Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling [1 ed.] 1119377382, 9781119377382

Table of contents :
Content: 1. Mathematical foundations 1: point-set concepts, set and measure functions, normed linear spaces, and integration --
2. Mathematical foundations 2: Probability, random variables, and convergence of random variables --
3. Mathematical foundations 3: Stochastic processes, Martingales, and Brownian motion --
4. Mathematical foundations 4: Stochastic integrals, Itô's integral, Itô's formula, and Martingale representation --
5. Stochastic differential equations --
6. Stochastic population growth models --
7. Approximation and estimation of solutions to stochastic differential equations --
8. Estimation of parameters of stochastic differential equations.

Citation preview

Stochastic Differential Equations

Stochastic Differential Equations An Introduction with Applications in Population Dynamics Modeling

Michael J. Panik Department of Economics and Finance, Barney School of Business and Public Administration West Hartford, CT, USA

This edition first published 2017 © 2017 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permision to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Michael J Panik to be identified as the author of this work, has been asserted in accordance with law. Registered Office John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty The publisher and the authors make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of fitness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or website is referred to in this work as a citation and/or potential source of further information does not mean that the author or the publisher endorses the information the organization or website may provide or recommendations it may make. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom. Library of Congress Cataloging-in-Publication Data Names: Panik, Michael J. Title: Stochastic differential equations : an introduction with applications in population dynamics modeling / Michael J. Panik. Description: 1st edition. | Hoboken, NJ : John Wiley & Sons, Inc., [2017] | Includes bibliographical references and index. Identifiers: LCCN 2016056661 (print) | LCCN 2016057077 (ebook) | ISBN 9781119377382 (cloth) | ISBN 9781119377412 (pdf) | ISBN 9781119377405 (epub) Subjects: LCSH: Stochastic differential equations. Classification: LCC QA274.23 .P36 2017 (print) | LCC QA274.23 (ebook) | DDC 519/.22–dc23 LC record available at https://lccn.loc.gov/2016056661 Cover image: © RKaulitzki/Getty Images Cover design by Wiley Set in 10/12pt Warnock by SPi Global, Pondicherry, India Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

v

Contents Dedication x Preface xi Symbols and Abbreviations

xiii

1

Mathematical Foundations 1: Point-Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration 1

1.1 1.1.1 1.1.2 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.9.1 1.9.2 1.10 1.10.1 1.10.2 1.10.3 1.10.4 1.10.5 1.10.6 1.10.7 1.11 1.11.1 1.11.2

Set Notation and Operations 1 Sets and Set Inclusion 1 Set Algebra 2 Single-Valued Functions 4 Real and Extended Real Numbers 6 Metric Spaces 7 Limits of Sequences 8 Point-Set Theory 10 Continuous Functions 12 Operations on Sequences of Sets 13 Classes of Subsets of Ω 15 Topological Space 15 σ-Algebra of Sets and the Borel σ-Algebra 15 Set and Measure Functions 17 Set Functions 17 Measure Functions 18 Outer Measure Functions 19 Complete Measure Functions 21 Lebesgue Measure 21 Measurable Functions 23 Lebesgue Measurable Functions 26 Normed Linear Spaces 27 Space of Bounded Real-Valued Functions 27 Space of Bounded Continuous Real-Valued Functions

28

vi

Contents

1.11.3 1.12 1.12.1 1.12.2 1.12.3 1.12.4 1.12.5 2

Some Classical Banach Spaces 29 Integration 31 Integral of a Non-negative Simple Function 32 Integral of a Non-negative Measurable Function Using Simple Functions 33 Integral of a Measurable Function 33 Integral of a Measurable Function on a Measurable Set 34 Convergence of Sequences of Functions 35 Mathematical Foundations 2: Probability, Random Variables, and Convergence of Random Variables 37

2.1 Probability Spaces 37 2.2 Probability Distributions 42 2.3 The Expectation of a Random Variable 49 2.3.1 Theoretical Underpinnings 49 2.3.2 Computational Considerations 50 2.4 Moments of a Random Variable 52 2.5 Multiple Random Variables 54 2.5.1 The Discrete Case 54 2.5.2 The Continuous Case 59 2.5.3 Expectations and Moments 63 2.5.4 The Multivariate Discrete and Continuous Cases 69 2.6 Convergence of Sequences of Random Variables 72 2.6.1 Almost Sure Convergence 73 2.6.2 Convergence in L p , p > 0 73 2.6.3 Convergence in Probability 75 2.6.4 Convergence in Distribution 75 2.6.5 Convergence of Expectations 76 2.6.6 Convergence of Sequences of Events 78 2.6.7 Applications of Convergence of Random Variables 79 2.7 A Couple of Important Inequalities 80 Appendix 2.A The Conditional Expectation E(X|Y) 81 3

Mathematical Foundations 3: Stochastic Processes, Martingales, and Brownian Motion 85

3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.1.1 3.2.2

Stochastic Processes 85 Finite-Dimensional Distributions of a Stochastic Process Selected Characteristics of Stochastic Processes 88 Filtrations of A 89 Martingales 91 Discrete-Time Martingales 91 Discrete-Time Martingale Convergence 93 Continuous-Time Martingales 96

86

Contents

3.2.2.1 Continuous-Time Martingale Convergence 97 3.2.3 Martingale Inequalities 97 3.3 Path Regularity of Stochastic Processes 98 3.4 Symmetric Random Walk 99 3.5 Brownian Motion 100 3.5.1 Standard Brownian Motion 100 3.5.2 BM as a Markov Process 104 3.5.3 Constructing BM 106 3.5.3.1 BM Constructed from N(0, 1) Random Variables 106 3.5.3.2 BM as the Limit of Symmetric Random Walks 108 3.5.4 White Noise Process 109 Appendix 3.A Kolmogorov Existence Theorem: Another Look Appendix 3.B Nondifferentiability of BM 110 4

109

Mathematical Foundations 4: Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation 113

4.1 Introduction 113 4.2 Stochastic Integration: The Itô Integral 114 4.3 One-Dimensional Itô Formula 120 4.4 Martingale Representation Theorem 126 4.5 Multidimensional Itô Formula 127 Appendix 4.A Itô’s Formula 129 Appendix 4.B Multidimensional Itô Formula 130 5 Stochastic Differential Equations 133 5.1 Introduction 133 5.2 Existence and Uniqueness of Solutions 134 5.3 Linear SDEs 136 5.3.1 Strong Solutions to Linear SDEs 137 5.3.2 Properties of Solutions 147 5.3.3 Solutions to SDEs as Markov Processes 152 5.4 SDEs and Stability 154 Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 159 5.A.1 Linear Homogeneous Variety 159 5.A.2 Linear Variety 161 Appendix 5.B Integrating Factors and Variation of Parameters 162 5.B.1 Integrating Factors 163 5.B.2 Variation of Parameters 164 6

6.1 6.2

Stochastic Population Growth Models

167 Introduction 167 A Deterministic Population Growth Model 168

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Contents

6.3 6.4 6.5

A Stochastic Population Growth Model 169 Deterministic and Stochastic Logistic Growth Models 170 Deterministic and Stochastic Generalized Logistic Growth Models 174 6.6 Deterministic and Stochastic Gompertz Growth Models 177 6.7 Deterministic and Stochastic Negative Exponential Growth Models 179 6.8 Deterministic and Stochastic Linear Growth Models 181 6.9 Stochastic Square-Root Growth Model with Mean Reversion 182 Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect 184 Appendix 6.B Reducible SDEs 189 7

Approximation and Estimation of Solutions to Stochastic Differential Equations 193

7.1 Introduction 193 7.2 Iterative Schemes for Approximating SDEs 194 7.2.1 The EM Approximation 194 7.2.2 Strong and Weak Convergence of the EM Scheme 196 7.2.3 The Milstein (Second-Order) Approximation 196 7.3 The Lamperti Transformation 199 7.4 Variations on the EM and Milstein Schemes 203 7.5 Local Linearization Techniques 205 7.5.1 The Ozaki Method 205 7.5.2 The Shoji–Ozaki Method 207 7.5.3 The Rate of Convergence of the Local Linearization Method 211 Appendix 7.A Stochastic Taylor Expansions 212 Appendix 7.B The EM and Milstein Discretizations 217 7.B.1 The EM Scheme 217 7.B.2 The Milstein Scheme 218 Appendix 7.C The Lamperti Transformation 219 8 Estimation of Parameters of Stochastic Differential Equations 221 8.1 Introduction 221 8.2 The Transition Probability Density Function Is Known 222 8.3 The Transition Probability Density Function Is Unknown 227 8.3.1 Parameter Estimation via Approximation Methods 228 8.3.1.1 The EM Routine 228 8.3.1.2 The Ozaki Routine 230 8.3.1.3 The SO Routine 233 Appendix 8.A The ML Technique 235 Appendix 8.B The Log-Normal Probability Distribution 238

Contents

Appendix 8.C

The Markov Property, Transitional Densities, and the Likelihood Function of the Sample 239 Appendix 8.D Change of Variable 241 Appendix A: A Review of Some Fundamental Calculus Concepts 245 Appendix B: The Lebesgue Integral 259 Appendix C: Lebesgue–Stieltjes Integral 261 Appendix D: A Brief Review of Ordinary Differential Equations 263 References 275 Index 279

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x

In Memory of John Snizik

xi

Preface The three common varieties of stochastic models that are typically used to study population dynamics are discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation (SDE) models. This book focuses on the third type—the use of (Itô) SDEs to determine a variety of population growth equations. While deterministic growth models have a rich history of applications in a multitude of fields, it should be obvious that such devices have their limitations in that they don’t adequately account for, say, environmental noise (e.g., weather, natural disasters, and epidemics) as incorporated in and mirrored by a stochastic process. As will be seen later on, SDEs represent mathematical models that combine deterministic and stochastic components of dynamic behavior; their solutions are stochastic processes that depict diffusive dynamics. A major difficulty associated with the derivation and application of SDEs is the up-front investment required of the reader in terms of preparation in the areas of mathematical statistics, real analysis and measure theory, stochastic calculus, and stochastic processes. Indeed, the scope of the requirements for studying SDEs can be rather daunting. To address any background deficiencies on the part of the reader, and to offer a comprehensive review for those who need a mathematical refresher, I have developed four detailed and very comprehensive chapters (Chapters 1–4) pertaining to the aforementioned areas of mathematics/statistics. These chapters are richer and much more extensive than what is normally included as a review section in virtually every book on SDEs. The perfunctory mathematical reviews typically offered are woefully inadequate for practitioners and consumers of SDE-based growth models. In addition, Appendices A–D are also offered for review purposes. These involve some basic calculus concepts (including the Riemann integral), the Lebesgue and Lebesgue–Stieltjes integrals, and ordinary differential equations. Given that the book is written for beginners or novices to the area of SDEs, it is particularly well suited for advanced undergraduates and beginning graduate students in the areas of economics, population studies, environmental sciences, engineering, and biological sciences who have had a few courses in the calculus

xii

Preface

and statistics but have not been exposed to the full spectrum of mathematical study received by mathematics majors proper. It is also designed for practitioners in the aforementioned fields who need a gentle introduction to SDEs via a thorough review of the mathematical apparatus needed for studying this discipline. That said, the book will certainly be challenging to most readers. However, it does contain all that is needed for the successful study and application of (Itô-type) SDEs. My sincere gratitude is extended to my colleagues in the Department of Economics and Finance at the University of Hartford. In particular, I have benefited considerably over the years from conversations with Bharat Kolluri, Farhad Rassekh, Rao Singamsetti, and Mahmoud Wahab. In addition, Alice Schoenrock accurately and steadfastly typed the entire manuscript. Her efforts are greatly appreciated. I would also like to thank Kathleen Pagliaro, Assistant Editor, for helping to expedite the publication process. A special note of appreciation goes to Susanne Steitz-Filler—Editor (mathematics and statistics) at John Wiley & Sons—for her professionalism, vision, and encouragement.

xiii

Symbols and Abbreviations

■ 2Ω Ω (Ω, A) (Ω, A, P) (Ω, A, F, P) (Ω, A, μ) (Ω, Ac, μc) (Ω, T) χ A(x) ω δij δx(A) μ μc μr μr μ∗ σ(C) τ Γ A A A Ā A1 A2

denoting end of example power set the universal set or space or sample space measurable space probability space filtered probability space measure space complete measure space topological space, where T is a topology on Ω the indicator function of set A outcome point in a sample space Ω the Dirac delta function the Dirac point mass function measure function the completion of μ the rth central moment the rth moment about zero outer measure function the σ-algebra generated by C stopping time null set element inclusion the gamma function the nth-order variance–covariance matrix autonomous SDE σ-algebra the complement of set A the closure of set A the product σ-algebra

xiv

Symbols and Abbreviations

Ãt Atw t ≥ 0 A−B A B A B A B A B AΔB a.e. AL ALH ALNS arg max a.s. B B BM CIR CKLS C(X) DCT d(E, F) d(x, y) ei EM E(X) (or μ) E|Xt|p E(X|Y) F f f p (f, g) f X Y f −1 Y X Gf H Hr I( f ) (or I(t)) In

the augmentation with respect to P of Atw the natural filtration generated by Wt t ≥ 0

t≥0

the difference between sets A, B the union of sets A, B the intersection of sets A, B set A is a proper subset of set B set A is a subset of set B the symmetric difference between sets A, B almost everywhere autonomous linear SDE autonomous linear homogeneous SDE autonomous linear narrow sense SDE the argument of the maximum almost surely Banach space the Borel σ-algebra Brownian motion Cox, Ingersol, and Ross SDE Chan, Karolyi, Longstaff, and Sanders SDE the set of all bounded continuous functions on a metric space X dominated convergence theorem the distance between sets E, F the distance between points x, y the ith unit vector Euler–Maruyama method the expectation of a random variable X pth-order moment the conditional expectation of X given Y, where X and Y are random variables filtration the norm of a function f the p-norm of a function f the inner product of functions f, g f is a point-to-point mapping from set X to set Y f − 1 is the inverse (point-to-point) mapping from set Y to set X the graph of f Hilbert space Hilbert space of random variables the Itô stochastic integral the identity matrix of order n

