0  0

1

as x → 0,

M(x) dx = +∞ x

(14.2.7)

(14.2.8)

14.2 Applications of the Main Theorems

321

are not simultaneously satisfied but the existence problem can be solved with the use of Theorem 14.1.2. Example 14.2.6

Let ν be a measure with the density

ν(dx) =

1 (ln 1x )γ + γ (ln 1x )γ −1 · · 1(0,1) (x), x2 (ln 1x )2γ

It can be checked that the function Uν is given by  x y2 · g(y)dy = x · Uν (x) = 0

1 (ln 1x )γ

γ > 1.

.

It is clear that the function M(x) :=

1 (ln 1x )γ

,

γ >1

varies slowly at zero and that (14.2.7) holds. However, condition (14.2.8) is not satisfied and thus Theorem 14.2.4 does not cover this case. We can explicitly show that J2 is bounded and use Theorem 14.1.2. We have z  1  +∞ 1 − e− x (ln x)γ + γ (ln x)γ −1  −zy J2 (z) = y(1 − e )g(y)dy = dx · x (ln x)2γ 0 1  ≤

+∞

1

Example 14.2.7

1 (ln x)γ + γ (ln x)γ −1 dx < +∞. · x (ln x)2γ

Let ν be given by ν(dy) =

1 1(0,1) (y) dy, y1+ρ

ρ ∈ (0, 2).

Then (a) if ρ ∈ (1, 2) then equation (14.1.1) has no regular solutions, (b) if ρ ∈ (0, 1), or ¯ ∗ < 1 then equation (14.1.1) has a regular solution. (c) ρ = 1 and λT Indeed, for ρ ∈ (0, 2) we have Uν (x) =

1 2−ρ x , 2−ρ

x ∈ (0, 1),

and thus (a) and (b) follow from Theorem 14.2.4. If ρ = 1 than (c) cannot be deduced from Theorem 14.2.4 because the function M(x) ≡ 1 does not tend to zero. However, we have  1  z  z 1 − e−v 1 1 − e−v  −zy 1 y(1 − e ) 2 dy = dv = dv, J2 (z) = v z v y 0 0 0 z

322

Analysis of Morton’s Equation

and consequently ln z d H = lim  ¯ ∗ J (z) z→∞ λT z→∞ 2 lim

1 z 1−e−z z

¯ ∗ · λT

= lim

z→∞

1 1 = > 1. ¯ ∗ (1 − e−z ) ¯ ∗ λT λT

This condition clearly implies (J2) and (c) follows from Theorem 14.1.2.

14.3 Proof of Theorem 14.1.1 14.3.1 Outline of the Proof The main idea of the proof of Theorem 14.1.1 on nonexistence is based on a majorizing function method for the equation (14.1.9), which will be shown in Section 14.3.2 to be equivalent to (14.1.1). This method was introduced by Morton in [95] to show that (14.1.1) with a Wiener noise does not have regular solutions. Since the proof of the theorem is rather involved, we outline first its general idea when λ(t, T) ≡ 1. Assume that we want to prove that a solution f1 of the equation f1 (t, T) = e

t 0

T

J (

s

f1 (s,u)du)ds

g1 (t, T),

(t, T) ∈ T ,

(14.3.1)

with a bounded deterministic nonnegative function g1 , explodes in some point (x, y) of the domain T , that is, lim(t,T)→(x,y) f1 (t, T) = +∞. Let us consider an auxiliary function f2 such that lim

(t,T)↑(x,y)

f2 (t, T) = +∞,

(14.3.2)

and for any 0 < δ < y the function f2 is a unique solution of the equation f2 (t, T) = e

t 0

T

R(

s

f2 (s,u)du)ds

g2 (t, T),

(t, T) ∈ Tx,y−δ ,

(14.3.3)

where Tx,y := {(t, T) ∈ T : 0 ≤ t ≤ x, 0 ≤ T ≤ y}, R : R+ → R+ , g2 : Tx,y → R+ is bounded. The function R should be related to J  such that the following estimation holds f1 (t, T) ≥ e

t 0

T

R(

s

f1 (s,u)du)ds

gˆ 1 (t, T),

(t, T) ∈ Tx,y−δ ,

(14.3.4)

where gˆ 1 is a bounded, nonnegative function on Tx,y . The function R for which the preceding estimation holds can be constructed because (J1) holds. The key step of the method is the relation between (14.3.3) and (14.3.4). The following implication is true gˆ 1 (t, T) ≥ g2 (t, T)

(t, T) ∈ Tx,y−δ

⇒

f1 (t, T) ≥ f2 (t, T)

(t, T) ∈ Tx,y−δ , (14.3.5)

which, in view of (14.3.2), yields the explosion of f1 at (x, y). The function f2 is called a majorizing function for f1 . Although our original equation (14.1.9) is not

14.3 Proof of Theorem 14.1.1

323

deterministic we will find a deterministic majorizing function that is valid for each ω ∈ . The property that gˆ 1 majorizes g2 will be obtained by imposing additional constraints on the initial condition f0 . To apply the method described in the preceding we need to find a point (x, y) ∈ T and the functions R and g2 such that the corresponding majorizing function explodes in (x, y). We examine functions R = Rα,γ : R+ −→ R+ given by   z ≥ 0, α > 0, γ ≥ 1 (14.3.6) R(z) = Rα,γ (z) := α ln3 γ (z + e2 ) , and

⎧   t T ⎪ ⎨ e− 0 Rα,γ s h(s,u)du ds · h(t, T) for (t, T) = (x, y), g2 (t, T) = g(t, T) := ⎪ ⎩ 0 for (t, T) = (x, y), (14.3.7)

¯ + := R+ ∪ {+∞} given by with h : Tx,y −→ R ⎧ 1 ⎨ x−t+y−T ϕ(t,T) h(t, T) := e , where ϕ(t, T) := ⎩ +∞

for (t, T) = (x, y), for (t, T) = (x, y),

(14.3.8)

where 0 < x < y ≤ T ∗ , and show that they satisfy all the required properties for certain parameters α and γ .

14.3.2 Equivalence of Equations (14.1.1) and (14.1.9) Proposition 14.3.1 Assume that f is a regular field and conditions (A1) and (A2) are satisfied. Then f is a solution of (14.1.1) if and only if it solves (14.1.9). Proof Let us notice, that for each T the solution f (t, T), t ∈ [0, T] of (14.1.1) is a stochastic exponential and by the Dol´eans-Dade exponential formula (see Theorem 4.4.6) it can be written in the form t

J

 T

 t 2 t λ(s,u)f (s−,u)du λ(s,T)ds+ 0 λ(s,T)dZ(s)− q2 0 λ2 (s,T)ds

f (t, T) = f0 (T) e $ · (1 + λ(s, T)Z(s))e−λ(s,T)Z(s) , 0

s

(t, T) ∈ T .

(14.3.9)

s≤t

Under assumptions (A1) and (A2) we can write equation (14.3.9) in the form t

f (t, T) = f0 (T) e ·e

0

 t  +∞

0 −1/λ¯

J





T s

 t 2 t λ(s,u)f (s−,u)du λ(s,T)ds+ 0 λ(s,T)dZ(s)− q2 0 λ2 (s,T)ds



ln(1+λ(s,T)y)−λ(s,T)y π(ds,dy)

,

(t, T) ∈ T ,

(14.3.10)

324

Analysis of Morton’s Equation

or equivalently as f (t, T) = a(t, T)e

t 0

J

 T s

 λ(s,u)f (s−,u)du λ(s,T)ds

(t, T) ∈ T .

,

(14.3.11)

We show now that we can replace f (s−, u) in (14.3.11) by f (s, u). To do this we prove that for each (t, T) ∈ T , 

t

J



0

 t  λ(s, u)f (s, u)du λ(s, T)ds = J

T

s

0

T

λ(s, u)f (s−, u)du λ(s, T)ds.

s

Let us start with the observation that for T ∈ [0, T ∗ ], moments of jumps of the process f (·, T) are the same as for a(·, T). Moreover, it follows from (14.1.10) that the set of jumps of a(·, T) is independent of T and is contained in the set Z := {t ∈ [0, T ∗ ] : Z(t) = 0}. Thus if s ∈ / Z then  T    λ(s, u)f (s, u)du λ(s, T) = J J s

T

λ(s, u)f (s−, u)du λ(s, T).

s

By Theorem 2.8 in Applebaum [2] the set Z is at most countable, so the assertion follows.

14.3.3 Auxiliary Results The following properties of the function J  will be needed in the sequel. Proposition 14.3.2 (i) If (A2) holds then J1 , J2 , J3 , and thus J  as well, are increasing, real-valued functions on the interval [0, +∞). (ii) J  is a Lipschitz function on [0, +∞) if and only if  ∞ y2 ν(dy) < +∞. (14.3.12) 1

They follow directly from the formulae (14.1.11) and J1 (z) =



0

−1/λ¯

y2 e−zy ν(dy),

J2 (z) =

 0

1

y2 e−zy ν(dy),

J3 (z) =



∞

y2 e−zy ν(dy).

1

(14.3.13) Proposition 14.3.3 Let α > 0, γ ≥ 1 be fixed constants such that αγ > 2 and γ T ∗ > 1. Choose (x, y) ∈ T such that 0 < x < y < α2 ∧ T ∗ and γ (y − x) > 1. Let the functions h, Rα,γ be given by (14.3.8) and (14.3.6), respectively. Then the function g : Tx,y −→ R+ defined by the formula (14.3.7) is continuous.

14.3 Proof of Theorem 14.1.1

325

In the proof of Proposition 14.3.3 we will use the following inequality    b  b   1 1 3 2 ln (14.3.14) p(x)dx + e ≥ ln3 p(x) + e2 dx, b−a a b−a a valid for any positive integrable function p on the interval (a, b), a < b. It can be proven by an application of the Jensen’s inequality to the concave function ln3 (z + e2 ). Proof of Proposition 14.3.3 We need to show the continuity of g only in the point (x, y). Thus consider any point (t, T) ∈ Tx,y that is close to (x, y), i.e. s.t. (t, T) = (x, y) and γ (T − t) > 1. Using the monotonicity of Rα,γ and (14.3.14) we obtain the following estimation −

e

t 0

≤e

Rα,γ

−α

−α

≤e

α

≤ e− T

 T

 h(s,u)du ds

s

t 0

t 0

· h(t, T) = e

   T ln3 γ s h(s,u)du+e2 ds

ln3



tT 0 s

T 1 T−s s

t 0

   T ln3 γ ( s h(s,u)du+e2 ) ds

· h(t, T) ≤ e

 h(s,u)du+e2 ds

ln3 h(s,u)duds

−α

−α

t 0

· h(t, T) ≤ e−α α

· h(t, T) = e− T

tT 0 s

ln3



γ (T−t)  T s T−s

t

T 1 0 T−s s

· h(t, T)

 h(s,u)du+e2 ds

ln3 (h(s,u)+e2 )duds

ϕ 3 (s,u)duds+ϕ(t,T)

· h(t, T) · h(t, T)

.