Symbols and Abbreviations

Iq J L L L2(μ) Lp(μ) LH limn ∞ inf Ai limn ∞ inf xn limn ∞ sup Ai limn ∞ sup xn ln L LNS LS M M2 , MCT ML N O ODE OU |P| P(A|B) P1 P2 PML R R∗ R+ Rn RS SBM SDE SO S(X)(or σ) V(X)(or σ 2) W t Wt t ≥ 0

the Bessel function of the first kind of order q the Jacobian determinant linear SDE likelihood function the class of real-valued square integrable functions the set of all measurable functions f such that | f |p is integrable linear homogeneous SDE the limit inferior of a sequence {Ai} of sets the limit inferior of a sequence {xn} of real numbers the limit superior of a sequence {Ai} of sets the limit superior of a sequence {xn} of real numbers log-likelihood function linear narrow sense SDE Lebesgue–Stieltjes Lebesgue measurable sets the space of all real-valued processes f on the product space Ω × , monotone convergence theorem maximum likelihood the collection of P-null sets f x = O g x is the quantity whose ratio to g(x) remains bounded as x tends to a limit ordinary differential equation Ornstein–Uhlenbeck mesh of partition P the conditional probability of event A given event B the product probability measure pseudo-maximum likelihood the set of real numbers the extended real numbers the non-negative real numbers the set of ordered n-tuples (x1, x2, …, xn) or n-dimensional Euclidean space Riemann–Stieltjes standard Brownian motion stochastic differential equation Shoji–Ozaki method the standard deviation of a random variable X the variance of a random variable X white noise Brownian motion or Wiener process

xv

xvi

Symbols and Abbreviations

x Xd X N 0,1 X N μ,σ Xt t Xt(w) (x, x) X ×Y

T

the norm of an n-tuple x = x1 , x2 ,…, xn the derived set of all limit points of set X the random variable X is standard normal the random variable X is normally distributed with mean μ and standard deviation σ stochastic process sample path of a stochastic process the inner product of an n-tuple x = x1 , x2 ,…, xn the product of sets X, Y

1

1 Mathematical Foundations 1 Point-Set Concepts, Set and Measure Functions, Normed Linear Spaces, and Integration

1.1

Set Notation and Operations

1.1.1

Sets and Set Inclusion

We may generally think of a set as a collection or grouping of items without regard to structure or order. (Sets will be represented by capital letters, e.g., A, B, C, ….) An element is an item within or a member of a set. (Elements are denoted by small case letters, e.g., a, b, c, ….) A set of sets will be termed a class (script capital letters will denote a class of sets, e.g., A,B, C,…); and a set of classes will be called a family. Let us define a space (denoted Ω) as a type of master or universal set—it is the context in which discussions of sets occur. In this regard, an element of Ω is a point ω. To define a set X, let us write X = x the x's possess some defining property , that is, this reads “X is the set of all elements x such that the x’s have some unique characteristic,” where “such that” is written “|.” The set containing no elements is called the empty set (denoted ϕ)—it is a member of every set. What about the size of a set? A set may be finite (it is either empty or consists of n elements, n a positive integer), infinite (e.g., the set of positive integers), or countably infinite (its elements can be put into one-to-one correspondence with the counting numbers). We next look to inclusion symbols. Specifically, we first consider element inclusion. Element x being a member of set X is symbolized as x X. If x is not a member of, say, set Y, we write x Y . Next comes set inclusion (a subset notation). A set A is termed a subset of set B (denoted A B) if B contains the same elements that A does and possibly additional elements that are not found in A. If A is not a subset of B, we write A ⊈ B. Actually, two cases are subsumed in A B: (1) either A B (A is then called a proper subset of B, meaning that B is a set that is larger than A; or (2) A = B (A and B contain exactly the same

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

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1 Mathematical Foundations 1

elements and thus are equal). More formally, A = B if and only if A B and B A. If equality between sets A and B does not hold, we write A B. 1.1.2

Set Algebra

Given sets A and B within Ω, their union (denoted A B) is the set of elements that are in A, or in B, or in both A and B. Here, we are employing the inclusive or. Symbolically, A B = x x A or x B (Figure 1.1a). The intersection of sets A and B (denoted A B) is the set of elements common to both A and B, that is, A B = x x A and x B (Figure 1.1b). The complement of a set A is the set of elements within Ω that lie outside of A (denoted A ). Here, A = x x A (Figure 1.1c). If sets A and B do not intersect and thus have no elements in common, then A and B are said to be disjoint or mutually exclusive and we write A B = Ø. The difference between sets A and B (denoted A − B) is the set of elements in A but not in B or A− B = A B . Thus, A− B = x x A and x B (Figure 1.1d). The symmetric difference between sets A and B (denoted AΔ B) is the

(a)

(b) Ω

(c) Ω

Ω A

A

B A∪B

(d)

A

Ω

B A∩B

(e)

A′

A′

Ω

A

B A–B

A

B AΔB

Figure 1.1 (a) Union of A and B, (b) intersection of A and B, (c) complement of A, (d) difference of A and B, and (e) symmetric difference of A and B.

1.1 Set Notation and Operations

union of their differences in reverse order or AΔ B = (A − B) (B − A) = B A (Figure 1.1e). A B A few essential properties of these set operations now follow. Specifically for sets A, B, and C within Ω: UNION A A = A, A Ω = Ω, A Ø = A A B = B A (commutative property) A (B C) = (A B) C (associative property) A B if and only if A B = B INTERSECTION A A = A, A Ω = A, A Ø = Ø A B = B A (commutative property) A (B C) = (A B) C (associative property) A B if and only if A B = A COMPLEMENT (A ) = A, Ω = Ø, Ø = Ω A A = Ω, A A = Ø A B =A B De Morgan’s laws A B =A B DIFFERENCE A − B = (A B) − B = A − (A B) (A − B) − C = A − (B C) = (A − B) (A − C) A − (B − C) = (A − B) (A C) (A B) − C = (A − C) (B − C) SYMMETRIC DIFFERENCE A Δ A = Ø, A Δ Ø = A A Δ B = B Δ A (commutative property) A Δ (B Δ C) = (A Δ B) Δ C (associative property) A (B Δ C) = (A B) Δ (A C) DISTRIBUTIVE LAWS (connect the operations of union and intersection) A (B C) = (A B) (A C) A (B C) = (A B) (A C)

3

4

1 Mathematical Foundations 1

If Ai ,i = 1, …,n is any arbitrary finite class of sets, then the extension of the union and intersection operations to this class can be written, respectively, as n i = 1 Ai

and

n i = 1 Ai

Hence, the union of a class of sets is the collection of elements belonging to at least one of them; the intersection of a class of sets is the set of elements common to all of them. In fact, given these notions, De Morgan’s laws may be extended to n i = 1 Ai

=

n i = 1 Ai

and

n i = 1 Ai

=

n i = 1 Ai

Furthermore, if Ai , i = 1, …, n and Bj , j = 1,…, m are two finite classes of sets Bj , then with Ai n i = 1 Ai

m j = 1 Bj

and

m j = 1 Bj

n i = 1 Ai

In addition, if Ai , i = 1 2,… represents a sequence of sets, then their union and intersection appears as ∞ i = 1 Ai

and

∞ i = 1 Ai ,

respectively.

1.2

Single-Valued Functions

Given two nonempty sets X and Y (which may or may not be equal), a singlevalued function or point-to-point mapping f: X Y is a rule or law of correspondence that associates with point x X a unique point y Y. Here, y = f(x) is the image of x under rule f. While set X is called the domain of f (denoted Df), the collection of those y’s that are the image of at least one x X is called the range of f and denoted Rf . Clearly the range of f is a subset of Y (Figure 1.2a). If Rf Y, then f is an into mapping. In addition, if Rf = Y (i.e., every y Y is the image of at least one x X or all the y’s are accounted for in the mapping process), then f is termed an onto or surjective mapping. Moreover, f is said to be one-to-one or injective if no y Y is the image of more than one f x2 ). Finally, f is called bijective if it is both x X (i.e., x1 x2 implies f x1 one-to-one and onto or both surjective and injective. If the range of f consists of but a single element, then f is termed a constant function. Given a nonempty set X, if Y consists entirely of real numbers or Y = R, then f: X Y is termed a real-valued function or mapping of a point x X into a unique real number y R.1 Hence, the image of each point x X is a real scalar y = f(x) R. 1 A discussion of real numbers is offered in Section 1.3.

1.2 Single-Valued Functions

(a) Y

X • x

f

y = f(x) • Rf Rf ⊆ Y

X = Df (b) X

Y f • x=

f –1(y)

x = Df = Rf–1

f –1

y = f (x) •

Y = Rf = Df–1

Figure 1.2 (a) f is an into mapping and (b) f is one-to-one and onto.

For sets X and Y with set A X, let f1: A Y be a point-to-point mapping of A into Y and f2: X Y be a point-to-point mapping of X into Y. Then f1 is said to be a restriction of f2 and f2 is termed an extension of f1 if and only if for each x A, f1(x) = f2(x). Let X1, X2, …, Xn represent a class of nonempty sets. The product set of X1, X2, …, Xn (denoted X1 × X2 × × Xn ) is the set of all ordered n-tuples (x1, x2, …, xn), where xi Xi for each i = 1, …, n. Familiar particularizations of this definition are R1 = R (the real line); R2 = R × R is the two-dimensional coordinate plane (made up of all ordered pairs (x1, x2), where both x1 R and x2 R); and R n = R × R × × R (the product is taken n times) depicts the collection of ordered n-tuples of real numbers. In this regard, for f a point-to-point mapping of X into Y, the subset Gf = x, y x X, y = f x Y of X × Y is called the graph of f. If the point-to-point mapping f is bijective ( f is one-to-one and onto), then its single-valued inverse mapping f − 1 Y X exists. Thus to each point Y, there corresponds a unique inverse image point x X such that y x = f −1 y = f −1 f x so that x is termed the inverse function of y. Here, the domain Df −1 of f −1 is Y, and its range Rf −1 is X. Clearly, f −1 must also be bijective (Figure 1.2b).

5

6

1 Mathematical Foundations 1

1.3

Real and Extended Real Numbers

We noted in Section 1.2 that a function f is real valued if its range is the set of real numbers. Let us now explore some of the salient features of real numbers— properties that will be utilized later on. The real number system may be characterized as a complete, ordered field, where a field is a set F of elements together with the operations of addition and multiplication. Moreover, both addition and multiplication are associative and commutative, additive and multiplicative inverse and identity elements exist, and multiplication distributes over addition. Set F is ordered if there is a binary order relation “ 0 there exists an index value nε such that n > nε implies xn −x < ε. If we think of the condition xn − x < ε as defining an open sphere of radius ε about x, then we can say that {xn} converges to x if for each open sphere of radius ε > 0 centered on x, there exists an nε such that xn is within this open sphere for all n > nε . Hence, the said sphere contains all points of {xn} from xnε on, that is, x is the limit of the sequence {xn} in Rn if, given ε > 0, all but a finite number of terms of the sequence are within ε of x. A point x R n is a limit (cluster) point of an infinite sequence {xk} if and only if there exists an infinite subsequence xk k K of {xk} that converges to x, that is, there exists an infinite subsequence {xk} such that limj ∞ xkj −x = 0 or xkj x as j ∞ . Stated alternatively, x is a limit point of {xk} if, given a δ > 0 and an index value k, there exists some k > k such that xk − x < δ for infinitely many terms of {xk}. What is the distinction between the limit of a sequence and a limit point of a sequence? To answer this question, we state the following: a. x is a limit of a sequence {xk} in Rn if, given a small and positive ε R, all but a finite number of terms of the sequence are within ε of x. b. x is a limit point of {xk} in Rn if, given a real scalar ε > 0 and given k, infinitely many terms of the sequence are within ε of x. Thus, a sequence {xk} in Rn may have a limit but no limit point. However, if a convergent sequence {xk} in Rn has infinitely many distinct points, then its limit is a limit point of {xk}. Likewise, {xk} may possess a limit point but no limit. In fact, if the sequence {xk} in Rn has a limit point x, then there is a subsequence xk k K of {xk} that has x as a limit; but this does not necessarily mean that the entire sequence {xk} converges to x.3 3 If xk = n = constant for all k, then {xk} converges to the limit n. But since the range of this sequence contains only a single point, it is evident that the sequence has no limit point. If xk = 1 k, then the sequence {xk} converges to a limit of zero, which is also a limit point. In addition, if xk = −1 k , then the sequence {xk} has limit points at ±1, but has no limit.