One can check that  t T 1 t −T 2 − Tt − ty + 2Ty + 2Tx − tx du ds = · , 3 2 (x − t + y − T)(x + y − 2t)(x + y − T)(x + y) 0 s (x − s + y − u) and thus α − T

 t

T

ϕ 3 (s, u)duds + ϕ(t, T)    αt −T 2 − Tt − ty + 2Ty + 2Tx − tx ϕ(t, T). = 1− 2T(x + y − 2t)(x + y − T)(x + y) 0

s

Passing to the limit we obtain lim (−T 2 − Tt − ty + 2Ty + 2Tx − tx) = y2 − x2 ,

t→x,T→y

lim (x + y − 2t) = y − x,

t→x,T→y

Hence lim

t→x,T→y

lim (x + y − T) = x.

t→x,T→y

   αt −T 2 − Tt − ty + 2Ty + 2Tx − tx α =1− < 0, 1− 2T(x − t + y − T)(x + y − 2t)(x + y − T)(x + y) 2y



and consequently limt→x,T→y g(t, T) = 0.

326

Analysis of Morton’s Equation

Proposition 14.3.4 Fix α > 0, γ ≥ 1 s.t. αγ > 2 and γ T ∗ > 1. Let 0 < x < y < α ∗ 2 ∧ T , γ (y − x) > 1, 0 < δ < y and g : Tx,y−δ −→ R+ be a bounded function. Assume that there exists a bounded function h : Tx,y−δ −→ R+ , which solves the equation t

h(t, T) = e

0

Rα,γ



T s

 h(s,u)du ds

· g(t, T),

∀(t, T) ∈ Tx,y−δ ,

(14.3.15)

where Rα,γ is given by (14.3.6). Then h is uniquely determined in the class of bounded functions on Tx,y−δ . For the proof of Proposition 14.3.4 we use the following auxiliary result. Lemma 14.3.5

Let 0 < t0 ≤ T0 < +∞ and define a set & ' A := (t, T) : t ≤ T, 0 ≤ t ≤ t0 , t ≤ T ≤ T0 .

If d : A −→ R+ is a bounded function satisfying  t T d(s, u)duds ∀(t, T) ∈ A, d(t, T) ≤ K 0

(14.3.16)

s

where 0 < K < ∞ then d(t, T) ≡ 0 on A. Proof that

Assume that d is bounded by a constant M > 0 on A. We show inductively

d(t, T) ≤ MK n

(tT)n , (n! )2

∀(t, T) ∈ A.

(14.3.17)

The formula (14.3.17) is valid for n = 0. Assume that it is true for some n and show that it is true for n + 1. We have the following estimation  t  T  t T (su)n n+1 1 d(t, T) ≤ K MK n duds = MK sn ( un du)ds (n! )2 (n! )2 0 0 s s = MK

n+1

= MK n+1

1 (n! )2





t

s 0

n

T n+1 − sn+1 n+1

 ds ≤ MK

n+1

1 (n! )2

n+1 1 tn+1 T n+1 n+1 (tT) . = MK (n! )2 (n + 1) (n + 1) ((n + 1)! )2

Letting n −→ ∞ in (14.3.17) we see that d(t, T) = 0. It can be verified that 

d := Rα,γ (0) =

3α(2 + ln γ )2 e2



t 0

sn

T n+1 ds n+1

14.3 Proof of Theorem 14.1.1

327

and that Rα,γ is concave. Thus |Rα,γ (z1 ) − Rα,γ (z2 )| ≤ d|z1 − z2 |,

z1 , z2 ≥ 0.

(14.3.18)

Proof of Proposition 14.3.4 Assume that h1 , h2 : Tx,y−δ −→ R+ are bounded solutions of (14.3.15). Then the function | h1 − h2 | is bounded and satisfies | h1 (t, T) − h2 (t, T) |≤| g | · | e t

−e

0

Rα,γ



 h2 (s,u)du ds

T s

|,

t 0

Rα,γ

 T s

 h1 (s,u)du ds

∀(t, T) ∈ Tx,y−δ ,

where | g |=

sup

(t,T)∈Tx,y−δ

| g(t, T) | .

As a consequence of the inequality | ex − ey |≤ max{ex , ey } | x − y | for x, y ∈ R we have  T  t  Rα,γ | h1 (t, T) − h2 (t, T) |≤ K h (s, u)du 1  0 s   T  − Rα,γ h2 (s, u)du  ds, ∀(t, T) ∈ Tx,y−δ , s

where K :=| g |

&  t R  T h (s,u)duds ' max e 0 α,γ s i < ∞.

sup

(t,T)∈Tx,y−δ i=1,2

In virtue of (14.3.18) we have  t | h1 (t, T) − h2 (t, T) |≤ dK 0

T

| h1 (s, u) − h2 (s, u) | duds,

∀(t, T) ∈ Tx,y−δ .

s

In view of Lemma 14.3.5, with t0 = min{x, y − δ}, T0 = y − δ, we have h1 (t, T) = h2 (t, T) for all (t, T) ∈ Tx,y−δ . Proposition 14.3.6 Fix α > 0, γ ≥ 1 s.t. αγ > 2, γ T ∗ > 1 and the function Rα,γ given by (14.3.6). Choose (x, y) s.t. 0 < x < y < α2 ∧ T ∗ , γ (y − x) > 1 and δ s.t. 0 < δ < y. Let f1 : Tx,y−δ −→ R+ , where be a bounded function satisfying inequality f1 (t, T) ≥ e

t 0

Rα,γ



T s f1 (s,u)du

 ds

· g1 (t, T),

∀(t, T) ∈ Tx,y−δ ,

(14.3.19)

where g1 : Tx,y−δ −→ R+ . Let f2 : Tx,y−δ −→ R+ be a bounded function solving equation f2 (t, T) = e

t 0

Rα,γ

 T s

 f2 (s,u)du ds

· g2 (t, T),

∀(t, T) ∈ Tx,y−δ ,

(14.3.20)

328

Analysis of Morton’s Equation

where g2 : Tx,y−δ −→ R+ is a bounded function. Moreover, assume that g1 (t, T) ≥ g2 (t, T) ≥ 0,

∀(t, T) ∈ Tx,y−δ .

(14.3.21)

Then f1 (t, T) ≥ f2 (t, T) for all (t, T) ∈ Tx,y−δ . Proof

Let us define the operator K acting on bounded functions on Tx,y−δ by Kk(t, T) := e

t 0

Rα,γ

 T s

 k(s,u)du ds

· g2 (t, T),

(t, T) ∈ Tx,y−δ .

(14.3.22)

Let us notice that in view of (14.3.19), (14.3.21) and (14.3.22) we have Kf1 (t, T) ≤ e

t 0

Rα,γ



T s f1 (s,u)du

 ds

· g1 (t, T) ≤ f1 (t, T),

∀(t, T) ∈ Tx,y−δ . (14.3.23)

It is clear that the operator K is order preserving, i.e. k1 (t, T) ≤ k2 (t, T) ∀(t, T) ∈ Tx,y−δ

⇒ Kk1 (t, T) ≤ Kk2 (t, T) ∀(t, T) ∈ Tx,y−δ . (14.3.24)

Let us consider the sequence of functions: f1 , Kf1 , K2 f1 ,. . . . In virtue of (14.3.23) and (14.3.24) we see that f1 ≥ Kf1 ≥ K2 f1 ≥. . . . Thus this sequence is pointwise convergent to some function f¯ and it is bounded by f1 , so applying the dominated convergence theorem in the formula Kn+1 f1 (t, T) = e

t 0

Rα,γ



T s

 Kn f1 (s,u)du ds

· g2 (t, T),

∀(t, T) ∈ Tx,y−δ

we obtain f¯ (t, T) = e

t 0

Rα,γ



T s

 f¯ (s,u)du ds

· g2 (t, T),

∀(t, T) ∈ Tx,y−δ .

Moreover, f¯ is bounded and thus, in view of Proposition 14.3.4, we have f¯ = f2 . As a consequence f1 ≥ f2 on Tx,y−δ . As a consequence of (A1)–(A3) we have the following regularity of the field a in (14.1.10): Proposition 14.3.7 Assume that the conditions (A1), (A2), (A3) are satisfied. Then the field {a(t, T); (t, T) ∈ T } given by (14.1.10) is bounded from below and above by strictly positive random constants. Moreover, a(·, T) is adapted and c`adl`ag on [0, T] for all T ∈ [0, T ∗ ] and a(t, ·) is continuous on [t, T ∗ ] for all t ∈ [0, T ∗ ]. A rather technical proof can be found in Barski and Zabczyk [7], proposition 2.3.

14.3 Proof of Theorem 14.1.1

329

14.3.4 Conclusion of the Proof Let us notice that for 0 < α˜ < a and any γ˜ ≥ 1 we have   α˜ ln3 γ˜ (z + e2 ) α˜ = < 1. lim 3 z→+∞ a a ln z Thus (J1) implies that for 0 < α˜ < a and any γ˜ ≥ 1 there exists β˜ ∈ R such that   ˜ J  (z) ≥ α˜ ln3 γ˜ (z + e2 ) + β˜ = Rα, z ≥ 0. (14.3.25) ˜ γ˜ (z) + β, ˜ γ˜ such that (14.3.25) holds and Now fix parameters α, ˜ β, 0 < α˜ < a,

γ˜ (λ ∧ 1) ≥ 1,

γ˜ ≥ 1,

λα˜ γ˜ (λ ∧ 1) > 2,

γ˜ (λ ∧ 1)T ∗ > 1.

It can be checked that for a constant c > 0 s.t. γ (c ∧ 1) ≥ 1 we have Rα,γ (cz) ≥ Rα,γ (c∧1) (z),

z ≥ 0.

(14.3.26)

Let us assume that there exists a regular solution of (14.1.9). Using (14.3.25), (14.1.7) and (14.3.26) the forward rate f satisfies the following inequality f (t, T) = e

t 0

λ

≥e

t

t

=e

T

J (

0

0

s

λ(s,u)f (s,u)du)λ(s,T)ds

a(t, T) ≥ e

   T Rα, ˜ ˜ γ˜ λ s f (s,u)du ds βt

Rα,γ

e a(t, T) ≥ e

 T s

t

t 0

0

Rα, ˜ γ˜



T s

Rλα, ˜ γ˜ (λ∧1)

 ˜ λ(s,u)f (s,u)du λ(s,T)ds+βt 

T s

a(t, T)

 f (s,u)du ds βt ˜

 f (s,u)du ds βt ˜

e a(t, T).

e a(t, T) (14.3.27)

˜ γ := γ˜ (λ ∧ 1) satisfy α > 0, γ ≥ 1, αγ > 2, The preceding constants α := λα, γ T ∗ > 1. Choose (x, y) ∈ T such that 0 < x < y < α2 ∧ T ∗ , γ (y − x) > 1 and fix ¯ + , Rα,γ : R+ −→ R+ , g : Tx,y −→ R+ three deterministic functions h : Tx,y −→ R given by (14.3.8), (14.3.6) and (14.3.7), respectively. It is clear that they satisfy the equation h(t, T) = e

t 0

Rα,γ

 T s

 h(s,u)du ds

· g(t, T),

∀(t, T) ∈ Tx,y .

(14.3.28)

In virtue of Proposition 14.3.3 the function g is continuous on Tx,y and thus bounded. It follows from Proposition 14.3.7 that if the constant K is sufficiently large, then with probability arbitrarily close to 1, ˜

eβt a(t, T) ≥ g(t, T),

∀(t, T) ∈ Tx,y .

(14.3.29)

Let us fix 0 < δ < y and consider inequality (14.3.27) and equality (14.3.28) on the set Tx,y−δ . Then the function h is continuous. In virtue of Proposition 14.3.6 we have 1

f (t, T) ≥ h(t, T) = e (x−t+y−T) ,

∀(t, T) ∈ Tx,y−δ .

330

Analysis of Morton’s Equation

For any sequence (tn , Tn ) ∈ Tx,y satisfying tn ↑ x, Tn ↑ y, define a sequence δn := y−Tn 2 . Then 1

∀(t, T) ∈ Tx,y−δn ,

f (t, T) ≥ e (x−t+y−T) ,

and consequently limn→∞ f (tn , Tn ) = +∞, which contradicts the assumption that f is regular. 

14.4 Proof of Theorem 14.1.2 We use again the equivalent equation (14.1.9) f = Af , where, for field f , Af (t, T) := a(t, T) · e

t 0

J

T



s

λ(s,u)f (s,u)du λ(s,T)ds

2

t

and t

a(t, T) := f0 (T)e

0

λ(s,T)dZ(s)− q2

0

 t  +∞

λ2 (s,T)ds+

,



0 −1/λ¯

(t, T) ∈ T

(14.4.1) 

ln(1+λ(s,T)y)−λ(s,T)y π(ds,dy)

.

The proof will be divided into two steps. We fix ω ∈  and treat (14.4.1) as a deterministic transformation with a positive and bounded function a. In Step 1, in Proposition 14.4.1, one shows that under the assumption (J2) the operator A preserves the boundedness of random fields on T . In Step 2, one shows that the limit of an increasing sequence of the iterations of the operator A on the initial function 0 is a solution of the equation. Step 1: Proposition 14.4.1 Assume that the function J  satisfies (J2). Then there exists a positive constant c such that if h(t, T) ≤ c,

∀(t, T) ∈ T

for a nonnegative function h, then Ah(t, T) ≤ c,

∀(t, T) ∈ T .

(14.4.2)

Proof Let us assume that h(t, T) ≤ c for all (t, T) ∈ T for some positive c. Using the fact that J  is increasing and λ positive, we have 

¯

Ah(t, T) ≤ a(t, T) · eJ (λcT

∗)

t 0

λ(s,T)ds

.

By Proposition 14.3.7, a(·, ·) is bounded by a positive constant K = K(ω) and we arrive at the inequality 

¯

Ah(t, T) ≤ KeJ (λcT

∗)

t 0

λ(s,T)ds

, (t, T) ∈ T .

14.4 Proof of Theorem 14.1.2

331

It is therefore enough to find a positive constant c such that  t  ¯ ∗ ln K + J (λcT ) · λ(s, T)ds ≤ ln c, (t, T) ∈ T . 0

J

If the function is negative on [0, +∞) then it is enough to take c = K. If J  takes positive values then it is enough to find a positive arbitrarily large constant c such that ¯ ∗ · J  (λcT ¯ ∗ ) ≤ ln c, (t, T) ∈ T . ln K + λT Existence of such c is an immediate consequence of the assumption (14.1.2).



Step 2: Part (i): The operator A is order-preserving, i.e. h1 ≤ h2

⇒

Ah1 ≤ Ah2 .

The sequence h0 ≡ 0, hn+1 := Ahn is thus monotonically increasing to h¯ and by the monotone convergence theorem we have ¯ T) = Ah(t, ¯ T), h(t,

∀(t, T) ∈ T .

Moreover, since h0 ≤ c, where c = c(ω) is given by Proposition 14.4.1, h¯ is bounded and thus (14.1.6) is satisfied. Moreover, by the boundedness of h¯ it follows that the process  t  T ¯ u)du λ(s, T)ds J λ(s, u)h(s, (14.4.3) 0

s

is continuous wrt. (t, T) ∈ T for fixed ω. It is also adapted wrt. t. If we replace ¯ u) in the preceding formula by any bounded field k(s, u) that is adapted wrt. s h(s, ¯ T) is adapted as a limit then (14.4.3) becomes adapted wrt. t. As a consequence, h(·, of the adapted sequence {hn (·, T)}. In virtue of Proposition 14.3.7, the field h¯ satisfies (14.1.4) and (14.1.5). Part (ii): The function J  is Lipschitz on [0, +∞) and therefore we can repeat all arguments from the proof of Proposition 14.3.4 and the result follows.

15 Analysis of the Morton–Musiela Equation

Existence and nonexistence results for the Morton-Musiela equation are established.

15.1 Formulation and Comments on the Results We are concerned here with the Morton–Musiela equation    x ∂ dr(t, x) = r(s, u)λ(u)du) λ(x)r(t, x) dt r(t, x) + J  ∂x 0 + λ(x)r(t−, x)dZ(t),

t ∈ [0, T ∗ ],

x ≥ 0,

(15.1.1)

already introduced in Section 12.3. The preceding λ(·) is a deterministic bounded function on [0, +∞). Our aim now is to provide conditions for the existence of solutions r(t) = r(t, x), 2,γ

x ≥ 0,

t ∈ [0, T ∗ ]

1,γ

of (15.1.1) in spaces L+ or H+ . As for the Morton equation in the previous chapter the central role here will be played by two logarithmic growth conditions on the function J  introduced already in Chapter 14 (see Section 14.1), i.e. For some a > 0, b ∈ R, J  (z) ≥ a(ln z)3 + b, for all z > 0,   ∗  ∗ ¯ (J2) lim sup ln z − λT J (z) = +∞, 0 < T < +∞,

(J1)

z→∞

where λ¯ := sup λ(x) < +∞. x≥0

We will show, roughly speaking, that the solutions of (15.1.1) explode if (J1) holds and solutions exist if (J2) is satisfied. For proving this we use the fact that equation (15.1.1) is equivalent to the Morton–Musiela operator equation, which will be introduced in the proofs. This equivalence allows one to use methods developed in the previous chapter.

15.1 Formulation and Comments on the Results

333

1,γ

Our first result on solutions in the state space H+ is of negative type. Theorem 15.1.1 (0)

λ is continuous and infx≥0 λ(x) = λ > 0, supx≥0 λ(x) = λ¯ < +∞, supp ν ⊆ [− λ1¯ , +∞),

(1) (3) (L0) (J1)

Assume that the conditions

λ, λ , λ , are bounded and continuous on R+ ,  +∞ yν(dy) < +∞, 1 J  (z) ≥ a(ln z)3 + b,

for some a > 0, b ∈ R, and all z > 0

are satisfied. 1,γ Then, for some k > 0 and all r0 (·) ∈ H+ such that r0 (x) ≥ k, ∀x ∈ [0, T ∗ ], the 1,γ equation (15.1.1) does not have solutions in H+ on the interval [0, T ∗ ]. 1,γ

As we proved in Theorem 13.1.8, equation (15.1.1) has local solutions in H+ . It follows thus from Theorem 15.1.1 that under (0), (1), (2), (3), (L0), (J1) and (L2), each such solution explodes. 2,γ 1,γ First, we formulate the main existence results in the spaces L+ and H+ and outline the methods of proving them. The proofs are given in the next subsection. Theorem 15.1.2 Assume that (0), (1) and the following conditions hold (2) λ are bounded and continuous on R+ , λ,+∞ yν(dy) < (L0) 1  +∞, ∗   ¯ J (z) = +∞, (J2) lim supz→∞ ln z − λT 0 < T ∗ < +∞. 2,γ

(a) If r0 ∈ L+ then there exists a solution to (15.1.1) taking values in the space 2,γ L+ . (b) Assume, in addition, that (3) λ, λ , λ , are bounded and continuous on R+ , Z has positive only, i.e. supp{ν} ⊆ [0, +∞) and  ∞ jumps 2 ν(dy) < ∞. y (L1) 1 1,γ If r0 ∈ H+ then there exists a unique solution to (15.1.1) taking values in the 1,γ space H+ .

15.1.1 Comments on the Results Let us recall important classes of L´evy processes for which (J1) or (J2) is satisfied. Condition (J1) is satisfied if the Wiener part of Z is not degenerated, i.e. q = 0, or Z admits negative jumps, i.e. ν((−∞, 0)) > 0 (see Theorem 14.2.1). Hence (J2) may be satisfied only if Z does not have the Wiener part nor negative jumps. In fact, if this is the case, then  1  +∞ y(1 − e−zy )ν(dy) + ye−zy ν(dy), J  (z) = −a + 0

1

334

Analysis of the Morton–Musiela Equation

and (J2) depends only on the behaviour of ν close to zero. To see this, note that  +∞ sup ye−zy ν(dy) < +∞, z ≥ 0, 1

z≥0

so only small jumps of Z are essential for (J2). Let us assume that for some ρ ∈ (0, +∞),  x y2 ν(dy) ∼ xρ · M(x), as x → 0 (15.1.2) 0

is satisfied, where M is a slowly varying function at 0 (see (14.2.1)). Then (a) if ρ > 1, then (J2) holds, (b) if ρ < 1, then (J1) holds, (c) if ρ = 1, the measure ν has a density and M(x) −→ 0

as x → 0,



1

and 0

M(x) dx = +∞, x

(15.1.3)

then (J2) holds (see Theorem 14.2.4). Recall that (J2) is satisfied in the case in which Z is a subordinator with drift (see Proposition 14.2.2). As explained in Example 14.2.3, the converse implication is, however, not true.

15.2 Proofs of Theorems 15.1.1 and 15.1.2 15.2.1 Equivalence Results In the proofs of the theorems we use Theorem 15.2.1 and Theorem 15.2.2 which state that equations (15.1.1) and (15.2.1) are equivalent under rather mild conditions. As their proofs are technical we refer the reader to Barski and Zabczyk [6]. A random field r(t, x), t ∈ [0, T ∗ ], x ≥ 0, is said to be a solution, in L2,γ , respectively in H 1,γ , to the Morton–Musiela operator equation: r(t, x) = a(t, x)e

t 0

 t−s+x

J (

0

λ(v)r(s,v)dv)λ(t−s+x)ds

,

x ≥ 0, t ∈ [0, T ∗ ],

(15.2.1)

where t

a(t, x) := r0 (t + x)e ·

$

0

2

λ(t−s+x)dZ(s)− q2

t 0

λ2 (t−s+x)ds

(1 + λ(t − s + x)(Z(s) − Z(s−))) e−λ(t−s+x)(Z(s)−Z(s−)) ,

0≤s≤t

x ≥ 0, t ∈ (0, T ∗ ],

(15.2.2)

if r(t, ·) , t ∈ [0, T ∗ ], is L2,γ −, respectively H 1,γ -valued, bounded and adapted process such that, for each t ∈ [0, T ∗ ], equation (15.2.1) holds for almost all x > 0,

15.2 Proofs of Theorems 15.1.1 and 15.1.2

335

in the case of L2,γ , and for all x ≥ 0, in the case of H 1,γ . We refer to Section 14.1 where a similar concept is introduced for Morton’s equation. 1,γ

Theorem 15.2.1 Let r be a solution of (15.1.1) in the state space H+ . Then r(·, ·) 1,γ is a solution of (15.2.1) in H+ . Theorem 15.2.2 Assume that conditions (0), (1) and  +∞ yν(dy) < +∞, (L0) 1 are satisfied. (a) If (2) λ, λ are bounded and continuous on R+ , 2,γ and r(·) is a bounded solution in L+ of (15.2.1), then r(·) is a c`adl`ag process 2,γ in L+ and solves (15.1.1). (b) If (3) λ, λ , λ , are bounded and continuous on R+ , Z has positive jumps i.e. supp{ν} ⊆ [0, +∞) and  ∞ only, 2 ν(dy) < ∞, y (L1) 1 1,γ 1,γ and r(·) is a bounded solution in H+ of (15.2.1), then r(·) is c`adl`ag in H+ and solves (15.1.1). 1,γ

As a consequence, equations (15.1.1) and (15.2.1) are equivalent in H+ , while 2,γ each solution of (15.2.1) in L+ solves also (15.1.1).