1.5 Limits of Sequences

A sufficient condition that at least one limit point of an infinite sequence {xk} in Rn exists is that {xk} is bounded, that is, there exists a scalar M R such that xk ≤ M for all k. In this regard, if an infinite sequence of points {xk} in Rn is bounded and it has only one limit point, then the sequence converges and has as its limit that single limit point. The preceding definition of the limit of the sequence {xn} explicitly incorporated the actual limit x. If one does not know the actual value of x, then the following theorem enables us to prove that a sequence converges even if its actual limit is unknown. To this end, we state first that a sequence is a Cauchy sequence if for each ε > 0 there exists an index value Nε/2 such that m, n > Nε 2 implies d xm , xn = xm −xn < ε.4 Second, Rn is said to be complete in that to every Cauchy sequence {xn} defined on Rn there corresponds a point x such that limn ∞ xn = x. Given these concepts, we may now state the Cauchy Convergence Criterion: Given that Rn is complete, a sequence {xn} in Rn converges to a limit x if and only if it is a Cauchy sequence, that is, a necessary and sufficient condition for {xn} to be convergent in Rn is 0 as m, n ∞ . that d xm , xn Hence, every convergent sequence on Rn is a Cauchy sequence. The implication of this statement is that if the terms of a sequence approach a limit, then, beyond some point, the distance between pairs of terms diminishes. It should be evident from the preceding discussion that a complete metric space is a metric space in which every Cauchy sequence converges, that is, the space contains a point x to which the sequence converges or limn ∞ xn = x. In this regard, it should also be evident that the real line R is a complete metric space as is Rn. We next define the limit superior and limit inferior of a sequence {xn} of real numbers as, respectively, a b

lim sup xn = lim

n

∞

n

∞

lim inf xn = lim

n

∞

n

∞

supxm and m≥n

13

inf xm m≥n

Hence, the limit superior of the sequence {xn} is the largest number x such that there is a subsequence of {xn} that converges to x—and no subsequence converges to a higher value. Similarly, the limit inferior is the smallest limit attainable for some convergent subsequence of {xn}—and no subsequence converges to a lower value. Looked at in another fashion, for, say, Equation (1.3a), a 4 That is, for ε > 0 there exists a positive integer Nε/2 such that m ≥ Nε/2 implies d xm , x < ε 2; and n ≥ Nε/2 implies d xn , x < ε 2. Hence, both m, n > Nε/2 imply, via the triangle inequality, that d xm ,xn ≤ d xm , x + d xn , x < 2ε + 2ε = ε.

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number x is the limit superior of a sequence {xn} if (1) for every x < x, we have x < xn for infinitely many n’s; and (2) for every x > x, we have x < xn for only finitely many n’s. Generally speaking, when there are multiple points around which the terms of a sequence tend to “pile up,” the limit superior and limit inferior select the largest and smallest of these points, respectively. We noted earlier that a sequence defined on a subset X of Rn is a function whose range is {xn}. If this function is bounded, then its range {xn} is bounded from both above and below. In fact, if {xn} is a bounded sequence of real numbers, then the limit superior and limit inferior both exist. It is also important to note that limn ∞ xn exists if and only if the limit superior and limit inferior are equal. We end this discussion of limits by mentioning that since any set of extended real numbers has both a supremum and an infimum, it follows that every sequence of extended real numbers has both a limit superior and a limit inferior.

1.6

Point-Set Theory

Let δ be any positive scalar. A δ-neighborhood of a point x0 Rn or sphere of radius δ about x0 is the set δ x0 = x x−x0 < δ,δ > 0 . A point x is an interior point of a set X in Rn if there exists a δ-neighborhood about x that contains only points of X. A set X in Rn is said to be open if, given any point x0 X, there exists a positive scalar δ such that δ x0 X. Hence, X is open if it contains only interior points. Moreover, a. Ø, δ(x0), and Rn are all open sets. b. Any union of open sets in Rn is open; and any finite intersection of open sets in Rn is open. Let X be a set in Rn. The complementary set of X, denoted X , is the collection of all points of Rn lying outside of X. A point x X is an exterior point of X in Rn if there exists a δ-neighborhood of x that contains only points of X . A point x is a boundary point of a set X in Rn if every δ-neighborhood of x encompasses points in X and in X . A set X in Rn is bounded if there exists a scalar M R such that x ≤ M for all x X. Stated alternatively, X is bounded if it has a finite diameter d X = sup x −y x, y X . A set X in Rn has an open cover if there exist a collection {Gi} of open n subsets from Rn such that X i G The open cover {Gi} of X in R is said to contain a finite subcover if there are finitely many indices i1, …, im for m which X j = 1 Gij . A point x is termed a point of closure of a set X in Rn if every δ-neighborhood of x contains at least one point of X, that is, δ x X ϕ. It is important to note

1.6 Point-Set Theory

that a point of closure of X need not be a member of X; however, every element within X is also a point of closure of X. A subset X of Rn is closed if every point of closure of X is contained in X. The closure of a set X in Rn, denoted X, is the set of points of closure of X. Clearly, a set X in Rn is closed if and only if X = X. A set X in Rn has a closed cover if there exists a collection {Gi} of closed subsets from Rn such that X iG Closely related to the concept of a point of closure of X is the notion of a limit (cluster) point of a set X in Rn. Specifically, x is a limit point of X if each δ-neighborhood about x contains at least one point of X different from x, that is, points of X different from x tend to “pile up” at x. So if x is a limit point of a set X in Rn, then X δ x is an infinite set—every δ-neighborhood of x contains infinitely many points of X. Moreover, a. If X is a finite set in Rn, then it has no limit point. b. The limit point of X need not be an element of X. c. The collection of all limit points of X in Rn is called the derived set and will be denoted Xd. Based on the preceding discussion, we can alternatively characterize a set X in Rn as closed if it contains each of its limit points or if X d X. In addition, we can equivalently state that the closure of a set X in Rn is X together with its collection of limit points or X = X X d . Furthermore, a. Ø, a single point, and Rn are all closed sets. b. Any finite union of closed sets in Rn is closed; any intersection of closed sets in Rn is closed. c. The closure of any set X in Rn is the smallest closed set containing X. d. A subset X in Rn is closed if and only if its complementary set X is open. e. A subset X in Rn is closed if and only if X contains its boundary. Let’s now briefly relate the concepts of a limit and a limit point of a sequence in Rn to some of the preceding point-set notions that we just developed. In particular, we shall take another look at the point of closure concept. To this end, a limit point (as well as a limit) of a sequence {xk} in Rn is a point of closure of a set X in Rn if X contains {xk}. Conversely, if x is a point of closure of a set X in Rn, then there exists a sequence {xk} in X (and hence also a subsequence xk k K in X) such that x is a limit point of {xk} (and thus a limit of xk k K ). Hence, the closure of X,X, consists of all limit points of convergent sequences {xk} from X. Similarly, we note that a subset X in Rn is closed if and only if every convergent sequence of points {xk} from X has a limit in X, that is, X is closed if for {xk} in X, limk ∞ xk = x X. Also, a set X in Rn is bounded if every sequence of points {xk} formed from X is bounded. In addition, if a set X in Rn is both closed and bounded, then it is termed compact. (Equivalently, a set X in Rn is compact

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if it has the finite intersection property: every finite subclass has a nonempty intersection.) We mention briefly the following: a. A closed subset of a compact set X in Rn is compact. b. The union of a finite number of compact sets in Rn is compact; the intersection of any number of compact sets in Rn is compact. c. A set X in Rn is compact if and only if it is closed and bounded. d. Any finite set of points in Rn is compact. e. If X in Rn is a set consisting of a convergent sequence {xk} and its limit x = limk ∞ xk , then X is compact. Conversely, if X in Rn is compact, every sequence {xk} has a convergent subsequence xk k K whose limit belongs to X. A set X in Rn is locally compact if each of its points has a δ-neighborhood with compact closure, that is, for each x X, δ x is compact. In this regard, any compact space is locally compact but not conversely, for example, Rn is locally compact but not compact.

1.7

Continuous Functions

For metric spaces X and Y with metrics d1 and d2, respectively, let f: X Y be a point-to-point mapping of X into Y. f is said to be continuous at a point x0 X if either a. for any ε > 0 there exists a δε > 0 such that d1 x,x0 < δε implies d2 f x ,f x0 < ε. (Note that the subscript on δ means that “δ depends upon the ε chosen.”); or b. for each ε-neighborhood of f(x0), ε(f(x0)), there exists a δε-neighborhood ε f x0 , that is, points “near” x0 are about x0, δε(x0), such that f δε x0 mapped by f into points “near” f(x0). In general, the point-to-point mapping f: X Y is continuous on X if it is continuous at each point of X. Theorems 1.7.1 and 1.7.2 provide us with a set of necessary and sufficient conditions for the continuity of a point-to-point mapping at a specific point x0 X and at any arbitrary x X, respectively. Specifically, we start with Theorem 1.7.1. Theorem 1.7.1 (continuity in terms of convergent sequences). For metric spaces X and Y, the point-to-point mapping f of X into Y is continuous at x0 X if and only if xk x0 implies f(xk) f(x0) for every subsequence {xk} in X. Hence, f is a continuous mapping of X into Y if it “sends convergent sequences in X into convergent sequences in Y.” Next comes Theorem 1.7.2.

1.8 Operations on Sequences of Sets

Theorem 1.7.2 (continuity in terms of open (resp. closed) sets). For metric spaces X and Y, let f be a point-to-point mapping of X into Y. Then, (a) f is continuous if and only if f −1 (A) is open in X whenever set A is open in Y; and (b) f is continuous if and only if f −1 (A) is closed in X whenever A is closed in Y. Thus, f is continuous if it “pulls open (resp. closed) sets back to open (resp. closed) sets,” that is, the inverse images of open (resp. closed) sets are open (resp. closed). We next consider Theorem 1.7.3 which states that continuous mappings preserve compactness. That is, Theorem 1.7.3 For metric spaces X and Y, let f be a continuous point-to-point mapping from X into Y. If A is a compact subset of X, then so is its range f(A). Next, let X be a subset of Rn. A continuous point-to-point mapping g R n X is termed a retraction mapping on Rn if g(x) = x for all x X. Here, X is called a retraction of Rn. If X is contained within an arbitrary subset A of Rn, then g A X is a retraction of A onto X if g(x) = x for all x X.

1.8

Operations on Sequences of Sets

Let Ai ,i = 1 2,…, represent a sequence of sets in a metric space X. If {Ai} is such that Ai Ai + 1 , i = 1 2,…, then {Ai} is said to be a nondecreasing sequence (if Ai Ai + 1 , then {Ai} is said to be an expanding sequence). In addition, if {Ai} is such that Ai Ai + 1 , i = 1 2,…, then {Ai} is called a nonincreasing sequence (if Ai Ai + 1 ,i = 1 2,…, then {Ai} is termed a contracting sequence). A monotone sequence of sets is one which is either an expanding or contracting sequence. If the sequence Ai , i = 1 2, …, in X is nondecreasing or nonincreasing, then its limit exists and we have the following: lim Ai =

∞ i = 1 Ai

if Ai is nondecreasing ;

lim Ai =

∞ i = 1 Ai

if Ai is nonincreasing

∞

i

n

∞

In addition, for any sequence of sets Ai , i = 1 2, …, in X, sup Ai =

∞ i = 1 Ai ,

inf Ai =

∞ i = 1 Ai ;

with sup inf

∞ i = 1 Ai ∞ i = 1 Ai

= sup sup Ai , i = 1 2, … , = inf inf Ai , i = 1 2,… ; and

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1 Mathematical Foundations 1

sup inf

∞ i = 1 Ai ∞ i = 1 Ai

≤ inf supAi , i = 1 2, … , ≥ sup inf Ai , i = 1 2, …

Let Ai ,i = 1 2,…, again depict a sequence of sets in X. Then there are subsets Ei Ai of disjoint sets, with Ej Ek = ϕ for j such that ∞ i = 1 Ei

k,

∞ i = 1 Ai

=

We next consider the concepts of the limit superior and limit inferior of a sequence of sets Ai , i = 1 2,…, in a metric space X. To this end, the limit superior of a sequence {Ai} is defined as lim supAi =

i

∞

∞ i=1

k ≥ i Ak

= A1 A2

A2 A3

= x X x Ai for infinitely many i Hence, lim sup Ai is the set S of points such that, for every positive integer i, there exists a positive integer k ≥ i such that S Ai; thus S consists of those points that belong to Ai for an infinite number of i values. Looked at in another fashion, if x S, then x is in all of k ≥ i Ak . Hence, no matter how large of an i value is chosen, you can find a k ≥ i for which x is a member of Ai. Similarly, the limit inferior of a sequence {Ai} is lim inf Ai =

i

∞

∞ i=1

k ≥ i Ak

= A1 A2

A2 A3

= x X x Ai for all but finitely many i Thus, lim inf Ai is the set I of points such that, for some positive integer i, I Ai for all positive integers k ≥ i; hence, I consists of those points that belong to Ai for all except a finite number of i values. Stated alternatively, if x I, then x is an element of k ≥ i Ak so that x Ai for k ≥ i—x must be in I with only finitely many exceptions, that is, for x I, there is an index value such that x is in every Ai in the remaining portion of the limit.5 5 Alternative definitions of the limit superior and limit inferior of a sequence of sets are the following. Again, let {Ai} be a sequence of sets in a metric space X. Then lim supAi = x X lim inf d x, Ai = 0 ;

i

∞

i

∞

lim inf Ai = x X d x, Ai = 0 ,

i

∞

where d(x, Ai) is the distance from x to Ai.

1.9 Classes of Subsets of Ω

We note briefly that if Ai , i = 1 2,…, is any sequence of sets in a metric space X, then lim inf Ai lim sup Ai. A sequence of sets {Ai} is convergent (or a subset A of X is the limit of {Ai}) if lim sup Ai = lim inf Ai = lim Ai = A

i

∞

i

∞

i

∞

Here, A is termed the limit set. In this vein, any monotone sequence of sets Ai ,i = 1 2,…, is convergent.

1.9

Classes of Subsets of Ω

1.9.1

Topological Space

We previously defined a metric space (Ω, μ) as consisting of the space Ω and a metric μ defined on Ω. Let A denote the class of open sets in the metric space. Then A satisfies the following conditions: i. Ø, Ω A. ii. If A1 ,A2 , …, An A, then ni= 1 Ai A (the intersection of every finite class of sets in A is itself a set in A). iii. If Aα A for α I, then α I Aα A (the union of every arbitrary class of sets in A is itself a set in A). Armed with properties (i)–(iii), let us generalize a metric space to that of a topological space. That is, given a nonempty space Ω and a given class A of subsets of Ω consisting of the “open sets” in Ω, a class T of subsets of Ω is called a topology on Ω if (i)–(iii) hold. (Thus, the class of open sets A determines the topology in Ω.) Hence, a topological space consists of Ω and a topology T on Ω and is denoted Ω, . A subset A of a topological space (Ω, T) is said to be (everywhere) dense if its closure Ā equals Ω, . Hence, A is dense if and only if (a) A intersects every nonempty set; or (b) the only open set disjoint from A is Ø. 1.9.2 σ-Algebra of Sets and the Borel σ-Algebra

A ring R is a nonempty class of subsets that contains Ø and is closed under the operations of union, intersection, and difference. A σ-ring is a ring R that is closed under countable unions and intersections, that is, if Ai R, i = 1, 2, …, R. then A = i∞= 1 Ai R and i∞= 1 Ai = A− i∞= 1 A −Ai Next, we can define a σ-algebra as a class of sets F that contains Ω and is a σ-ring. More formally, for Ω a given space, a σ-algebra on Ω is a family F of subsets of Ω that satisfies the following conditions: i. Ω F. ii. If a set A

F, then its complement A

F, where A = Ω − A.

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1 Mathematical Foundations 1

iii. If Ai i ≥ 1 also in F.

∞ i = 1 Ai

F, then

F, that is, countable unions of sets in F are

∞ ∞ Note that since Ω F, we must have Ω = Ø F; with i = 1 Ai = i = 1 Ai , it follows that F is closed under countable intersections as well. Note also that if Ai i ≥ 1 F, then limi ∞ Ai F, limi ∞ sup Ai F, and limi ∞ inf Ai F. The pair (Ω, F) is called a measurable space and the sets in F are termed (F-) measurable sets. Given a family C of subsets of Ω, there exists a smallest σ-algebra σ(C) on Ω that contains C, is contained in every σ-algebra that contains C, and is unique. Here, σ(C) is termed the σ-algebra generated by C and is specified as

σ C =

Hj Hj a σ-algebra on Ω, C

Hj

(For instance, if C = {E}, E Ω, then σ (C) = {Ø, E,E , Ω}.) Now, if Ω = Rn and C is a family of open sets in Rn, then B n = σ C is called the Borel σ-algebra on Ω and an element B Bn is called a Borel set. Hence, the Borel σ-algebra on Ω is the smallest σ-algebra generated by all the open subsets of Rn; and the class of Borel sets B n in Rn is the σ-algebra generated by the open sets in Rn. In fact, the class of half-open intervals in Rn generates the σ-algebra B n of Borel sets in Rn. Borel sets also include all open and closed sets, all countable unions of closed sets, among others. Since σ-algebras will be of paramount importance in our subsequent analysis (especially in our review of the essentials of probability theory), let us consider Example 1.1. Example 1.1 To keep the analysis manageable, suppose Ω = 1,2,3,4 . Then possibly, F = {Ø,{1,2},{3,4},Ω}. Does F satisfy (i)–(iii) given earlier? If so, then F is a legitimate σ-algebra. Specifically, 1. As constructed, Ø F and Ω F. Hence, (i) holds. 2. Ø = Ω,Ω = Ø 1,2 = 3,4 , 3,4 = 1,2 . Clearly each of these subsets is a member of F and thus (ii) is valid. 3. Since F contains four disjoint subsets, partition the index set I = 1,2,3,4 into four disjoint subsets according to I1 = i A i = Ø I3 = i Ai = 3,4

Ø I2 = i Ai = 1,2 Ø Ø , and I4 = i Ai = Ω

Ø

Then ∞ i = 1 Ai

= =

i I

Ai

i I1

Ai

i I2

Ai

i I3

=Ø 1,2 3,4 Ω F and thus F is a σ-algebra on Ω. ■

Ai

i I4

Ai

1.10 Set and Measure Functions

1.10

Set and Measure Functions

1.10.1

Set Functions

We previously defined the concept of a point-to-point function or mapping as a rule f that associates with a point x from a nonempty set X a unique point y = f(x) in a nonempty set Y, where X, a set of points, was called the domain of the function. Now, let’s consider a real-valued function whose domain is a class of sets, that is, we have a function of sets rather than a function of points. In this regard, consider a function μ: C R∗, where C is a nonempty class of sets and R∗ denotes the set of extended real numbers. Thus μ is a rule that associates with each set E C a unique element μ(E), which is either a real number or ±∞. Some important types of set functions follow. First, a set function μ: C R∗ is said to be (finitely) additive if i. μ(Ø) = 0; and ii. for every finite collection E1, E2, …, En of disjoint sets (Ej Ek = Ø, j n μ Ei R∗ . such that ni= 1 Ei C, we have μ ni= 1 Ei = i=1

k) in C

Remember that the domain C of μ is a finitely additive class of sets {Ei, i = 1, …, n} n μ Ei is defined in R∗. (If Ω = R, C is the class of all finite intervals of and i=1 R, and if E is taken to be (a, b) or (a, b] or [a, b) or [a, b], then μ(E) = b − a.) It should be evident that a suitable domain of definition of an additive set function μ is a ring R since, if Ei R, i = 1,…, n, then ni= 1 Ei R. So if C is a ring, then the set function μ: R∗ is additive if and only if μ(Ø) = 0 and, if Ej and Ek are disjoint sets in C, then μ Ej Ek = μ Ej + μ Ek . In this regard, suppose μ: C R∗ is an additive set function defined as a ring C with sets Ej, Ek C. Then i. ii. iii. iv.

if Ej Ek and μ(Ej) is finite, then μ(Ek − Ej) = μ(Ek) − μ(Ej) ≥ 0; if Ej Ek and μ(Ej) is infinite, then μ(Ej) = μ(Ek); if Ej Ek and μ(Ek) is finite, then μ(Ej) is finite; and if μ(Ek) = +∞, then μ(Ej) −∞.

Next, a set function μ: C pletely additive) provided

R∗ is termed σ-additive (or countably or com-

i. the domain of μ is a σ-ring of sets C; ii. μ(Ø) = 0; and iii. for any disjoint sequence E1, E2, … of sets in C such that ∞ i = 1 Ei C, we have μ

∞ i = 1 Ei

∞

=

μ Ei

R∗

i=1

Here, the domain C of μ is a countably additive class of sets {Ei, i = 1, 2, …} and ∞ μ Ei R∗ is defined in the extended real numbers. Clearly, a σ-additive i=1

17

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1 Mathematical Foundations 1

set function is also (finitely) additive, though the converse is not generally true. R∗ implies However, if C is a finite class of sets, then the additivity of μ: C σ-additivity. A set function μ: R∗ is said to be σ-finite if, for each set E C, there is a sequence of sets Ei C, i = 1, 2, …, such that E = i∞= 1 Ei and μ(Ei) < +∞ for all i. As this definition reveals, additivity is not a property of σ-finite set functions. For instance, consider the Borel σ-algebra in Rn that is generated by the collection of all “cubes” of the form C = a1 ,b1 × a2 ,b2 × × an ,bn , with bi > ai ,i = 1, …, n. Then, μ C = ni= 1 bi − ai . Here, μ is σ-finite since R n = i∞= 1 − i, i n . A set function μ defined on C is nondecreasing if μ(Ek) ≥ μ(Ej) whenever Ej Ek; it is nonincreasing if μ(Ek) ≤ μ(Ej) when Ej Ek; and it is said to be monotone if it is either nondecreasing or nonincreasing. Now, if μ is additive and nondecreasing (resp. nonincreasing), then it is everywhere non-negative (resp. nonpositive). In fact, the reverse implication holds, that is, if μ is additive and everywhere non-negative (resp. nonpositive), then it is also nondecreasing (resp. nonincreasing). 1.10.2

Measure Functions

+

Let R denote the set of non-negative real numbers together with +∞, that is, ∗ R + = x R x ≥ 0 . A measure function on a σ-ring C is any non-negative R+. (For any subset A C we assume that σ-additive set function μ: C −∞ < μ A < +∞.) Note that since a measure function μ on C is non-negative, it must also be nondecreasing. A Borel measure is a measure function μ on the σ-algebra B of Borel subsets of a given topological space (Ω, T), that is, μ B

0, +∞

A couple of important characteristics of measure functions are as follows: i. If μ is a measure function on C and if {Ei, i = 1, 2, …} is any sequence of sets ∞ μ Ei . from C, then μ is countably subadditive or μ i∞= 1 Ei ≤ i=1 ii. If μ is a measure function on C and if {Ei, i = 1, 2, …} is any sequence of sets from C with μ i∞= 1 Ei < +∞, then μ limi ∞ Ei = limi ∞ μ Ei . We next examine the continuity of set functions. To this end, suppose R is a ring and the set function μ: R R∗ is additive with μ(A) > −∞ for all sets A R: i. μ is continuous from below at A if limi ∞ μ Ei = μ A for every monotone increasing sequence {Ei} in R that converges to A. ii. μ is continuous from above at A if limi ∞ μ Ei = μ A for every monotone decreasing sequence {Ei} in R for which μ(Ei) < +∞ for some i. iii. μ is continuous at A R if it is continuous at A from both above and below. Moreover, under the aforementioned assumptions, if

1.10 Set and Measure Functions

iv. μ is σ-additive (and thus additive) on R, then μ is continuous at A for all sets A R. Given two classes C and D of subsets of Ω, with C D, and set functions μ: R∗ respectively, τ is termed an extension of μ if, for all A R∗ and τ: D C, τ(A) = μ(A); and μ is called a restriction of τ to C. In later chapters, we shall be concerned with issues pertaining to the convergence of sequences of random variables. To adequately address these issues, we need to be able to define measures on countable unions and intersections of “measurable sets.” To accomplish this task, we need to assume that the collection of measurable sets is a σ-algebra F, that is, F contains Ω and is a σ-ring. This requirement enables us to confine our analysis, for the most part, to “measure spaces,” where a measure space is a triple (Ω, F, μ) consisting of a space Ω, a σ-algebra F on Ω (a collection subsets of Ω), and μ: F R+ is a measure on F.

C

1.10.3

Outer Measure Functions

Suppose C is the class of all subsets of a space Ω. Then μ∗: C measure function on Ω if

R+ is an outer

i. μ∗(Ø) = 0; ii. μ∗ is nondecreasing (i.e., for subsets Ej Ek, μ(Ej) ≤ μ(Ek)); and iii. μ∗ is countably subadditive—i.e., for any sequence {Ei, i = 1, 2, …} of subsets of Ω, μ∗

∞ i = 1 Ei

≤

∞

μ∗ Ei

14

i=1

In sum, μ∗ is said to be non-negative, monotone, and countably subadditive. We note that every measure on the class of all subsets of Ω is an outer measure on Ω; and, in defining an outer measure on Ω, no “additivity” requirement was in effect. Given that μ∗ is an outer measure on Ω, a subset E is said to be measurable with respect to μ∗, or simply μ∗-measurable, if for every set A Ω, μ ∗ A = μ∗ A E + μ∗ A E

15 ∗

(given that A = A E A E ). Thus, a subset E of Ω is μ -measurable if it partitions a set A Ω into two subsets, A E and A E , on which μ∗ is additive. As this definition reveals, a set E is not innately measurable—its measurability depends upon the outer measure employed. That is, to define the measurability of a set E, we start with an arbitrary set A and we examine the effect of E on the outer measure of A, μ∗(A). If E is measurable, then it is sufficiently “well-behaved” in that it does not partition A in a way that compromises the additivity of μ∗, that is, if we partition A into A E and A E , then the outer measures of A E and A E add up correctly to μ∗ (A).

19

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1 Mathematical Foundations 1

Since μ∗ is countably subadditive, we have, from (1.4), μ∗ A = Ω. Hence, E is μ∗-measurable if μ∗ A E + μ∗ A E for all sets A, E and only if μ∗ A ≥ μ∗ A E + μ∗ A E

16

for every set A Ω. Since this inequality holds for any set A for which μ∗ (A) = +∞, it follows that a necessary and sufficient condition for E to be μ∗-measurable is that μ∗(A) < +∞ for every A Ω. Some important properties of outer measures are the following: i. ii. iii. iv.

If E Ω is μ∗-measurable, then E is also μ∗-measurable. If μ∗(E) = 0, then E Ω is measurable. Any finite union of μ∗-measurable sets in Ω is μ∗-measurable. If {Ei, i = 1, 2, …} is a sequence of disjoint μ∗-measurable sets in Ω and if ∞ ∗ μ A Ej . G = j∞= 1 Ej , then for any set A Ω, μ∗ A G = j=1

v. Any countable union of μ∗-measurable sets in Ω is μ∗-measurable. vi. Any countable union of disjoint μ∗-measurable sets in Ω is μ∗-measurable. vii. If {Ei, i = 1, 2, …} is a sequence of disjoint μ∗-measurable sets in Ω and Ω, μ∗ A Gn = if, for each n, Gn = nj= 1 Ej , then, for each set A n j=1

μ∗ A Ej .