15.2.2 Proof of Theorem 15.1.1 Assume to the contrary that r is a solution of (15.1.1) on [0, T ∗ ] in the space H+ . Then the solution in the natural frame f (t, T) = r(t, T − t), 0 ≤ t ≤ T ≤ T ∗ satisfies Morton’s equation: 1,γ

 f (t, T) = f0 (T) +

t

J 0

 +





T

λ(v − s)f (s, v)dv λ(T − s)f (s, T)ds

s t

λ(T − s)f (s−, T)dZ(s),

(15.2.3)

0

a particular form of equation 14.1.1 studied in Section 14.1 as here λ(·) depends on one variable only. Assumptions (0), (1), (3), (L0) imply the conditions (A1)– (A3) required in Section 14.1. We check the regularity conditions (14.1.4)–(14.1.6). 1,γ Since r is adapted and c`adl`ag in H+ , it follows that (a) f (·, T) is adapted and c`adl`ag for each T ∈ [0, T ∗ ], (b) f (t, ·) is continuous.

336

Analysis of the Morton–Musiela Equation

Using Lemma 15.2.3 and the fact that r is bounded on [0, T ∗ ], as a c`adl`ag process in 1,γ H+ , we obtain sup

t∈[0,T ∗ ],x≥0

| r(t, x) |= sup sup | r(t, x) |≤ 2 t∈[0,T ∗ ] x≥0

 1 1 2 sup | r |H 1,γ < +∞, + γ t∈[0,T ∗ ]

which clearly implies that (c)

sup

0≤t≤T≤T ∗

f (t, T) < +∞.

It follows, however, from Theorem 14.1.1 that, under (J1), for sufficiently large k > 0 there is no solution of (15.2.3) in the class of regular random fields satisfying (a)–(c), a contradiction.  Lemma 15.2.3

If r ∈ H 1,γ then  1 1/2 sup |r(x)| ≤ 2 | r |H 1,γ . γ x≥0

Proof

Integrating by parts yields  x  x x dr(y) y r(y) dy, dy = yr(y)0 − dy 0 0

and thus

  |xr(x)| ≤  ≤

x

y 0

  dr(y)   x  r(y) dy dy +  dy 0

1/2  +∞  dr(y) 2 1/2 e−γ y y2 dy eγ y dy dy 0 0  +∞ 1/2  +∞ 1/2 + e−γ y dy eγ y r2 (y) dy 

+∞

0

≤

0

 2 1/2  1 1/2 | r |H 1,γ + | r |L2,γ . 3 γ γ

In particular, lim r(x) = 0.

x→+∞

Moreover,

  |r(x) − r(0)| =  ≤

x 0



 dr    x dr    e−γ y/2 eγ y/2  (y) dy (y) dy ≤ dy dy 0

+∞ 0

1/2  e−γ y dy 0

+∞

eγ y

2 1/2  1 1/2 ≤ | r |H 1,γ . (y) dy dy γ

 dr

15.2 Proofs of Theorems 15.1.1 and 15.1.2

337

Consequently, | r(0) |≤

 1 1/2 γ

| r |H 1,γ ,

and therefore  1 1/2

sup |r(x)| ≤ 2 x≥0

γ

| r |H 1,γ .

15.2.3 Proof of Theorem 15.1.2 Proof of the existence part Define the operator K, acting on functions of two variables, by K(h)(t, x) = a(t, x)e

t 0

J

 t−s+x 0

 λ(v)h(s,v)dv λ(s,t−s+x)ds

x ≥ 0, t ∈ [0, T ∗ ], (15.2.4)

,

where a(t, x) is given by (15.2.2). Then the equation (15.2.1) can be compactly written in the form r(t, x) = K(r)(t, x),

t ∈ [0, T ∗ ],

x ≥ 0.

Let us consider the sequence of random fields h0 ≡ 0,

hn+1 := Khn ,

n = 1, 2, . . . .

(15.2.5)

Let us write a in the form a(t, x) = r0 (t + x)I1 (t, x)I2 (t, x), where



t

I1 (t, x) :=

λ(t − s + x)dZ(s),

t ∈ [0, T ∗ ], x ≥ 0,

(15.2.6)

0

 t I2 (t, x) := 0

+∞  − 1¯ λ

 ln(1 + λ(t − s + x)y) − λ(t − s + x)y π(ds, dy),

t ∈ [0, T ∗ ], x ≥ 0.

(15.2.7)

One can show that under (2) the field b, defined by b(t, x) := I1 (t, x)I2 (t, x), is bounded, i.e. sup

t∈[0,T ∗ ],x≥0

¯ b(t, x) < b,

(15.2.8)

¯ where b¯ = b(ω) > 0 (for details see Barski and Zabczyk [6, p. 2681–2682]).

338

Analysis of the Morton–Musiela Equation 2,γ

It can be shown by induction that if r0 ∈ L+ then hn (t) is a bounded process in 2,γ L+ for each n. Indeed assume this for hn and show for hn+1 . In view of (15.2.8) we have ¯

hn+1 (t, x) ≤ r0 (t + x) b¯ eλ

t

0 |J

¯

λT ≤ r0 (t + x) b¯ e

∗

  ( t−s+x λ(v)h (s,v)dv)|ds n 0

 ¯ J  ( √λ

γ



supt |hn (t)|L2,γ )

,

2,γ

and thus hn+1 (t) is bounded in L+ . It follows from the assumption λ > 0 and the fact that J  is increasing that the sequence {hn } is monotonically increasing and thus there exists h¯ : [0, T ∗ ] × [0, +∞) −→ R+ such that ¯ x), lim hn (t, x) = h(t,

0 ≤ t ≤ T ∗ , x ≥ 0.

n→+∞

(15.2.9)

Passing to the limit in (15.2.5), by the monotone convergence, we obtain ¯ x) = Kh(t, ¯ x), h(t,

0 ≤ t ≤ T ∗ , x ≥ 0.

It turns out that properties of the field h¯ strictly depend on the growth of the 2,γ ¯ is a bounded process in L+ function J  . In the following text we show that h(t) , 2,γ ∗ ¯ i.e. h(t), t ∈ [0, T ] is a non-exploding solution of (15.2.1) in L+ . Additional 1,γ ¯ is bounded in H+ assumptions guarantee that h(t) and that the solution is unique. We need an auxiliary result. Proposition 15.2.4 Assume that J  satisfies (J2). If r0 ∈ L+ then there exists a positive constant c1 such that if 2,γ

sup | h(t) |L2,γ ≤ c1 +

t∈[0,T ∗ ]

then sup | Kh(t) |L2,γ ≤ c1 . +

t∈[0,T ∗ ]

By elementary arguments using (15.2.8), for any t ∈ [0, T ∗ ], we have  +∞  t   t−s+x λ(v)h(s,v)dv)λ(t−s+x)ds γ x |r0 (t + x)b(t, x)|2 e2 0 J ( 0 e dx | Kh(t, ·) |2 2,γ =

Proof

L+

0

≤ b¯ 2



+∞

2J 



|r0 (t + x)| e 2

¯ √λ γ

·supt |h(t)|

2,γ L+

t 0

λ(t−s+x)ds

eγ x dx

0

≤ b¯ 2 · | r0 |2 2,γ · L+

2J 

sup s∈[0,t],x≥0

e



¯ √λ γ

·supt |h(t)|

2,γ L+

t 0

λ(t−s+x)ds

.

15.2 Proofs of Theorems 15.1.1 and 15.1.2 This implies ¯ | r0 | 2,γ · sup | Kh(t) |L2,γ ≤ b· L +

t

+

J

sup

t∈[0,T ∗ ],s∈[0,t],x≥0

e



¯ √λ γ

339

·supt |h(t)|

2,γ L+

t 0

λ(t−s+x)ds

,

and thus it is enough to find a constant c1 such that  t    ¯  λc1 ¯ J √ λ(t − s + x)ds ≤ ln c1 . (15.2.10) sup ln b· | r0 |L2,γ + + γ 0 t∈[0,T ∗ ],s∈[0,t],x≥0 ¯ | r0 | 2,γ . If J  takes positive values If J  (z) ≤ 0 for each z ≥ 0 then we put c1 = b· L+ then it is enough to find large c1 such that    ¯ λc ¯ | r0 | 2,γ ≤ ln c1 − λT ¯ ∗J √ 1 . ln b· L+ γ Existence of such c1 is a consequence of (J2). Continuation of the proof of Theorem 15.1.2: ¯ x) is adapted for each x ≥ 0 as a pointwise limit, we only need to show Since h(·, 2,γ 1,γ ¯ that h(t) is a bounded process in L+ , resp. H+ . Then h¯ solves (15.1.1) in virtue of Theorem 15.2.2. (a) Let c1 be a constant given by Proposition 15.2.4. By the Fatou lemma we have  +∞  +∞ ¯ x) |2 eγ x dx ≤ sup lim inf | h(t, | hn (t, x) |2 eγ x dx ≤ c21 , sup t∈[0,T ∗ ] n→+∞ 0

t∈[0,T ∗ ] 0

¯ is bounded in L+ . and hence h(t) (b) In view of (a) we need to show that hx (t) is bounded in L2,γ . Differentiating the equation h¯ = Kh¯ yields 2,γ

h¯  (t, x) = r0 (t + x)b(t, x)F1 (t, x) + r0 (t + x)bx (t, x)F1 (t, x) + r0 (t + x)b(t, x)F1 (t, x)F2 (t, x), where F1 (t, x) := e

t 0



 t−s+x

J ( t

F2 (t, x) :=

J

0





0

¯ λ(v)h(s,v)dv)λ(t−s+x)ds t−s+x

,

¯ ¯ t − s + x)ds λ(v)h(s, v)dv λ2 (t − s + x)h(s,

0

 +

t

J 0



 0

t−s+x

λ(v)h(s, v)dv λx (t − s + x)ds.

Assumption (3) implies that b(·, ·) and bx (·, ·) are bounded on (t, x) ∈ [0, T ∗ ]× 1γ [0, +∞). Since r0 ∈ H+ , it is enough to show that sup

t∈[0,T ∗ ],x≥0

F1 (t, x) < +∞,

sup

t∈[0,T ∗ ],x≥0

F2 (t, x) < +∞.

340

Analysis of the Morton–Musiela Equation We have sup

t∈[0,T ∗ ],x≥0

  ¯ J  √λ γ F1 (t, x) ≤ e

¯ supt |h(t)|

2,γ L+

 λ¯ T ∗

< +∞.

It follows from Proposition 14.2.2 that (J2) excludes the Wiener part of the noise as  +∞   well as negative jumps. Thus J reduces to the form J (z) = 0 y2 e−zy ν(dy) and 0 ≤ J  (0) < +∞ due to the assumption (L1). Since J  is decreasing, the following estimation holds  t ¯ t − s + x)ds h(s, sup F2 (t, x) ≤ J  (0)T ∗ λ¯ 2 sup t∈[0,T ∗ ],x≥0

t∈[0,T ∗ ],x≥0 0

 + T ∗ J 



λ¯ ¯ | 2,γ √ sup | h(t) L+ γ t



  · sup λ (x), x≥0

and it is enough to show that h¯ is bounded on {(t, x), t ∈ [0, T ∗ ], x ≥ 0}. In view of the fact that h¯ = Kh¯ we obtain    λ¯ T ∗ J  √1 supt |h(t)| ¯ 2,γ γ L+ ¯ x) ≤ sup r0 (x) · h(t, sup sup b(t, x) · e < +∞. t∈[0,T ∗ ],x≥0

t∈[0,T ∗ ],x≥0

x≥0

Proof of the uniqueness part We need the following auxiliary result which is a simple modification of Lemma 14.3.5. Lemma 15.2.5

Let d : [0, T ∗ ] × [0, +∞) → R+ be a bounded function satisfying  t  t−s+x d(t, x) ≤ C d(s, v)dvds, (15.2.11) 0

0

where C > 0 is a fixed constant. Then d(t, x) = 0 for all (t, x) ∈ [0, T ∗ ] × [0, +∞). To start the proof of the theorem assume that r1 , r2 are two solutions of the 1,γ equation (15.1.1) in H+ . Then they are bounded processes in H 1,γ and, in view of Theorem 15.2.1, satisfy (15.2.1). Define 0 ≤ t ≤ T ∗ , x ≥ 0.

d(t, x) :=| r1 (t, x) − r2 (t, x) |,

Denote B := supt∈[0,T ∗ ],x≥0 b(t, x). The following estimation holds d(t, x) ≤ r0 (t + x)b(t, x) e

t 0

 t−s+x

J (

t

+e ) ≤ sup r0 (x) · B · e x≥0

0

0

λ(s,v)r1 (s,v)dv)λ(s,t−s+x)ds

 t−s+x

J (

0

λ(s,v)r2 (s,v)dv)λ(s,t−s+x)ds



¯ λ¯ T ∗ J  ( √λγ supt |r1 (t)|



2,γ L+

)

!



¯ λ¯ T ∗ J  ( √λγ supt |r2 (t)|

+e

*

2,γ L+

)

< +∞,

15.2 Proofs of Theorems 15.1.1 and 15.1.2

341

and thus d is bounded on [0, T ∗ ] × [0, +∞). In view of the inequality | ex − ey |≤ ex∨y | x − y |; x, y ≥ 0 and the fact that J  is decreasing with 0 ≤ J  (0) < +∞ due to assumption (L1), we have &    '   max 0t J  0t−s+x λ(s,v)r1 (s,v)dv λ(s,t−s+x)ds; 0t J  0t−s+x λ(s,v)r2 (s,v)dv λ(s,t−s+x)ds

d(t, x) ≤ sup r0 (x)·Be x≥0

 t  t−s+x  ·  J  λ(s, v)r1 (s, v)dv λ(s, t − s + x)ds 0

−

 t

0

J

 t−s+x

0

0

  λ(s, v)r2 (s, v)dv λ(s, t − s + x)ds & 

≤ sup r0 (x) · Be

'

  

¯ ∗ max J  √λ¯ supt |r1 (t)| 2,γ ;J  √λ¯ supt |r2 (t)| 2,γ  λT γ γ L L +

+

·

x≥0

· J  (0)λ¯ 2 =C

 t  t−s+x

0 0  t  t−s+x

| r1 (s, v) − r2 (s, v) | dvds

d(s, v)dvds, 0

0

(t, x) ∈ [0, T ∗ ] × [0, +∞).

It follows from Lemma 15.2.5 that r1 = r2 on [0, T ∗ ] × [0, +∞).

Appendix A

Following Itˆo [73] and Kunita [84], we present and prove the general martingale representation theorem when the filtration is generated by a general L´evy process without the Wiener part.

A.1 Martingale Representation for Jump L´evy Processes We assume that Z is of pure jump type,     yπ˜ (ds, du) + Z(t) = at + t |u|≤1

t |u|>1

yπ(ds, du),

t ≥ 0,

and Ft = σ {Z(s), s ≤ t} is its natural filtration. Let ν be the intensity measure and π˜ the compensated random measure of Z. Denote by 1,2 the class of predictable processes g(s, y), s ∈ [0, T ∗ ], y ∈ U, satisfying  T∗  (| g(s, y) |2 ∧ | g(s, y) |)dsν(dy) < +∞, P − a.s. 0

U

For g ∈ 1,2 , one defines the integral  t  t  t g(s, y)π˜ (ds, dy) := g1{|g|≤1} π˜ (ds, dy) + g1{|g|>1} π˜ (ds, dy), 0

U

0

U

0

U

which is a local martingale. In this setting Theorem 6.1.1 has the following form. Theorem A.1.1 Let M be a real-valued P-local martingale on [0, T ∗ ] adapted to the filtration Ft = σ {Z(s), s ≤ t}. Then there exists a process ψ ∈ 1,2 such that  t Mt = M0 + ψ(s, y)π˜ (ds, dy), t ∈ [0, T ∗ ]. (A.1.1) 0

U

Moreover, the process ψ is unique, i.e. if ψ  ∈ 1,2 and (A.1.1) holds with ψ  , then ψ = ψ ,

dP × dt × dν − a.s.

A.1 Martingale Representation for Jump L´evy Processes

343

If M is a square integrable martingale then (A.1.1) holds and  t  2 2 2 EMt = EM0 + E | ψ(s, y) | dsν(dy) < +∞. 0

U

The proof will be divided into several parts. First, one introduces chaos processes and establishes the representation theorem for them in the form of the so-called multiple Itˆo–Wiener chaoses. Then one treats the case of square integrable martingales. If M is a square integrable martingale then Mt = E(MT ∗ |Ft ), t ∈ [0, T ∗ ] and MT ∗ is a square integrable random variable and thus an element of the space L2 (, FT ∗ , P). One first represents X as the orthogonal series  MT ∗ = Xk k=0

of the k-multiple Itˆo–Wiener chaoses. The sum of the representations is the required representation in the square integrable case. The final part is devoted to the general case when M is a general local martingale. It is technically rather complicated and starts from the case when M is positive (see Kunita [84]).

A.1.1 Multiple Chaos Processes Denote by ν¯ the product measure dtν(du). It follows from the basic properties of the compensated random measure that if E1 , E2 , . . . , Ek , are disjoint subsets of [0, T ∗ ] × U such that  dt ν(du) = ν(E ¯ j ) < +∞, j = 1, 2, . . . , k, Ej

then the random variables  T∗  1Ej (s, u)π˜ (ds, du) = π˜ (Ej ), j = 1, . . . , k, 0

U

are independent with zero expectations. The product k  T∗  $ 1Ej (s, u)π˜ (ds, du) j=1 0

U

is the simplest example of the k-multiple Itˆo–Wiener integral, also called k-multiple chaos,     ... f (t1 , u1 ), . . . , (tk , uk ) π˜ (dt1 , du1 ) . . . π˜ (dtk , duk ), Ik (f ) = [0,T ∗ )×U

[0,T ∗ )×U

(A.1.2)

344

Appendix A

when   f (t1 , u1 ), . . . , (tk , uk ) = 1Ei1 (t1 , u1 ) · . . . · 1Eik (tk , uk ).   The concept can be extended to all functions f (t1 , u1 ), . . . , (tk , uk ) belonging to the Hilbert space Lk2 equipped with the norm,   | f |L 2 =

1/2



  f 2 (t1 , u1 ), . . . , (tk , uk ) ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk )

...

k

[0,T ∗ )×U

.

[0,T ∗ )×U

(A.1.3) The extension is defined in an obvious way for the so-called special elementary functions of order k that are of the form    f (t1 , u1 ), . . . , (tk , uk ) := ai1 ,...,ik 1Ei1 (t1 , u1 ) · . . . · 1Eik (tk , uk ), (A.1.4) (i1 ,...,ik )∈I

where E1 , . . . , Em , m ≥ k, is an arbitrary sequence of disjoint subsets of [0, T ∗ ] × U, I stands for the set of all sequences (i1 , . . . , ik ) with pairwise different elements from {1, 2, . . . , m} and ai1 ,...,ik ∈ R. Namely, for such f one defines the k-multiple integral of order k     Ik (f ) = ... f (t1 , u1 ), . . . , (tk , uk ) π˜ (dt1 , du1 ) . . . π˜ (dtk , duk ), [0,T ∗ ]×U

[0,T ∗ ]×U

by the formula Ik (f ) :=



ai1 ,...,ik π(E ˜ i1 ) . . . π˜ (Eik ).

(i1 ,...,ik )∈I

One checks directly that if f is a special elementary function of order k, then E|Ik (f )|2 = |f |L2 . k

(A.1.5)

To perform extension to arbitrary f ∈ Lk2 , note that special elementary functions are dense in Lk2 and therefore for arbitrary f ∈ Lk2 there exists a sequence (fn ) of special elementary function converging, in the sense of Lk2 , to f . It follows from (A.1.5) that also random variables Ik (fn ) converge, in the sense of L2 (, FT ∗ ), to a random variable. The limiting random variable is independent of the choice of the approximating sequence and is identified as Ik (f ). We establish now the main properties of the multiple integrals needed in the proof of the chaos expansion theorem. k  A function g : [0, T ∗ × U → R, is called symmetric if it is invariant under permutations of its arguments, i.e.     g (t1 , u1 ), . . . , (tk , uk ) = g (tp(1) , up(1) ), . . . , (tp(k) , up(k) )

A.1 Martingale Representation for Jump L´evy Processes

345

for any permutation p of the set {1, 2, . . . , k}. It is clear that Ik (f ) is invariant after any permutation of its arguments. The following symmetrization gˆ of g is clearly a symmetric function    1   gˆ (t1 , u1 ), . . . , (tk , uk ) := g (tp(1) , up(1) ), . . . , (tp(k) , up(k) ) . k! p The preceding sum is taken over all permutations p of the set {1, 2, . . . , k}. The following proposition is crucial for the extension. Proposition A.1.2

i) For arbitrary functions f , g belonging to Lk2 ,

Ik (f ) = Ik (fˆ ),    E Ik (f )Ik (g) = k!

(A.1.6) 

  fˆ (t1 , u1 ), . . . , (tk , uk )

...

[0,T ∗ ]×U

[0,T ∗ ]×U

 · gˆ (t1 , u1 ), . . . , (tk , uk ) ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk ) (A.1.7) 

(fˆ , gˆ )L2 .

= k!

k

ii) If f ∈ Lk2 , g ∈ Ll2 , k = l,

then

(A.1.8)   E Ik (f )Il (g) = 0.

(A.1.9)

Proof We can assume that f , g are special elementary functions. First, we prove (A.1.6). If f is of the form (A.1.4) then   1 fˆ (t1 , u1 ), . . . , (tk , uk ) = k!



ai1 ,...,ik

(i1 ,...,ik )∈I



1Ei1 (tp(1) , up(1) ) . . . 1Eik (tp(k) , up(k) )

p

and therefore Ik (fˆ ) =

1 k!



ai1 ,...,ik

(i1 ,...,ik )∈I





π˜ (Ei1 ) . . . π˜ (Eik )

p

= Ik (f ). Now we prove (A.1.7). Let g be of the form    bj1 ,...,jk 1Ej1 (t1 , u1 ) · . . . · 1Ejk (tk , uk ). g (t1 , u1 ), . . . , (tk , uk ) = (j1 ,...,jk )∈I

Then   E Ik (f )Ik (g) = E





(i1 ,...,ik )∈I

ai1 ,...,ik π(E ˜ i1 ). . .π˜ (Eik )·

 (j1 ,...,jk )∈I

bj1 ,...,jk π˜ (Ej1 ). . . π(E ˜ jk ) .