Why are outer measures important? Simply because they are useful for constructing measure functions. That is, given that the outer measure μ∗ has as its domain the class of all subsets of the space Ω, a restriction of μ∗ to a “smaller” domain always generates a measure function. In this regard, suppose μ∗ is an outer measure function on Ω and let be the class of μ∗-measurable sets. Then is a completely additive class (a σ-algebra) and the restriction of μ∗ to is a measure function μ. An outer measure μ∗ is said to be regular if, for every subset A Ω, there is a ∗ μ -measurable set E A such that μ∗(E) = μ∗(A). (Here, E is said to be a measurable cover for A.) Thus, an outer measure is regular if it effectuates measurable sets in a manner that guarantees that every set A Ω has a measurable cover E. Key properties of regular outer measures are the following: i. If μ∗ is a regular outer measure on Ω and {Ei, i = 1, 2, …} is an increasing sequence of sets, then μ∗ limi ∞ Ei = limi ∞ μ∗ Ei . ii. If μ∗ is a regular outer measure on Ω for which μ∗ (Ω) < +∞, then a subset Ω is measurable if and only if μ∗ Ω = μ∗ Ω E + μ∗ Ω E = E ∗ μ E + μ∗ E . (This result follows from Equation (1.5) with A = Ω, since (1.5) must hold for any set A.) Next, let Ω be a metric space. An outer measure μ∗ on Ω is a metric outer measure if μ∗(Ø) = 0; μ∗ is nondecreasing and countably subadditive; and μ∗

1.10 Set and Measure Functions

is additive on separated sets (i.e., for subsets E and F in Ω with d(E, F) > 0, μ∗(E F) = μ∗E + μ∗(F)).6 We note briefly the following: i. If μ∗ is a metric outer measure, then any closed set is measurable. ii. If μ∗ is a metric outer measure, then every Borel set is measurable (since the class of μ∗-measurable sets contains the open sets, and thus contains B, the class of Borel sets). 1.10.4

Complete Measure Functions

Given a measure function μ: C R+, the class C of subsets of Ω is complete with respect to μ if E F, F C, and μ(F) = 0 implies E C. Now, if μ: C R+ is such that C is complete with respect to μ, then μ is said to be complete. Hence, μ is complete if its domain contains all subsets of sets of measure zero, that is, every subset of a set of measure zero is measurable. For a measure space (Ω, F, μ), the completion of F, denoted Fc , with respect to a measure μ on F involves all subsets A Ω such that there exist sets E, F F, with E A F, and μ(F − E) = 0. The completion of μ, μc, is defined on Fc as μc(A) = μc (E) = μc(F); it is the unique extension of μ to Fc . For A Fc , μc A = inf μ F F F, A F = sup μ E E F,E A . The complete measure space (Ω, Fc , μc) is thus the completion of (Ω, F, μ). In fact, (Ω, Fc , μc) is the smallest complete measure space that contains (Ω, F, μ). If a measure μ is obtained by restricting an outer measure μ∗ to , the class of sets of Ω that are μ∗-measurable, then μ is a complete measure. In fact, any measure generated by an outer measure is complete. 1.10.5

Lebesgue Measure

In what follows, our discussion will focus in large part on a class M of open sets (containing Ø) in Ω = R. This will then facilitate our development of the Lebesgue integral. Let us express the length of a bounded interval I (which may be open, closed, or half-open) with endpoints a and b, a < b, as l I = b − a. Our objective herein is to extend this “length” concept to arbitrary subsets of R, for example, for a subset E R, the notion of the “length of E” is simply its measure μ(E). In particular, we need to explore the concept of Lebesgue measure of a set E, μ(E), and specify the family of Lebesgue measurable sets. Our starting point is the concept of Lebesgue outer measure.

6 For sets E, F R n , the distance between sets E, F is d E, F = inf If E F Ø ,d E, F = 0.

x −y x E, y F .

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1 Mathematical Foundations 1

R, the Lebesgue outer measure μ∗ (E) is defined as

For each subset E n

μ∗ E = inf

l Ii

Ii is a sequence of open intervals with E

n i = 1 Ii

i=1

What is the significance of this expression? Suppose E can be covered by multiple sets of open intervals, where the union of each particular set of open intervals contains E. Since the total length of any set of intervals can overestimate the measure of E (it may contain points not in E), we need to take the greatest lower bound of the lengths of the interval sets in order to isolate the covering set whose length fits E as closely as possible and whose constituent intervals do not overlap. Given the discussion on outer measures in Section 1.10.3, it follows that the key properties of Lebesgue outer measures are the following: i. For every set E R,0 ≤ μ∗ E ≤ +∞ . ii. μ∗ is nondecreasing. iii. μ∗ is countably subadditive, that is, for any sequence Ei , i = 1 2,… of subsets of R, μ∗

∞ i = 1 Ei

n

≤

μ∗ Ei

i=1 ∗

iv. μ generalizes or extends the concept of “length” in that μ∗ I = l I . How does the concept of Lebesgue outer measure translate to the notion of Lebesgue measure itself? In order to transition from μ∗ (E) to μ(E), we need an additional condition on E. Specifically, a set E R is Lebesgue measurable if for every set A R, μ∗ A = μ∗ A E + μ∗ A E

15

This requirement is not new (see the discussion underlying Equation (1.5) of Section 1.10.3). As explained therein, if for every A the partition of A induced by E (the sets A E and A E ) has outer measures that correctly add up to the outer measure of A itself, then set E is “well-behaved” in that E does not adversely impact or distort the outer measure of A when E is used to partition A. The upshot of all this is that, under (1.5), μ∗ (E) yields μ(E). That is, if E is Lebesgue measurable, then the Lebesgue measure of E is defined to be its outer measure μ∗ (E) and simply written as μ(E). As far as the properties of Lebesgue measure μ(E) are concerned, they mirror those of μ∗ (E) (see properties (i)–(iv)), but with one key exception—property (iii) involving countable subadditivity is replaced by countable additivity: if Ei ,i = 1 2, … is a sequence of disjoint subsets of R, then iii μ

∞ i = 1 Ei

∞

= i=1

μ Ei

1.10 Set and Measure Functions

How should the family of Lebesgue measurable sets (denoted M) be defined? Clearly, we need to specify the largest family M of subsets of R for which μ: M R+ and properties (i), (ii), (iii) , and (iv) hold. Hence, the family of Lebesgue measurable sets M encompasses the collection of all open intervals as well as all finite unions of intervals on the real line. Then for E M, μ(E), the Lebesgue measure of E, is the total length of E when E is decomposed into the union of a finite number of disjoint intervals. We note in passing that Ø and R are Lebesgue measurable with μ (Ø) = 0 and μ R = +∞, respectively; open and closed intervals of real numbers are Lebesgue measurable; every open set and every closed set is Lebesgue measurable; every Borel set (which includes countable sets, open and closed intervals, all open sets and all closed sets) is Lebesgue measurable; any countable set of real numbers has Lebesgue measure equal to zero; if E is Lebesgue measurable, then so is E ; and if Ei , i = 1 2, … is a sequence of Lebesgue measurable sets, then ∞ ∞ i = 1 Ei and i = 1 Ei are Lebesgue measurable sets. a, b ,a < x1 < x2 < x3 < b. Example 1.2 Let E = a, b M with x1 , x2 ,x3 Consider the set A = E x1 , x2 , x3 . For the measure function μ: M R+, μ A = μ a, x1

x1 , x2

x2 , x3

x3 , b

= μ a, x1 + μ x1 , x2 + μ x2 , x3 + μ x3 ,b = x1 − a + x2 − x1 + x3 −x2 + b − x3 = b −a = μ E

1.10.6

■

Measurable Functions

Let (X, D) and (Y, G) be measurable spaces, where D is a σ-algebra on X and is a σ-algebra on Y, respectively. A measurable function is a mapping f: X Y such that f −1 (G) D for every set G G. Clearly, the measurability of f depends upon D and G and not on the particular measures defined on these σ-algebras. As this definition indicates, measurable functions are defined in terms of inverse images of sets. (Thus, measurable functions are mappings that occur between measurable spaces in much the same way that continuous functions are mappings between topological spaces.) To elaborate on this notion, if f: Ω G and A , let f − 1 A = x Ω f x A and call f −1 A the inverse image of set A under rule f. (Note: f − 1 A contains all of the points in the domain Ω of f mapped by f into A; it does not denote the inverse function of f.) Key properties of the inverse image of A are the following: i. f −1 A = f −1 A for all A G. ii. If A, B G, then f −1 A B = f −1 A f −1 B . G, then f − 1 k∞= 1 Ak = k∞= 1 f −1 Ak . iii. If Ak

23

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1 Mathematical Foundations 1

In addition, if C is a collection of subsets of G, let f −1 C = f −1 A A C . In this regard, if f: Ω G and C is a collection of subsets of G, then (a) for C a σ-algebra on G, f −1 C is a σ-algebra on Ω; and (b) f −1 F C = F f − 1 C , (Ω, F) a measurable space. A measurable mapping g X Y on a measure space (Ω, F, μ) is measure preserving if μ g − 1 A = μ A for all measurable sets A. We previously termed the σ-algebra generated by intervals (open, closed, halfopen) in R the Borel σ-algebra B. In this regard, if (Ω, F) is a measurable space, the mapping f: Ω R is F-measurable if f − 1 B F for every Borel set B R. In fact, the collection of sets f − 1 B , where B is contained within the Borel subsets of R, is a σ-algebra on Ω. In addition, if the collection C of Borel subsets of R generates the Borel σ-algebra, then f: Ω R is F-measurable if and only if f −1 C F. Equivalently, the mapping f: Ω R is F-measurable if and only if the set x Ω f x ≤ a is measurable (i.e., it is a member of F) for all a R. (Note: “≤” can be replaced by “.”) The indicator or characteristic function of a set A Ω is defined as 1, x A;

χA x =

17

0, x A

If A, B are two subsets of Ω, then χA

B

= min χ A , χ B = χ A χ B ;

χA

B

= max χ A , χ B = χ A + χ B − χ A χ B ; and

χ A = 1 −χ A Moreover, if Ai , i = 1,…, n, and Bj , j = 1, …, m, are subsets of Ω and X = n

xχ i = 1 i Ai

m

yχ , j = 1 j Bj

and Y = n

then

m

xi yj χ Ai

X Y=

Bj

i=1 j=1

In addition, if Ai ni= 1 and Bj partition of Ω, and thus n

m j=1

are partitions of Ω, then Ai Bj

all i, j

is also a

m

xi + yj χ Ai

X +Y =

Bj

i=1 j=1

If F is a σ-algebra on Ω, then Ø and Ω are members of F. In addition, with set A F, it follows that A F. Hence, Fχ A = Ø , A, A ,Ω , and thus χ A is F-measurable if and only if A F. Note also that if X and Y are F-measurable functions on Ω, then X + Y , X −Y ,X Y , and cX(c a real scalar) are all F-measurable. Suppose Y x 0 for all x Ω. Then X/Y is also F-measurable.

1.10 Set and Measure Functions

Suppose f Ω R∗ is a measurable function with A = x f x ≥ 0 and B = x f x ≤ 0 . If f + = f χ A and f − = −f χ B , then the positive part of f is defined as f

+

= max f x ,0 =

f x , f x ≥ 0; 0,f x < 0;

and the negative part of f is defined as f − = max − f x ,0 =

−f x , f x ≤ 0; 0, f x ≥ 0,

where f + and f − are both positive functions on Ω. With F a σ-algebra on Ω and f is measurable, sets G = f − 1 x x ≥ 0 and H = f −1 x x ≤ 0 are in the σ-algebra generated by F (denoted Ff ). Hence, f + ,f − and f are all Ff -measurable. The upshot of this discussion is that an arbitrary measurable function f can be written in a canonical way as the difference between two positive measurable functions as f = f + −f − . In addition, f = f + + f − . A function Φ: Ω R defined on a measurable space (Ω, F) is a simple function if there are disjoint measurable sets A1, …, An and real scalars c1, …, cn such that n

ci χ Ai

Φ=

18

i=1

Clearly, Φ takes on finitely many, finite values ci ,i = 1,…,n. Since simple functions are measurable, any measurable function may be approximated by simple functions. In fact, for f: Ω R+ a non-negative measurable function, there is a monotone increasing sequence {Φi} of simple functions that converges pointwise to f. Given a measure space (Ω, F, μ), if Ω = ni= 1 Ai and the sets Ai are disjoint, then these sets are said to form a (finite) dissection of Ω. They are said to form an F-dissection if Ai F,i = 1,…, n. A function f: Ω R is termed F-simple n c χ , where the Ai’s, i = 1, …,n, form an if it can be expressed as f x = i = 1 i Ai F-dissection of Ω. Thus, f(x) takes on a constant value ci on the set Ai, given that the Ai’s are disjoint subsets of F. A sequence of measurable functions { fn} from a measure space (Ω, F, μ) to R∗ converges pointwise to a function f: Ω R∗ if limn ∞ fn x = f x for every x Ω. Moreover, f itself is measurable. A sequence {fn} converges pointwise a.e.7 to f if it converges pointwise to f except on a set M of measure zero. 7 A set of measure zero is a measurable set M such that μ(M) = 0. A property or condition that holds for all x Ω−M, where M is a set of measure zero, is said to hold almost everywhere (abbreviated “a.e.”) or “except on a set of measure zero.” Note: a subset of a set of measure zero need not be measurable; but if it is measurable, then it must have measure zero.