346

Appendix A

One writes (j1 , . . . , jk ) & (i1 , . . . , ik ) if (j1 , . . . , jk ) can be obtained from (i1 , . . . , ik ) by permutation of its elements. Taking into account that π˜ (Ei ), π˜ (Ej ) are independent for disjoint sets Ei , Ej and     E π˜ (Ei ) = 0, E (π˜ (Ei ))2 = ν¯ (Ei ), i = 1, 2, . . . , m, we have   E π(E ˜ i1 ) . . . π˜ (Eik ) · π˜ (Ej1 ) . . . π˜ (Ejk )   ˜ ik ))2 if (j1 , . . . , jk ) & (i1 , . . . , ik ), = E (π˜ (Ei1 ) . . . π(E and 0 otherwise. Consequently,     ai1 ,...,ik · E Ik (f )Ik (g) = E (i1 ,...,ik )∈I



=

 2  bj1 ,...,jk · π˜ (Ei1 ) . . . π˜ (Eik )

 (j1 ,...,jk )&(i1 ,...,ik )



ai1 ,...,ik ·

(i1 ,...,ik )∈I

bj1 ,...,jk · ν¯ (Ei1 ) . . . ν¯ (Eik ).

(j1 ,...,jk )&(i1 ,...,ik )

(A.1.10) However,       fˆ (t1 , u1 ), . . . , (tk , uk ) · gˆ (t1 , u1 ), . . . , (tk , uk ) ν(dt ... ¯ 1 , du1 ) . . . ν¯ (dtk , duk ) E1

Ek

1 = (k! )2

 ... E1

·

 

  Ek



ai1 ...ik 1Ei1 (tp(1) , up(1) ) . . . 1Eik (tp(k) , up(k) )

p (i1 ,...,ik )∈I

bj1 ...jk 1Ej1(tq(1) , uq(1)) . . . 1Ejk (tq(k) , uq(k) ) ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk ).

q (j1 ,...,jk )∈I

For the product 1Ei1 (tp(1) , up(1) ) . . . 1Eik (tp(k) , up(k) ) · 1Ej1 (tq(1) , uq(1) ) . . . 1Ejk (tq(k) , uq(k) ) to be different from zero, one must have that (j1 , . . . , jk ) & (i1 , . . . , ik ). In addition, the permutation q should be such that the product is equal to 1Ei1 (tp(1) , up(1) ) . . . 1Eik (tp(k) , up(k) ). Therefore   ... [0,T ∗ ]×U

[0,T ∗ ]×U

  fˆ (t1 , u1 ), . . . , (tk , uk )

  · gˆ (t1 , u1 ), . . . , (tk , uk ) ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk )

=

1 (k! )2

A.1 Martingale Representation for Jump L´evy Processes 347     ... ai1 ...ik 1Ei1 (tp(1) , up(1) ) . . . 1Eik (tp(k) , up(k) ) [0,T ∗ ]×U

[0,T ∗ ]×U



·

p (i1 ,...,ik )∈I

bj1 ...jk ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk )

(j1 ,...,jk )&(i1 ,...,ik )

=

1 (k! )2





  ai1 ...ik k! ν¯ (Ei1 ) . . . ν¯ (Eik )

(i1 ,...,ik )∈I

bj1 ...jk .

(j1 ,...,jk )&(i1 ,...,ik )

Taking into account (A.1.10), the proof of (A.1.7) is complete. To prove (A.1.9) assume that k < l. Then ! E π˜ (Ei1 ) . . . π˜ (Eik ) · π˜ (Ej1 ) . . . π˜ (Ejl ) = 0, because in the sequence Ej1 , . . . , Ejl there exists a set, say E¯j , disjoint from all the others and the expectation of the product is a multiple of E[π(E ˜ ¯j )] = 0.

A.1.2 Representation of Chaoses Let f ∈

Lk2 .

The following integral      t ... f (t1 , u1), (t2 , u2), . . . ,(tk , uk) π˜ (dt1 , du1) . . . π(dt ˜ k , duk ), Ik (f ) := [0,t]×U [0,t]×U

[0,t]×U

is well defined for any t ∈ [0, T ∗ ] because it can be represented as the multiple integral Ikt (f ) = Ik (Ft ), with     Ft (t1 , u1 ), . . . , (tk , uk ) = f (t1 , u1 ), . . . , (tk , uk ) 1[0,t] (t1 )1[0,t] (t2 ) . . . 1[0,t] (tk ) and Ft ∈ Lk2 . Now we show that the process t → Ikt (f ), t ∈ [0, T ∗ ] is a square integrable martingale and can be represented as a stochastic integral. Theorem A.1.3 For arbitrary f ∈ Lk2 there exists a predictable process Rf (t, u), t ∈ [0, T ∗ ], u ∈ U such that  t Ikt (f ) = Rf (s, u)π˜ (ds, du), t ∈ [0, T ∗ ], (A.1.11) 0

and    E (Ikt (f ))2 = E

 t 0

U

U

 (Rf (s, u))2 ds ν(du) =| f |2k

t ∈ [0, T ∗ ].

(A.1.12)

348

Appendix A

The arguments in the proof will involve iterated Itˆo–Wiener integrals of the function f . More precisely, we prove that  t Ikt (f ) = Rf (tk , uk )π˜ (dtk , duk ), t ∈ [0, T ∗ ], U

0

where Rn , n = 1, 2, . . . , k is a sequence of processes iteratively defined by     (A.1.13) R1 (t1 , u1 ), . . . , (tk , uk ) := f¯ (t1 , u1 ), . . . , (tk , uk ) ,  tn+1      Rn+1 (tn+1 , un+1 ), . . . , (tk , uk ) := Rn (tn , un ), . . . , (tk , uk ) π(dt ˜ n , dun ), 0

U

(A.1.14) where f¯k is the truncated symmetrization of f defined by     f¯k (t1 , u1 ), . . . , (tk , uk ) := k! 1k (t1 , . . . , tk ) fˆ (t1 , u1 ), . . . , (tk , uk ) . The preceding k is a simplex given by k := {(t1 , t2 , . . . , tk ) : 0 < t1 < t2 < · · · < tk ≤ T}. Moreover, E

 t 0

! 2 Rk (tk , uk ) dtk ν(duk ) < +∞.



U

We need also the following auxiliary result. Proposition A.1.4 For f ∈ Lk2 , Ikt (f ) = Ikt (fˆ ) = Ikt (f¯ ),

t ∈ [0, T ∗ ].

Proof To simplify notation we assume that t = T ∗ . In view of Proposition A.1.2 only the latter identity requires a proof. It is sufficient to show that fˆ¯ = fˆ . Note that    1  fˆ¯ (t1 , u1), . . . , (tk , uk) = k! 1k (tp(1) , . . . , tp(k) )fˆk (tp(1) , up(1) ), . . . , (tp(k) , up(k) ) . k! p However, for each permutation p,     fˆ (tp(1) , up(1) ), . . . , (tp(k) , up(k) ) = fˆ (t1 , u1 ), . . . , (tk , uk ) , and therefore     fˆ¯ (t1 , u1 ), . . . , (tk , uk ) = fˆ (t1 , u1 ), . . . , (tk , uk ) 1k (tp(1) , . . . , tp(k) ) p

  = fˆ (t1 , u1 ), . . . , (tk , uk ) , because for arbitrary vector (s1 , . . . , sk ) with different coordinates (sp(1) , . . . , sp(k) ) ∈ k for exactly one permutation.

A.1 Martingale Representation for Jump L´evy Processes

349

Proof of Theorem A.1.3 We prove the result for special elementary functions of order k, with supports in k (see (A.1.4)) where Ej = [aj , bj ) × Uj ,

bj ≤ aj+1 , j = 1, 2, . . . , k − 1, Uj ⊆ U,

(A.1.15)

which are dense in Lk2 . We restrict to functions of the form k  $  1Ej (tj , uj ). f (t1 , u1 ), . . . , (tk , uk ) =

(A.1.16)

j=1

Notice that for f ∈ Lk2 satisfying (A.1.16), (A.1.15) the truncated symmetrization is identical with the original function, i.e. f = f¯ = k! fˆ 1k .    f Starting from R1 (t1 , u1 ), . . . , (tk , uk ) = f (t1 , u1 ), . . . , (tk , uk ) we obtain  t2   f R2 (t2 , u2 ), . . . , (tk , uk ) = 1E1 (t1 , u1 ) · . . . · 1Ek (tk , uk ) π˜ (dt1 , du1 ) 0

U

= π(E ˜ 1 ) · 1E2 (t2 , u2 ) · . . . · 1Ek (tk , uk ). Further integration provides a general formula for any n = 1, 2, . . . , k   Rfn (tn , un ), . . . , (tk , uk ) = π˜ (E1 ) · . . . · π˜ (En−1 ) · 1En (tn , un ) · . . . · 1Ek (tk , uk ). (A.1.17) As a consequence we obtain  t  t  f Rk tk , uk π˜ (dtk , duk ) = π˜ (E1 ) · . . . · π˜ (Ek−1 ) 1Ek (tk , uk ) π˜ (dtk , duk ) 0

U

U

0

 t

= π˜ (E1)· . . . · π˜ (Ek−1)

0

U

1[ak ,bk ]×Uk (tk , uk)π(dt ˜ k , duk)

= π˜ (E1 ) · . . . · π˜ (Ek−1 )π˜ ([t ∧ ak , t ∧ bk ] × Uk ). However, we can calculate directly    Ikt (f ) = ... 1E1 (t1 , u1 ) · . . . · 1Ek (tk , uk )π˜ (dt1 , du1) . . . π(dt ˜ k , duk ) [0,t]×U [0,T]×U

[0,T]×U

= π˜ (E1 ) · . . . · π˜ (Ek−1 )π˜ ([t ∧ ak , t ∧ bk ] × Uk ),

(A.1.18)

which yields (A.1.11). To prove (A.1.12) we proceed similarly as in Proposition A.1.2. We use the independence of random variables π˜ (E1 ), . . . , π˜ (Ek ) which, in view of (A.1.17), yields

350  f E Rk tk , uk

2 !

Appendix A

2 !  = E π˜ (E1 ) · . . . · π˜ (Ek−1 ) · 1Ek (tk , uk ) = ν¯ (E1 ) · . . . · ν¯ (Ek−1 ) · 1Ek (tk , uk ),

and  t 0

U

E



!

2 f Rk tk , uk

dtk ν(duk ) = ν¯ (E1 ) · . . . · ν¯ (Ek−1 )

 t 0

U

1Ek (tk , uk )dtk ν(duk )

= ν¯ (E1 ) · . . . · ν¯ (Ek−1 ) · ν([t ∧ ak , t ∧ bk ] × Uk ). (A.1.19) Finally, by (A.1.18) and (A.1.19), we obtain ! E (Ikt (f ))2 = ν¯ (E1 ) · . . . · ν¯ (Ek−1 ) · ν([t ∧ ak , t ∧ bk ] × Uk ) =E

 t   0

U

 2 f Rk tk , uk dtk ν(duk ) .

In fact, let f be an arbitrary element of Lk2 and fn a sequence of functions considered f in the preceding converging to f . Then Rkn is a sequence of predictable fields such that the sequence  t f f (Rkn (s, u) − Rkm (s, u)2 dsν(du) = |fn − fm |2k (A.1.20) E 0

U

fn

tends to 0 as m, n → +∞. Therefore a subsequence Rk j converges almost surely, on the product space,  × [0, T ∗ ] × U to a predictable field g with the property  t f (Rkn (s, u) − g(s, u))2 dsν(du) = |fn − g|2k . (A.1.21) E U

0

f

Thus it is enough to define Rk = g.

A.1.3 Chaos Expansion Theorem We prove now the chaos expansion theorem, from which the representation theorem for square integrable martingales will be derived. Let Lˆ k2 denote the Hilbert space of symmetric functions f of 2k-variables ¯ equipped with the norm (t1 , u1 ), . . . , (tk , uk ) with (ti , ui ) ∈ [0, T ∗ ] × U =: U,     | f |2k = k! . . . f 2 (t1 , u1 ), . . . , (tk , uk ) ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk ). ¯k U

Since Lˆ k2 is a closed subspace of Lk2 , it is also a Hilbert space. The transformation Ik : Lˆ k2 −→ H := L2 (, FT∗ , P)

A.1 Martingale Representation for Jump L´evy Processes

351

is an isometry and this is why in the sequel we will use the analysis Lˆ k2 instead of Lk2 . By Hk we denote the subspace of H consisting of all integrals Ik (f ), f ∈ Lˆ k2 . It follows that Hk is a closed subspace of H. Let H0 be the real line with usual metric. Theorem A.1.5 The Hilbert space H is a direct sum of the subspaces Hk , k = 0, 1, . . ., i.e.  ⊕Hk . (A.1.22) H= k≥0

That is, arbitrary X ∈ L2 (, FT ∗ , P) can be represented as the sum X = E(X) +

+∞ 

Ik (fk ),

(A.1.23)

k=1

where (fk ) is a sequence of symmetric functions from the spaces Lk2 , k = 1, 2, . . .. The series in (A.1.23) consists of orthogonal random variables and converges in mean square. Proof It follows from Proposition A.1.2, formula (A.1.9), that the subspaces Hk , k = 0, 1, . . . , are mutually orthogonal and the identity (A.1.22) means, in addition,   +∞    2 2 E[X ] = (EX) + k! . . . fk2 (t1 , u1 ), . . . , (tk , uk ) ν¯ (dt1 , du1 ) . . . ν¯ (dtk , duk ). k=1



¯k U

It is clear that k≥0 ⊕Hk is a closed subspace of H and therefore it is sufficient to show that it is dense in H. Taking into account that the σ -field FT∗ is generated by the L´evy process Z(t), t ∈ [0, T ∗ ], it is not difficult to see that random variables of the form f (π(E1 ), . . . , π(Em )), where f is a bounded continuous function and sets Ej , j = 1, 2, . . . , m are pairwise disjoint, consitute a dense set in the space H. Indeed, a function of Z can be approximated by a function of its increments. The same is true if we replace functions f by linear combinations of polynomials Y = π(E1 )p1 π(E2 )p2 . . . π(Ek )pk . Taking into account that elements of Hm can be approximated by linear combinations of Z = π(F1 )π(F2 ) . . . π(Fm ), with pairwise disjoint sets F1 , . . . , Fm , it is enough to show that random variables Y can be approximated in H by linear combinations of random variables of the form Z.

352

Appendix A

< To do so, set κ := ν¯ ( Ej ). We can assume that κ < +∞. Choose ε ∈ (0, κ) and let F1 , F2 , . . . , Fm be a subdivision of E1 , . . . , Ek so fine that ε ν¯ (Fj ) ≤ , j = 1, 2, . . . , m. (A.1.24) κ Note that sets E1 , . . . , Ek are covered by disjoint families (Fj1 (1) , . . . , Fjm1 (1) ),. . . , (Fj1 (k) , . . . , Fjmk (k) ), and therefore Y is a sum of elements of the form π(Fl(1) )k1 . . . π(Fl(r) )kr with l(1) < l(2) < . . . < l(r) ≤ m. Therefore Y=



(A.1.25)

π(Fl(1) )k1 . . . π(Fl(r) )kr ,

R

 where the sum  is over a finite family R of sequences of the form (l(1), . . . , l(r)), (k1 , . . . , kr ) satisfying (A.1.25). Since the random variables π(G) take values 0, 1, 2, . . . we have that  π(Fl(1) ) . . . π(Fl(r) ) =: V. Y≥ R

However, Y is strictly greater than V if π(Fl ) ≥ 2 for some l. Consequently, P(Y > V) = P(π(Fl ) ≥ 2 for some l) ≤

m 

P(π(Fl ) ≥ 2) ≤

l=1

≤

ε κ

m 

m  

2

ν¯ (Fl )

l=1



 ν¯ (Fl ) ≤ ε.

l=1

Thus Y can be approximated in probability by V and thus, for appropriate subsequence, also almost surely and, since Y ≥ V, by Lebesgue’s dominated convergence theorem also in H.

A.1.4 Representation of Square Integrable Martingales We prove here Theorem A.1.1 for the case in which M is a square integrable martingale and Z is a L´evy process without Gaussian part. Proof

Let X := MT ∗ . By Theorem A.1.5, for some sequence fk ∈ Lˆ k2 , X = E(X) +

+∞  k=1

∗

IkT (fk ),

A.1 Martingale Representation for Jump L´evy Processes

353

and, consequently, Mt = E[X | Ft ] = E(X) +

+∞ 

∗

E[IkT (fk ) | Ft ].

k=1

In view of Theorem A.1.3 ∗



IkT (fk ) =



T∗ 0

f

U

Consequently, ∗

E[IkT (fk ) | Ft ] =

Rk (s, u)π˜ (ds, du).

 t

f

U

0

Rk (s, u)π˜ (ds, du),

and Mt = E(M0 ) +

+∞  t   k=1 0

f

U

Rk (s, u)π˜ (ds, du),

with the series on the right side converging in mean square because converges in H. But E[(X − E(X))2 ] =



+∞    ∗ E (IkT (fk ))2 < +∞, k=1

T∗

and the integrals Ik (fk ), k = 1, 2, . . . are orthogonal, i.e.   ∗ ∗ E IkT (fk )IlT (fl ) = 0, for k = l. Therefore, for k = l    T∗  f Rk (s, u)π˜ (ds, du) · E 0

=E

U

 0

T∗

 U

T∗ 0

 U

 f Rl (s, u)π˜ (ds, du)

 f f Rk (s, u)Rl (s, u) ds ν(du) = 0.

Thus +∞     T∗ 2 E | Ik (fk ) | = E k=1

T∗

  +∞

0

U k=1

 f (Rk )2 (s, u) ds ν(du) < +∞,

and the sum +∞  k=1

f

Rk (s, u)

∗

IkT (fk )

354

Appendix A f

of predictable integrands Rk has a subseries converging dP × ds × du – almost surely to a predictable g. Consequently, +∞  t  

f

U

k=1 0

Rk (s, u)π˜ (ds, du) =

 t 0

g(s, u)π˜ (ds, du). U

A.1.5 Representations of Local Martingales To prove the general version of Theorem A.1.1 we start with auxiliary results dealing with strictly positive local martingales. Proposition A.1.6 Let g, h be predictable processes such that h ∈ 2 is bounded and g(t, y)h(t, y) = 0, t ∈ [0, T ∗ ], y ∈ U. Assume that A(t), t ∈ [0, T ∗ ] is a continuous process of finite variation. Then the process α given by  t

Xt

αt := e , Xt := 0

g(s, y)π(ds, dy) +

 t

U

0

t ∈ [0, T ∗ ]

h(s, y)π˜ (ds, dy) − At ,

U

is a local martingale if and only if the following two conditions are satisfied eg − 1 ∈ 1 ,  t  t g(s,y) At = (e − 1)dsν(dy) + (eh(s,y) − 1 − h(s, y))dsν(dy), U

0

0

(A.1.26) t ∈ [0, T ∗ ].

U

(A.1.27) Proof have

Application of the Itˆo formula provides the integral representation of α. We 

αt = 1 +

t

αs− dXs +

0

=1+

 t 0

+

 t 0

1 2



t 0

αs− d X c , X c s +

αs− g(s, y)π(ds, dy) +



(αs − αs− − αs− Xs )

s∈[0,t]

 t

U

 αs− h(s, y)π˜ (ds, dy) − U

0

t

αs− dAs 0

αs− (eg(s,y)+h(s,y) − 1 − g(s, y) − h(s, y))π(ds, dy), t ∈ [0, T ∗ ].

U

Since g(s, y)h(s, y) = 0 we can split the last integral into the form  t 0

αs− (eg(s,y) −1− g(s, y))π(ds, dy) + U

 t 0

U

αs− (eh(s,y) − 1 − h(s, y))π(ds, dy).

A.1 Martingale Representation for Jump L´evy Processes

355

Then, by compensation, we finally obtain  t  t h(s,y) αt = 1 + αs− (e − 1)π˜ (ds, dy) − αs− dAs 0

 t

+

0

U

0

U

 t

+

U

0

αs− (eg(s,y) − 1)π(ds, dy) t ∈ [0, T ∗ ].

αs− (eh(s,y) − 1 − h(s, y))dsν(dy),

(A.1.28)

Now we show the sufficiency of (A.1.26) and (A.1.27). In view of (A.1.26) we can rearrange (A.1.28) to the form  t  t h(s,y) αs− (e − 1)π˜ (ds, dy) + αs− (eg(s,y) − 1)π˜ (ds, dy) αt = 1 + U

0



t

−

αs− dAs +

0

+

0

 t

0

 t

U

αs− (eg(s,y) − 1)ν(dy)ds U

αs− (eh(s,y) − 1 − h(s, y))dsν(dy),

t ∈ [0, T ∗ ].

U

0

Further, (A.1.27) implies that the last two lines in the preceding disappear and, consequently, α is a local martingale. To show the necessity of (A.1.26) and (A.1.27) let us write (A.1.28) in the form  t αs− (eh(s,y) − 1)π˜ (ds, dy) αt − 1 − U

0



t

=−

αs− dAs +

0

+

 t 0

 t

αs− (eg(s,y) − 1)π(ds, dy)

U

0

αs− (eh(s,y) − 1 − h(s, y))dsν(dy),

t ∈ [0, T ∗ ].

(A.1.29)

U

Since the left side is a local martingale and the right side is a finite variation process it follows from Proposition 4.2.9 that the right side belongs to Aloc . It follows from Proposition 4.2.8 that  t  t αs− dAs + αs− (eh(s,y) − 1 − h(s, y))dsν(dy) ∈ Aloc , − 0

which implies that

0

 t 0

U

αs− (eg(s,y) − 1)π(ds, dy) ∈ Aloc .