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1 Mathematical Foundations 1

If (Ω, F, μ) is a complete measure space and {fn} converges pointwise a.e. to f, then f is measurable. We note briefly that if the measurable space is (Rn, B n ), a B n -measurable function is termed a Borel-measurable function. 1.10.7

Lebesgue Measurable Functions

Suppose (X, D) and (Y, G) are measurable spaces, with X and Y equipped with the σ-algebras D and G, respectively. Then, as indicated in Section 1.10.6, the function f X Y is measurable if the anti-image of E under f is in D for every E G, that is, f −1 E = x X f x E D for all E G. Let us now get a bit more specific. Suppose (R, L) and (R, B) are measurable spaces, with L the σ-algebra of Lebesgue measurable sets and B the Borel σ-algebra on R. (Remember that B is the smallest σ-algebra containing all the open sets.) The function f R R is Lebesgue measurable if the anti-image of B under f is a Lebesgue measurable subset of R for every Borel subset B of R, that is, f −1 B = x R f x B L for all B B. (Clearly, the domain and range of f involve different σ-algebras defined on the same set R.) In very basic terms, for a bounded interval I, a function f I R is Lebesgue measurable if, for every open set B R, the anti-image f −1 B is measurable in I. An important alternative way of specifying a function that is Lebesgue measurable is the following. If (X, L) is a measurable space, then f X R is Lebesgue measurable if and only if f −1 a, +∞ = x X f x > a L for all a R (Figure 1.3). (Note: equivalent statements involve “>” being replaced by “ ≥ ” or “ 0 there exists an Nε 2 > 0 such that for all m, n > Nε 2 , fm x −fn x < ε . Thus, B is complete in that, for each x B and every Cauchy sequence {fn} defined on B, there exists a well-defined continuous limit function f x = limn ∞ fn x so that {fn(x)} converges pointwise to f(x). In fact, any normed space with the property that every Cauchy sequence defined on it is convergent is complete. The preceding discussion enables us to conclude that B constitutes an important type of function space, namely, a Banach space—a complete normed linear (metric) space. We next turn to another type of function space that is also a Banach space. 1.11.2

Space of Bounded Continuous Real-Valued Functions

A key property that the elements of a function space can possess is continuity. To explore this characteristic, let’s assume at the outset that the set of all realvalued functions is defined on a metric space X. Furthermore, given B above (as defined earlier, B is the subset of X containing all bounded real-valued functions), let C(X) B denote the set of all bounded continuous functions defined on X. It should thus be evident that a. if f, g are continuous real-valued functions defined on X, then, pointwise, f + g and αf (α a real scalar) are also continuous;

1.11 Normed Linear Spaces

b. C(X) is a linear subspace of the linear (metric) space B; and c. C(X) is a closed8 subset of the linear (metric) space B. But remember that B is a Banach space and, since a closed linear subspace of a Banach space is also a Banach space, it follows that C(X), the set of all bounded continuous real-valued functions defined on a metric space X with norm (1.10) is a Banach space. 1.11.3

Some Classical Banach Spaces

1. Let Rn denote the set of all vectors or ordered n-tuples x = x1 ,x2 ,…,xn of real numbers. For elements x = x1 , x2 ,…, xn and y = y1 , y2 ,…,yn in Rn, let us define, coordinatewise, addition and scalar multiplication as x + y = x1 + y1 , x2 + y2 , …, xn + yn , αx = αx1 , αx2 , …, αxn , α a real scalar, respectively. In addition,, the zero (or null) element 0 = (0, 0, …, 0) (also an n-tuple) and − x = −x1 , −x2 , …, − xn are elements in Rn. For any element x Rn, let us define the norm of x, x , by 1 2

n

x =

xi

2

1 11

i=1

(See Section 1.4 for a discussion of the properties of this norm.) Given (1.11), it is evident that Rn can be characterized as a normed linear (metric) space (also called n-dimensional Euclidean space—since (1.11) is the Euclidean norm). Moreover, it is complete with respect to the metric (1.2) and thus amounts to a complete normed linear space or Banach space. p 2. Let L μ denote the set of all measurable functions f defined on a measure space (Ω, F, μ) and having the property that |f(x)|p is integrable, with p-norm f

p

=

f x

p

1 p

dμ x

, 1 ≤ p < +∞

1 12

For p 1, +∞ , L p μ is complete with respect to (1.12) and thus constitutes a Banach space.

8 Suppose f B with f C X (the closure of C(X)). Let d be the metric on X, with ε > 0 given. Since f is in C X , there exists a function f0 in C(X) such that f −f0 < ε implies f x −f x0 < ε for each x X, where ε is proportional to ε. With f0 continuous at x0, there exists a δε > 0 such that d x,x0 = x −x0 < δε implies f x0 − f0 x0 < ε . Since x −x0 < δε implies that f x −f x0 < ε, we see that f is continuous at x0 (x0 arbitrary). Hence, f C X = C X so that C(X) must be closed (Simmons, 1963, p. 83).

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1 Mathematical Foundations 1

3. Let H be an n-dimensional linear (vector) space with the inner product norm defined by 1

x = x, x 2 ,

1 13

where (x, x) is the inner product9 defined by n

xi 2 , x H

x,x = i=1

(A linear space equipped with an inner product is called an inner product space.) Clearly, H is a normed linear space and is complete with respect to the norm given by (1.13); it will be called a Hilbert space—a complete normed inner product space. Although H is always a Banach space whose norm is determined by an inner product , (i.e., f = f ,f 1 2 for all f in the space), the converse does not generally hold. So what is the essential difference between a Banach space and a Hilbert space? The difference is in the source of the norm, that is, for a Banach space, the norm is defined B 0, +∞ for all points x, y (and scalar c) satisfying the directly as properties outlined earlier in footnote 2; and for a Hilbert space, the norm is defined by an inner product (Equation (1.13)). The inner product is not defined on a Banach space. The ordered n-tuples (vectors) x, y R n are said to be orthogonal if n x y = 0. Elements x and y within a Hilbert space (H) are orthogox, y = i=1 i i nal if (x, y) = 0 and orthonormal if, in addition, x = y = 1. An orthonormal set in H is a nonempty subset of H that consists of mutually orthogonal unit vectors ei ,i = 1 2, …. (A unit vector ei has a “1” as its ith component and “0’s” elsewhere.) That is, an orthonormal set is a non-empty subset ei ,i = 1 2,… of H with the properties (1) ei , ej = 0,i j; and (2) ei = 1 for all i. An orthonormal sequence ei , i = 1 2, … in H is complete if the only member of H that is orthogonal to every ei is the null vector 0 (which contains all zero components). (Stated alternatively, we cannot find a vector e such that {{ei}, e} is an orthonormal set that properly contains ei , i = 1 2,… .) Suppose {e1, e2, …, en} is a finite orthonormal set in H. If x is any vector in H, then 1.

n i=1

x,ei

2

≤ x 2;

9 The inner product (x, x) satisfies the following conditions: a. b. c. d.

For x H, (x, x) ≥ 0 and (x, x) = 0 if and only if x = 0 (positive semidefiniteness). For x, y H, x, y = y, x (symmetry). For x, y H, with a, b real scalars, ax1 + bx2 , y = ax1 , y + bx2 , y (linear in its first argument). For x, y H, x, y ≤ x y (Cauchy–Schwarz inequality).

1.12 Integration n

2. x = 3.

i=1 n

x−

x, ei ei ; and

i=1

x, ei ei , ej = 0 for each j.

An orthonormal basis for a Hilbert space (H) is a basis10 consisting of nonzero orthonormal vectors. Such vectors are linearly independent and span H in the sense that every element in H can be written as a linear combination of the basis vectors. In fact, every Hilbert space contains a maximal orthonormal set that serves as a basis. Suppose in (1.12), we set p = 2. Then the class of real-valued square-integrable functions L μ = f 2

2

1 2

2

=

f dμ x

< +∞

1 12 1

is a Hilbert space. In addition, if (Ω, F, μ) is a measure space and the functions f, g L2 (μ), then the inner product of f and g is f , g = fgdμ,

1 14

where f g ≤ f g . The space L2(a, b), the collection of Borel measurable real-valued square b integrable functions f on (a, b) (i.e., a f t 2 dt < +∞ , is a Hilbert space. For this space, the inner product is f , g = and

metric

f −g =

b a

are,

f t −g t

respectively, 2

dt

1 2

b a f

t g t dt, and the associated norm f

2

=

b a

f t

2

dt

1 2

and d f ,g =

. (Here, the functions f, g are considered equal

if they differ on (a, b) only on a set of measure zero.)

1.12

Integration

Our approach in this section is to first define the integral of a non-negative simple function. We then define the integral of a non-negative measurable 10 To review, a vector x R n is a linear combination of the vectors xj R n ,j = 1,…,m, if there m exists scalars λj , j = 1,…, m, such that x = λ x . A set of vectors is linearly independent if the j=1 j j trivial combination 0x1 + + 0xn is the only linear combination of the xj which equals the null vector. (The set of vectors xj ,j = 1,…,m is said to be linearly dependent if there exists scalars λj , j = 1, …, m, not all zero such that

m

λx j=1 j j

= 0.) The vectors xj , j = 1, …, m, span Rn if every element

of Rn can be written as a linear combination of the xj’s. Hence, the xj’s constitute a spanning set for Rn. A basis for Rn is a linearly independent set of vectors from Rn which spans Rn. Thus, every vector in Rn can be expressed as a linear combination of the basis vectors.

31

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function via an approximation by simple functions. Next comes the definition of the integral of a measurable function, followed by the specification of the integral of a measurable function on a measurable set. In what follows, (Ω, F, μ) is taken to be a measure space. However, if Ω = R admits the Borel σ-algebra, F is the σ-algebra of Lebesgue measurable sets in R, and the measure μ F 0, +∞ is given by μ E = μ∗ E , E F, then the integrals defined below are also Lebesgue integrals. (Readers not familiar with the Lebesgue integral are encouraged to read Appendix B to this chapter along with Taylor (1973) before tackling this section.) 1.12.1

Integral of a Non-negative Simple Function

Recall (Section 1.10.5) that a non-negative simple function has the form n

Ø x =

ci χ Ei x , ci ≥ 0, i = 1, …, n,

1 15

i=1

where the indicator function χ Ei is defined as χ Ei x =

1,x Ei ; 0, x Ei

The integral of a non-negative simple function with respect to μ is defined in terms of the integral operator “ ” as n Ω

Ø dμ =

ci μ E i ,

1 16

i=1

where Ei = x Ø x = ci ,

Ω

χ Ei dμ =

dμ = μ Ei < +∞ and the sum on the Ei

right-hand side of (1.16) is well defined since each of its terms is non-negative. (It is important to note that since the specification of a simple function in terms of indicator functions is not unique, this definition of the integral is independent of the actual specification used.) For the simple function given in Equation (1.15), suppose set A F is measurable. Then the integral of a non-negative simple function over a set A is defined as n

Ø dμ = A

ci μ Ei A

1 16 1

i=1

As far as the essential properties of the integral operator “ ” are concerned, it is linear as well as order preserving on the class of non-negative simple n cχ functions. That is, given two non-negative simple functions Ø = i = 1 i Ei and ψ =

m

dχ , j = 1 j Fj

the simple function

1.12 Integration

Ø +ψ =

Ω

n

m

i=1

j=1

Ø + ψ dμ =

Ω

ci + dj χ Ei Ø dμ +

Ω

Fj ,

and thus

ψdμ; “ ” is linear

1 17

while, for Ø ≥ ψ,

Ω

Ø dμ ≥

Ω

ψdμ

“ ” is order preserving or monotonic

1 18

1.12.2 Integral of a Non-negative Measurable Function Using Simple Functions

Suppose the non-negative function f: Ω R+ is measurable. It was noted in Section 1.10.5 that there exists a monotone increasing sequence {fn} of simple functions that converge pointwise to f. Given that

Ω

fn dμ is defined

for all n, and the said sequence is monotonic, it follows that the limit of Ω

fn dμ is an element of R+. Hence, we may define the operation of integration

for non-negative measurable functions as

Ω

Since

Ω

f dμ = lim

∞ Ω

n

1 19

fn dμ

fdμ may be finite or infinite in R+, we may conclude that a non-negative

measurable function f is integrable with respect to a measure μ if the limit in (1.19) is finite. In addition, if f ≥ 0 is measurable, then integration for nonnegative measurable functions over a set A is defined as Ø dμ < +∞

f dμ = sup Ø

A

1 19 1

A

where the supremum is taken over all simple functions Ø with 0 ≤ Ø ≤ f. 1.12.3

Integral of a Measurable Function

Suppose the function f: Ω R+ is measurable. Then, as indicated earlier in Section 1.10.5, so are f + and f − . If f + and f − are integrable with respect to μ, then f = f + −f − itself is integrable with respect to μ, and thus

Ω

fdμ =

Ω

f + dμ−

Ω

f − dμ

1 20

33

34

1 Mathematical Foundations 1

so that this expression defines integration for the class of integrable measurable functions. Also, for set A F, f is integrable over a set A if f dμ =

f + dμ +

A

1.12.4

A

f − dμ < +∞

1 20 1

A

Integral of a Measurable Function on a Measurable Set

F. Suppose f χ A dμ is defined (e.g., either fχ A is non-negative and

Let set A

measurable, or fχ A is measurable and integrable). Then fdμ = A

A

f χ A dμ

1 21

Thus, f is integrable over a set A if fχ A is integrable. (Note that if A R∗ is integrable over A with

μ A = 0, then f: Ω

F and

fdμ = 0 ) A

∗

For a measure space (Ω, F, μ) and f: Ω R an integrable function with respect to μ over Ω, some additional properties of the integral operator “ ” are the following: i. For A and B disjoint sets in F, fdμ =

fdμ +

A B

fdμ

A

B

ii. |f| is integrable and iii. For a constant c iv. If f ≥ 0, then ∗

Ω

Ω

fdμ =

Ω

f dμ.

R, cf is integrable and

Ω

cfdμ = c

Ω

fdμ.

fdμ ≥ 0; but if f ≥ 0 and fdμ = 0, then f = 0 a.e.

v. If g: Ω R is integrable with respect to μ over Ω, then if f = g a.e., it follows that Ω

fdμ =

vi. If sets A, B

Ω

gdμ

F with A

fdμ ≤

B and f ≥ 0, then A

fdμ B

vii. Let μ be the counting measure on Ω = {1, 2, 3, …} and define the measurable function f: Ω R as f j = aj , j Ω. Then 11

11 Let (Ω, F, v) be a measure space. The counting measure v on Ω is defined as v(A) = number of elements in A F. This measure is finite if Ω is a finite set; it is σ-finite if Ω is countable.

1.12 Integration ∞ Ω

f j dμ j =

aj j=1

This integral is well-defined if f ≥ 0, or if the sum on the right-hand con∞ ∞ a is convergent, then a is termed verges absolutely. (If j=1 j j=1 j absolutely convergent.) In either instance we say that f is integrable with respect to μ. F if and only if |f | is viii. A measurable function f is integrable on A integrable on A. ix. If f is integrable on set A F, if g is measurable, and if g ≤ f a.e. on A, then g gdμ ≤

is integrable on A and A

fdμ A

x. If f is any function and for set A

F, if μ A = 0, then

fdμ = 0. A

1.12.5

Convergence of Sequences of Functions

Let (Ω, F, μ) be a measure space. A sequence of functions { fn}, where fn Ω R +, converges pointwise to a function f Ω R + if limn ∞ fn x = f x for every x Ω. Here f is termed a limiting function. The sequence { fn} converges pointwise a.e. to f if it converges pointwise to f on Ω − A, where A F is a set of measure zero. In this regard, let { fn} be a sequence of functions that converges pointwise to a limiting function f. When can we legitimately conclude that Ω fn dμ converges to Ω f dμ? Two conditions that guarantee the convergence of the integrals Ω fn dμ are (1) the monotone convergence of the sequence { fn}; and (2) a uniform bound on { fn} by an integrable function. To set the stage for a discussion of the first condition, let us define a sequence of functions { fn}, where fn Ω R +, as monotone increasing if f1 x ≤ ≤ fn x ≤ … for every x Ω. We then have Theorem 1.12.1. Theorem 1.12.1 (Lebesgue) Monotone Convergence Theorem (MCT) Let { fn} be a monotone increasing sequence of non-negative measurable func0, +∞ on a measure space (Ω, F, μ) and let f Ω 0, +∞ be the tions fn Ω pointwise limit of { fn} or f x = limn ∞ fn x . Then lim

n

∞ Ω

fn dμ =

Ω

f dμ

(Note that if f is integrable on Ω limn ∞ Ω fn dμ < +∞ , this theorem posits the convergence of the integrals Ω fn dμ to Ω fdμ. If f is not integrable on Ω, then

35

36

1 Mathematical Foundations 1

possibly fn is integrable for all n and of the MCT is Corollary 1.12.1.

Ω fn dμ

+∞ as n

+∞.) A consequence

Corollary 1.12.1 (corollary to the MCT). Let { fn}, f Ω sequence of non-negative measurable functions and set f =

0, +∞ , be a f . Then n=1 n ∞

∞ Ω

fdμ =

n=1 Ω

fn dμ

A generalization of the MCT is provided by Lemma 1.12.1. Lemma 1.12.1 Fatou’s lemma Let {fn} be a sequence of non-negative measurable functions fn Ω measure space (Ω, F, μ). Then lim inf fn dμ ≤ lim inf

n

Ω

∞

n

∞

Ω

0, ∞ on a

fn dμ

As this lemma indicates, the limit of the integrals on the right-hand side of this inequality is always at least as large as the integral of the limit function on the left-hand side. The second aforementioned condition is incorporated in Theorem 1.12.2. Theorem 1.12.2 (Lebesgue) Dominated Convergence Theorem (DCT) Let {fn} be a sequence of integrable functions, where fn: Ω R∗, on a measure space (Ω, F, μ) that converge pointwise to a limit function f: Ω R∗. If there is an 0, ∞ such that fn x ≤ g x for all x Ω (and indeintegrable function g Ω pendent of n), then f is integrable and lim

∞ Ω

n

fn dμ =

Ω

fdμ

We close this section with an additional useful Theorem 1.12.3. Theorem 1.12.3 Let {An} be a sequence of disjoint measurable sets with A = n∞= 1 An and let f be a non-negative measurable function that is integrable on each set An. Then f is integrable on A if and only if so that ∞

fdμ = A

fdμ n = 1 An

∞ n=1

fdμ < +∞, An

37

2 Mathematical Foundations 2 Probability, Random Variables, and Convergence of Random Variables

2.1

Probability Spaces

Let us define a random experiment as a class of occurrences that can happen repeatedly, for an unlimited number of times, under essentially unchanged conditions. A random phenomenon is a happening such that, on any given trial of a random experiment, the outcome is unpredictable. The outcome of a random experiment is called an event. For instance, consider the following game of chance: we roll a fair pair of six-sided dice. (Clearly, this is a process that has random outcomes.) Consider the event “the sum of the faces showing is nine.” Let’s denote this event as A. What is the probability that event A occurs? To answer this question, we need to examine the set of all possible outcomes that can obtain on any roll of a pair of dice. This set is called the sample space (Ω) and has the form Ω = ωij = i, j i, j = 1, …, 6 (see Figure 2.1). Clearly, this set has as its elements 36 simple events (points or ordered pairs). Are there other events in Ω besides A? (Since A is composed of more than one simple event, it will be termed a compound event.) Obviously, the answer is “yes.” How many possible subsets of Ω can exist? The answer to this question is provided by the power set of Ω, 2Ω, which is the set of all subsets of Ω.1

1 For each set A χA ω =

Ω, let the indicator function of A be denoted as χ A Ω

0,1 or

1,ω A;

0,ω A Thus, 2Ω is actually a mapping from Ω to {0,1}. However, since the set {0,1} has but two elements, this mapping is written more succinctly as 2Ω. 2Ω has the following basic properties: (a) Ω 2Ω ; (b) if A 2 Ω then A 2Ω ; and (c) if A, B 2Ω , then A B 2Ω . Classes of subsets of Ω with these properties are called “algebras.”

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

38

2 Mathematical Foundations 2

Die 2 ω66

ω16 6 5 4 3

A

2

X

i+j=7

1 ω11

ω61 Die 1

1

2

3

4

5

6

Figure 2.1 Event A = ω36 , ω45 , ω54 , ω63 .

We previously specified event A as “the sum of the faces showing is nine.” The number 9 is actually the value of a random variable defined on Ω, that is, a random variable X is a real-valued function defined on the elements of a sample space Ω. Here, X (the sum of the faces) takes a point ωij Ω i,j = 1, …, 6 and maps it into a real number according to the rule i + j (Figure 2.1). Hence, the role of a random variable is to define the possible outcomes of a random experiment—it tells us how to get some real numbers out of our random experiment. In this case, X assumes the 11 values {2, 3, …, 12}. To find the probability of event A, P(A), we need the probability measure P 2Ω 0,1 , a set function that ascribes to each event A Ω its probability value P(A). To assign some number P(A) to each A 2Ω , let us employ the concept of relative frequency or fn A = nA n, where nA is the number of simple events in A and n is the total number of simple events within Ω. Since A = ωi, j = i,j i + j = 9 = ω36 , ω45 , ω54 , ω63 (see Figure 2.1), it is readily determined that fn A = 4 36 = P A . (Note that the relative frequency concept is applicable because the n = 36 simple events in Ω are equally likely.) It should be intuitively clear that P must satisfy two conditions: (1) P Ω = 1 (Ω is called the certain event); and (2) if A, B 2Ω are disjoint events A B = Ø , then P must be finitely additive or P A B = P A + P B . (More on all this shortly.) Given that Ω contains a finite number of simple events, the triple (Ω, 2Ω, P) completely describes the given random experiment. But what about the

2.1 Probability Spaces

circumstance in which a random experiment generates a countably infinite or uncountable number of outcomes? In these circumstances, we need to devise a more comprehensive analytical representation of a random experiment. To this end, a general description of a random process is incorporated in a particular type of measure space called a probability space Ω,A,P , where Ω, A is a measurable space; Ω is the sample space of the 0,1 random experiment; A is a σ-algebra of event subsets of Ω; and P A is a probability measure on A. In this regard, the σ-algebra A must satisfy the following conditions: 1. Ω A 2. If event A A, then A A (A denotes the non-occurrence of A) 3. If events Ai A, then i Ai A. In addition, the probability measure must satisfy the following conditions: a. For every event A A, P A ≥ 0. b. P Ω = 1 (Ω is treated as the “certain event”) c. If Ai , i = 1 2,… is a countably infinite sequence of pairwise disjoint events in A Aj Ak = Ø , j k , then P i Ai = Σ i P Ai , that is, P is σ-additive. (If Ai ,i = i,…, n is a finite sequence of such events in A, then P is termed finitely additive.) As corollaries to properties (a)–(c) of P P Ø = 0 (think of Ø as the “impossible event”); P A = 1 −P A (the probability that event A does not occur is 1 minus the probability that it does occur). In addition, if events A, B A, then A B implies P A ≤P B What about the random variable X defined on the probability space Ω, A, P ? For its specification, let’s first focus on the following generalized mapping between measurable spaces. Let Ω1 , A1 and Ω2 , 2 be measurable spaces. A mapping X Ω1 Ω2 that assigns to every element ω Ω1 the image ω = X ω Ω2 is said to be A 1 − A 2 -measurable. This mapping is called a Ω2 -valued random variable on Ω1 , A1 if the pre-images (or anti-images) of measurable sets in Ω2 are measurable sets in Ω1, that is, for B A 2 , ω X ω B = X −1 B A 1 . The set A 1 X of pre-images of measurable sets in Ω2 is itself a σ-algebra in Ω1 and is the smallest σ-algebra with respect to which X is measurable—it is termed the σ-algebra generated by X in Ω1. Now, back to our original question. Let Ω, A, P be a probability space and let R, U be a measurable space. A random variable X on Ω,A is a real-valued function from Ω to R such that, for all B U, X − 1 B A. (Clearly, X is a A − U -measurable map.) Thus, as required, pre-images of measurable sets in the range of X are measurable sets in A.

39

40

2 Mathematical Foundations 2

How are probabilities defined on Ω, A ? Given that the probability measure P is a set function with domain Ω and A is an event within the range of X, let X A denote the subset X − 1 A = ω X ω A of Ω. For P X A to exist, X − 1 A must lie within the domain Ω of P, that is, X −1 A A. A few additional points are in order. First, Ω,A, P is a complete probability space if A contains all subsets E of Ω having P-outer measure (P∗(∙)) zero, where P∗ E = inf P F E

F, F A

In fact, any probability space can be made complete simply by augmenting A by all sets of measure zero and extending P accordingly.2 Next, for C a class of subsets of Ω, the smallest σ-algebra containing C, σ C , is termed the σ-algebra generated by C—it is formed as the intersection of all σ-algebras containing C. Now, let C be an algebra of subsets of Ω with P C 0,1 a probability measure on Ω,C . If P has the following properties, 1. P Ω = 1. 2. For pairwise disjoint events Ai

C, with

i Ai

C, P is σ-additive,

then P is uniquely extended to a probability measure on Ω, σ C . Suppose Ω, B is a topological space. The Borel σ-algebra B is the σ-algebra generated by the collection of open sets in Ω. As indicated in Section 1.9.2, B is the smallest σ-algebra containing all open sets on Ω. An element B B is termed a Borel set. For instance, if a set A R is open, then A B (A is Borel). If the set C R is closed, then R–C is open, and thus R− C B so that C B. Thus, every closed set is a Borel set. In addition, countable intersections of open sets and countable unions of closed sets are Borel. Let Ω, , P be a probability space and R, B a topological space. A mapping X Ω R which is A − B -measurable (i.e., for an event B B, X −1 B = ω X ω B A) is a real-valued random variable. Then for any such X, P X B = P ω X ω B = P X − 1 B , B B, defines a probability measure on R,B . Suppose Ω1 , B is a topological space, where B is a Borel σ-algebra. If f Ω1 ,B Ω2 , C and f −1 ω B for ω C, then the mapping f is said to be Borel measurable. For Ω, A, P a probability space, a function f Ω R is called A-measurable if f − 1 B = ω f ω B A for all open (Borel) sets B R. Let’s consider a few of example problems. 2 For P a probability measure on Ω,A , the completion of A, A c = A Ω , there exist events B1, B2 A such that B1 A B2 and P(B1 − B2) = 0} is a σ-algebra on Ω containing A. Once P is extended to Ac (the completion of P, Pc, is defined on A c as Pc(A) = Pc(B1) = Pc(B2)), we say that the complete probability space Ω, A c ,Pc is the completion of (Ω, P).

2.1 Probability Spaces

Figure 2.2 The sample space Ω.