U

In view of Theorem 4.2.13 there exists a compensator of the preceding process t and it has the form 0 U αs− (eg(s,y) − 1)dsν(dy) which means that (A.1.26) holds. Compensation of the right side of (A.1.29) gives that the process

356



t

−

αs− dAs +

0

+

 t 0

U

0

U

 t

Appendix A αs− (eg(s,y) − 1)dsν(dy) αs− (eh(s,y) − 1 − h(s, y))dsν(dy),

t ∈ [0, T ∗ ]

is a local martingale. Since it is also a continuous process of finite variation it follows from Proposition 4.2.10 that it disappears, which means that (A.1.27) is satisfied. For the second auxiliary result we need the following result on continuous compensators of jump processes. Proposition A.1.7 Let X be a process adapted to the filtration generated by a L´evy process Z with c`adl`ag paths satisfying X(t) = 0 If the process

⇒

Z(t) = 0.



Y(t) :=

(A.1.30)

h(s, X(s)),

s∈[0,t];X(s)=0

where h :  × R+ × U → R, is adapted and of locally integrable variation then its compensator is continuous. Proof

Let us define the function f by X(s) = f (s, Z(s)),

s ≥ 0.

By (A.1.30) we have f (s, 0) = 0 and hence we can represent Y in the form  h(s, f (s, Z(s))) Y(t) = s∈[0,t];f (s,Z(s))=0



=

h(s, f (s, Z(s)))

s∈[0,t];Z(s)=0;f (s,Z(s))=0



=

h(s, f (s, Z(s)))1{f (s,Z(s))=0}

s∈[0,t];Z(s)=0

=

 t 0

U\{0}

h(s, f (s, y))1{f (s,y)=0} π(ds, dy),

t ≥ 0,

where π(ds, dy) stands for the jump measure of Z. It follows that the compensator of Y is given by  t h(s, f (s, y))1{f (s,y)=0} dsν(dy), t ≥ 0, 0

U\{0}

which is clearly continuous.

A.1 Martingale Representation for Jump L´evy Processes

357

Lemma A.1.8 Let α be a local martingale such that αt > 0, t ∈ [0, T ∗ ] and α0 = 1. Then there exists a unique process ψ, where eψ − 1 ∈ 1,2 , such that αt = eXt , where

 t

ψ1 (s, y)π(ds, dy) −

Xt := U

0

+ −

t ∈ [0, T ∗ ],

0

 t

(eψ1 (s,y) − 1)dsν(dy) U

ψ2 (s, y)π˜ (ds, dy)

0

U

0

U

 t

 t

(A.1.31)

(eψ2 (s,y) − 1 − ψ2 (s, y))dsν(dy),

t ∈ [0, T ∗ ]

and ψ1 (s, y) = ψ(s, y)1{|ψ(s,y)|>1} ,

ψ2 (s, y) = ψ(s, y)1{|ψ(s,y)|≤1} ,

s ∈ [0, T ∗ ], y ∈ U.

However, if eψ − 1 ∈ 1,2 then the process α given by (A.1.31) is a strictly positive local martingale. Before we present the proof let us comment on the formulation of the assertion. Remark A.1.9 The condition eψ − 1 ∈ 1,2 is equivalent to the pair of conditions eψ1 − 1 ∈ 1 ,

ψ2 ∈ 2 ,

and thus the integrals in the definition of the process X make sense. Remark A.1.10 Application of the Itˆo formula shows that (A.1.31) is equivalent to the fact that α solves the Dol´eans-Dade equation (see Theorem 4.4.6)  t αt = 1 + αs− (eψ(s,y) − 1)π˜ (ds, dy), t ∈ [0, T ∗ ]. 0

U

Proof of Lemma A.1.8 The process Xt := ln αt , t ∈ [0, T ∗ ] is a semimartingale with jump times equal to these of α. Since, for each n = 1, 2, . . . the process  Pnt := Xs 1{11} = g(s, y)π(ds, dy), t ∈ [0, T ∗ ]. U

0

s∈[0,t]

Now let us cut off from X the jumps exceeding 1 and define  t  Xt0 = Xt − Xs 1{|Xs |>1} = Xt − g(s, y)π(ds, dy), 0

s∈[0,t]

t ∈ [0, T ∗ ].

U

Since | Xt0 |≤ 1 it follows from Theorem 4.3.3 that it is a special semimartingale and the martingale part M in its canonical decomposition Xt0 = Mt + At ,t∈[0,T ∗ ] is a local square integrable martingale. Again application of Theorem A.1.1 for local square integrable martingales yields  t h(s, y)π˜ (ds, dy), t ∈ [0, T ∗ ] Mt = 0

U

for some h ∈ 2 . Notice also that h is bounded because the jumps of M are bounded. Moreover, by Proposition A.1.7, the process A is continuous. Finally we obtain  t  t Xt = h(s, y)π˜ (ds, dy) + g(s, y)π(ds, dy) + At , t ∈ [0, T ∗ ], 0

U

U

0

(A.1.32) where g(s, y)h(s, y) = 0 because X 0 and Writing

t 0 U

g(s, y)π(ds, dy) do not jump together.

t ∈ [0, T ∗ ],

αt = eXt ,

(A.1.33)

we are in the position to apply Proposition A.1.6, which provides that eg − 1 ∈ 1 and  t  t g(s,y) −At = (e − 1)dsν(dy) + (eh(s,y) − 1 − h(s, y))dsν(dy), t ∈ [0, T ∗ ]. 0

U

0

U

Plugging this into (A.1.32) then coming back to (A.1.33) and defining ψ(s, y) := h(s, y) + g(s, y) we obtain the result. The converse implication follows from Remark A.1.10.

A.1 Martingale Representation for Jump L´evy Processes

359

Proof of Theorem A.1.1 Let M be a martingale. For ε > 0 define two strictly positive martingales M 1,ε (t) := E[MT+∗ | Ft ] + ε, M 2,ε (t) := E[MT−∗ | Ft ] + ε, t ∈ [0, T ∗ ]. Application of Lemma A.1.8 and Remark A.1.10 to the normalized martingales Mt2,ε M02,ε

1

Mt1,ε , M01,ε

2

provides the existence of processes ψ 1 , ψ 2 where eψ − 1, eψ − 1 ∈ 1,2 ,

such that Mt1,ε Mt2,ε

= =

M01,ε M02,ε

+

 t 0

+

 t

U

U

0

1,ε ψ Ms− (e

1 (s,y)

− 1)π˜ (ds, dy),

2,ε ψ Ms− (e

2 (s,y)

− 1)π˜ (ds, dy),

It follows that Mt = Mt1,ε − Mt1,ε = M0 +

 t 0

φ(s, y)πds, ˜ dy,

t ∈ [0, T ∗ ].

t ∈ [0, T ∗ ],

(A.1.34)

U

1

2

1,ε ψ (s,y) 2,ε ψ (s,y) with ψ(s, y) := Ms− (e − 1) − Ms− (e − 1) and it is clear that ψ ∈ 1,2 . If M is a local martingale with localizing sequence {τn } then, in view of the first part of the proof, for each Mtn := M(t ∧ τn ) we have  t n Mt = M0 + ψ n (s, y)π˜ (ds, dy), t ∈ [0, T ∗ ], (A.1.35) U

0

for ψ n ∈ 1,2 . From uniqueness we conclude that there exists a process ψ such that ψ(s, y)1{[0,τn ]} (s) = ψ n (s, y), Since for each n hold  τn ∧T ∗   0

U T∗ 

 = 0

s ∈ [0, T ∗ ], n = 1, 2, . . . .

 | ψ(s, y) |2 ∧ | ψ(s, y) | dsν(dy) 

 | ψ n (s, y) |2 ∧ | ψ n (s, y) | dsν(dy) < +∞,

U

we see that ψ ∈ 1,2 . Letting n → +∞ in (A.1.35) we obtain the assertion.

Appendix B

The concept of generators of semigroups of linear transformations is recalled and their forms for the semigroups corresponding to stochastic equations are derived.

B.1 Semigroups and Generators A family S(t), t ≥ 0, of linear, bounded transformations from a Banach space E into E satisfying S(0) = I, S(t + s)x = S(s)(S(t)x),

s, t ≥ 0, x ∈ E,

lim S(t)x = x, x ∈ E

(B.1.1) (B.1.2)

t↓0

is called a C0 , or strongly continuous, semigroup. The infinitesimal generator of the C0 -semigroup S(t), t ≥ 0, is defined by Ax := lim h↓0

S(h)x − x , h

x ∈ D(A),

where 

 S(h)x − x D(A) := x ∈ H : ∃ lim . h↓0 h A core of a linear operator A with domain D(A) is any linear subset  of D(A) such that the closure of the set {(f , A(f )) : f ∈ } is exactly {(f , A(f )) : f ∈ D(A)}. Generators are uniquely determined by their values on a core. Let A be a linear operator on a Hilbert space H with dense domain D(A). Then the domain D(A∗ ) of its adjoint operator A∗ consists of all y ∈ H such that there exists z ∈ H that for all x ∈ D(A), Ax, y = x, z . If this is the case then A∗ y = z.

B.1 Semigroups and Generators

361

If (S(t)) is a C0 -semigroup on a Hilbert space H, then its adjoint semigroup (S∗ (t)) is determined by the formula S∗ (t)y, x = y, S(t)x , x, y ∈ H, t ≥ 0. Its generator is the adjoint operator A∗ . Transition semigroups (Pt ) of Markovian processes evolving in closed subsets K of Rn are usually studied on various spaces of functions defined on K, like Cb (K), the space of all bounded continuous functions on K. When K = [0, +∞) the transition semigroup will be considered here on the space C0 ([0, +∞)) of all bounded continuous functions vanishing at +∞.

B.1.1 Generators for Equations with L´evy Noise Consider solutions X x (t) on Rn of the following stochastic equation: dX(t) = F(X(t))dt + G(X(t−))dZ(t),

X(0) = x ∈ Rn ,

t > 0,

(B.1.3)

where F : Rn −→ Rn , G(x), for each x : Rn is a linear transformation from Rd into Rn and Z is a L´evy process in Rd with characteristic triplet (a, Q, ν). According to the L´evy–Itˆo decomposition (see (5.2.3)), Z(t) = at + W(t) +

 t 0

|y|≤1

y π˜ (ds, dy) +

 t 0

|y|>1

y π(ds, dy),

t > 0. (B.1.4)

Let Pt be the transition semigroup of the solution X x Pt f (x) := E(f (X x (t))),

t ≥ 0, x ∈ Rn , f ∈ Cb (Rn ).

Assume that Pt is a strongly continuous semigroup on E, a Banach subspace of Cb (Rn ). If there exists the limit 1 (Ph f (x)) − f (x)) h

(B.1.5)

as h → 0, uniformly in x and, as a function of x, it does belong to E then one says that f is in the domain D(A) of the generator of the transition semigroup and the limit is, by definition, the value of the generator A on f and is denoted by Af (x). If, instead of uniform convergence one requires existence of the limit for each x then the limit is called the weak generator of the transition semigroup or, with some abuse of language, of the equation.

362

Appendix B

Here is a typical result on generators and its proof can be easily adapted to equations acting on some subsets K of Rn . The proof is based on the Itˆo formula f (X(t)) − f (X(0)) =

n  

t

∂f (X(s−))dX j (s) ∂xj j=1  t 2 ∂ f 1  + (X(s−))d[X j , X l ]c (s) 2 0 ∂xj ∂xl 0

(B.1.6)

1≤j,l≤n

+

 &

f (X(s)) − f (X(s−)) −

0