Outcome on flip 2

T

ω2

ω4

H

ω1

ω3

Outcome on flip 1 H

T

Example 2.1 (A finite number of outcomes) Suppose a random experiment consists of flipping a fair coin twice (equivalently, we can simultaneously flip two coins once). The sample space is Ω = ω1 , ω2 , ω3 , ω4 (Figure 2.2). There are many possible sets A that can satisfy the properties of a σ-algebra. The smallest is A = Ø ,Ω . (Clearly, Ø , Ω A; Ø = Ω A; Ω = Ø A; and Ø Ω A.) If it is required A, then the smallest σ-algebra containing that the simple events ω1 , ω2 {ω1}, {ω2} is A = Ø ω1 , ω2 , ω1 , ω2 , ω3 , ω4 , ω1 ,ω3 , ω4 , ω2 ,ω3 ,ω4 ,Ω , the σ-algebra generated by {ω1}, {ω2}. The largest σ-algebra containing all four simple events in Ω is A = Ø , ω1 , ω2 , ω3 , ω4 , ω1 ,ω2 , ω1 ,ω3 , ω1 , ω4

ω2 , ω3

ω2 , ω4

ω3 , ω4 ,

ω1 ,ω2 ,ω3 , ω1 , ω2 , ω4 , ω1 ,ω3 ,ω4 , ω2 ,ω3 ,ω4 ,Ω

For the probability measure defined on Ω, take P ωi = 1 4, i = 1 2,3 4 (since the ωi are equally likely). Then, for instance, P ω1 = 1 −P ω1 = 3 4, P ω1 ω2 = P ω1 + P ω2 = 1 2, and P 4i = 1 ωi = triple Ω,A, P constitutes a probability space. ■

4 i=1

P ωi = 1. Thus, the

Example 2.2 (A continuous set of outcomes) Let a random experiment consist of selecting a real number x between 0 and 1 inclusive. Hence, Ω = x 0 ≤ x ≤ 1 = 0,1 . Suppose we define A as 0,1 . Then the σ-algebra A is the set generated by all half-open interx a, b vals (a, b]; it consists of all intervals of the form (a, b], all unions of the same, and the complements of all the effected sets. This σ-algebra is the Borel σ-algebra on Ω. For our probability measure P on A, let’s define P(A) as the probability that x 0,1 belongs to A A. Then P(A) = b − a. As usual, the probability space is Ω,A, P . ■

41

42

2 Mathematical Foundations 2

Example 2.3 (An infinite number of outcomes) Suppose our random experiment involves tossing a single six-sided (fair) die repeatedly until a 4 shows for the first time. Here, Ω = ω1 ,ω2 ,… , where ωi is the outcome where the first i − 1 tosses show 1 or 2 or 3 or 5 or 6 and the ith toss shows 4. Then A = Ø , ω1 , ω2 , … ω1 , ω2 , ω1 ,ω3 , …,} is a σ-algebra of event subsets of Ω so that ωi A, i = 1 2,…. If our probability measure on A is defined as P ωi = 1 6 i , i = 1 2, …, then the triple Ω,A, P represents the probability space. ■

2.2

Probability Distributions

Suppose Ω,A,P is a probability space and R, U is a measurable space. As was noted in Section 2.1, a random variable X is a real-valued function defined on the sample space Ω, that is, X Ω R. For each outcome, ω Ω,X assigns the value X ω R. Consider the (cumulative) distribution function F x = P ω Ω X ω ≤ x of the random variable X. Since X is taken to be Ω − U -measurable, the set A x = ω Ω X ω ≤ x must be a subset of A for each x R. Clearly, F x = P A x . Let’s assume that the random variable X is discrete—it assumes either a finite or a countably infinite number of values, that is, X ω x1 ,x2 ,… R,ω Ω. To display the possible values of a discrete random variable X, we utilize its 0,1 , where p xj = P X = xj . Here, probability mass function p x1 , x2 ,… we require that p xj ≥ 0 and Σ j p xj = 1,j = 1 2,…. In addition, the distribution function can now be written as F x =P X ≤x =

p xj

21

xj < x

As to the properties of F(x), a. F(x) is monotone nondecreasing (i.e., if x ≤ y, then F x ≤ F y ). b. F −∞ = lim F x = lim P X ≤ x = 0 and x

−∞

x

−∞

F +∞ = lim F x = P X < +∞ = 1 x

+∞

c. F(x) is continuous to the right or limy x + F y = F x + 0 = F x , where y means that y approaches x from the right.

x+

Since X is a discrete random variable, F increases by “jumps” that occur at points of discontinuity of F. Moreover, if F x + = limy x + F y and F x− = limy x− F y denote, respectively, right and left limits of F at x, then F x− ≤ F x ≤ F x + .

2.2 Probability Distributions

Now, suppose F is given. Then we can recover p from the difference p x = F x + − F x−

22

Example 2.4 Let our random experiment consist of flipping a (fair) coin once. The associated probability space is Ω, A, P , where Ω = H,T , A = Ø , H , T ,Ω , and P H = P T = 1 2. In addition, with X discrete on Ω, define X(H) = 0 and X(T ) = 1 so that, as required, X ω = x1 = 0,x2 = 1 and P 0 = P 1 = 1 2. Then X’s probability mass and distribution functions are provided in Table 2.1 and illustrated in Figure 2.3. Note also that 1 1 p 0 = F 0 + −F 0− = −0 = ; 2 2 1 1 p 1 = F 1 + −F 1− = 1− = 2 2

■

Table 2.1 The probability mass and distribution function for X. X

p(x)

F(x)

0

1/2

1/2

1/2

1

1

1

(a) p(x)

(b) F(x) 1 0, x < 0;

1

F(x) =

2

1

1

2

,0 ≤ x < 1;

1, x ≥ 1.

2

0

1

X

X 0

1

Figure 2.3 (a) Probability mass function of X and (b) distribution function for X.

43

44

2 Mathematical Foundations 2

Table 2.2 The probability mass and distribution function for X. X

p(x)

F(x)

1

1/6

1/6

2

1/6

2/6

3

1/6

3/6

4

1/6

4/6

5

1/6

5/6

6

1/6

1

1

Example 2.5 Suppose our random experiment consists of rolling a single (fair) six-sided die. Here, Ω = ω1 , ω2 ,…, ω6 , where ωi is the face showing, i = 1, …, 6. Then A= Ø , ω1 ,…, ω6 , ω1 ,…, ω6 , ω2 , ω3 , ω4 , ω5 , ω6 ,…, Ω and P ωi = 1 6 for all i. The triple Ω, A, P thus constitutes our probability space. Let the value of the discrete random variable X correspond to the face showing or X ω x1 ,…,x6 or X ωi = i, i = 1,…,6. Table 2.2 houses the values of X’s probability mass function p(x) along with the values of its distribution function F(x). These functions are illustrated in Figure 2.4. ■ If the random variable X is continuous (it can assume any value over some range), then there exists a piecewise continuous non-negative function p(x), called the probability density function of X, such that F t =P X ≤t =

t −∞

2 2a

p x dx

(Note that since p x ≥ 0 for all real x +∞

−∞

−∞ , +∞ , we require that

p x dx = 1.) Here, (2.2a) represents the (cumulative) distribution func-

tion of X (Figure 2.5). Key properties of F(t) are the following: a. 0 ≤ F t ≤ 1. b. F a ≤ F b when a < b (F is nondecreasing in t). c. F −∞ = lim F t = 0 and F +∞ = lim F t = 1. t

−∞

t

+∞

d. F is everywhere continuous from the right at each t. e. For a < b, P a ≤ x ≤ b = F b −F a =

b

p x dx. a

f. If F has a point of discontinuity at t, then P(X = t) is the size of the jump exhibited by F at t; but if F is continuous at t, then P X = t = 0. With F a continuous function of t, its derivative exists at every point of continuity of p(x) and, at each such point, dF t dt = p t . (Thus, the integrand p(x) in (2.2) must be the value of the derivative of F at x.) So if we know X’s

2.2 Probability Distributions

(a) p(x)

1 6

X 1

2

3

4

5

6

(b) F(x) 1 5 4 3 2 1

6 6 F(x) = 6 6 6

0, x < 1; 1 ,1 ≤ x < 2; 6 2 ,2 ≤ x < 3; 6 3 ,3 ≤ x < 4; 6 4 ,4 ≤ x < 5; 6 5 ,5 ≤ x < 6; 6 1, x ≥ 6. X

1

2

3

4

5

6

Figure 2.4 (a) Probability mass function for X and (b) distribution function for X. Figure 2.5 Distribution function of X.

F(t) 1

F(a) = P(X ≤ a)

a

t

45

46

2 Mathematical Foundations 2

probability density function p(x), then we can determine its distribution function using (2.2). Conversely, if the distribution function F(t) is given, we can recover the probability density function p(x) at each of its points of continuity by determining dF t dt = p t . Example 2.6 Suppose Ω = α, β = x α ≤ x ≤ β . For random variable X(x) = x, α,β .” Then A is the let A be the event “a randomly chosen value of x a, b set generated by all intervals of the form (a, b), all unions of the same, and the complements of the resulting sets. Let p(x) be a uniform probability density function defined by 0,x ≤ α; 1 , α < x < β; px = β−α 0, x ≥ β as shown in Figure (2.6a). Now, for a, b α, β , a < b, P a≤X ≤b =

b

p x dx = a

b− a β−α

In addition, α

t

F t = =

−∞

p x dx =

−∞

t

0 dx +

α

1 dx β−α

t −α , β−α

or

0,t < α; t −α , α ≤ t ≤ β; F t = β−α 1, t > β as shown in Figure (2.6b). Note that F α = 0 and F β = 1. Moreover, for α < t < β, dF t dt = 1 β −α , the probability density function. ■ (a)

(b)

p(x)

F(t)

p (a ≤ X ≤ b) 1 β−a

1

α

a

b

β

X

α

β

t

Figure 2.6 (a) Probability density function for X and (b) distribution function for X.

2.2 Probability Distributions

Example 2.7 Let the sample space be specified as Ω = x −∞ < x < +∞ and let the σ-algebra A contain all intervals of the form (a, b) for a, b R along with all countable unions of the same and complements of the resultant sets. For random X(x) = x, let A be an event in A and suppose X’s probability density function is of the normal form or X N x; μ, σ , with x−μ 2 1 23 e − 2 σ2 , μ R, σ > 0 σ 2π (As μ and/or σ is varied, we get an entirely new normal density function, that is, (2.3) represents a two-parameter family of normal curves.) Then

p x;μ,σ =

b x− μ 2 1 e − 2 σ2 dx σ 2π a as shown in Figure (2.7a). The distribution function corresponding to (2.3) appears as

P a≤X ≤b =

(a) p(a ≤ X ≤ b)

p(x; μ, σ)

–x

(b)

0

a

μ

b

x

F(t; μ, σ)

1 F(b)

P(a ≤ X ≤ b) = F(b) – F(a)

0.5 F(a)

–t

0

a

μ

b

t

Figure 2.7 (a) Normal probability density function and (b) normal distribution function.

47

48

2 Mathematical Foundations 2

(a)

p(z; 0, 1) P

–z

a−μ σ

a−μ≤ Z ≤ b−μ σ σ

z

b−μ σ

0 F(z; 0, 1)

(b)

1 F

b−μ σ

P(a ≤ X ≤ b) = F

F

–t

0.5

a−μ σ

a−μ σ

0

b−μ a−μ −F σ σ

t

b−μ σ

Figure 2.8 (a) Standard normal probability density function and (b) standard normal distribution function. t

F t;μ,σ =

−∞

p x; μ, σ dx =

1 σ 2π

t −∞

e−

x− μ 2 2 σ2

dx

24

as shown in Figure (2.7b). If we define a new random variable as Z = X −μ σ, then (2.3) is transformed to standard normal form or 1 − 1 z2 p z;0,1 = e 2 231 2π as shown in Figure (2.8a). Under this transformation, P a≤X ≤b =

b− μ σ a −μ σ

=P

f z;0,1 dz

a− μ b −μ ≤Z≤ , σ σ

2.3 The Expectation of a Random Variable

where again Z

N z;0,1 . The distribution function corresponding to (2.3.1) is

F t; μ, σ = F

t −μ ;0,1 σ

1 = 2π

t −μ σ

−∞

241 e

1 2

− z2

dz

as shown in Figure (2.8b). Then P a≤X ≤b =F

b −μ a− μ ;0,1 − F ;0,1 σ σ

■

2.3

The Expectation of a Random Variable

2.3.1

Theoretical Underpinnings

Let X be a real-valued simple random variable defined on the probability space Ω, A, P and suppose that measurable X has, say, a finite number of values X ω R, ω Ω, with x1 ,…xn n

xi χ Ai ,

X= i=1

where 1,ω Ai ;

χ Ai ω =

0,ω Ai ,

−1

ωi = X xi ,Ai = ω Ω X ω = xi , i = 1,…, n, and the events {A1, …, An} form an A-measurable partition of Ω (i.e., for events Ai A, i = 1,…, n, Ai Aj = Ø . i j, and ni= 1 Ai = Ω). The integral of a simple random variable X with respect to the measure P is defined as n Ω

X ω dP ω =

Ω

Ω

xi P Ai

25

i=1

A, E χ Ai = P Ai ) and termed the expectation of X

(since, for any event Ai and denoted E X =

XdP =

XdP =

xdP x

251

R

Thus, the expectation and integral operators are mutually interchangeable. Here, X is said to be “P-integrable” if the integral in (2.5.1) is finite. (See also Equation (C.1).)

49

50

2 Mathematical Foundations 2

If X is a non-negative random variable on Ω, A,P , define E X =

Ω

XdP = sup

Ω

YdP Y ≤ X, Y simple

26

In fact, for non-negative X, there exists, via the monotone convergence theorem, a nondecreasing sequence of non-negative simple random variables Xn ,n ≥ 1 (i.e., Xn ω ≤ Xn + 1 ω , ω Ω) such that X ω = limn ∞ Xn ω ,ω Ω. We can then define E X = limn ∞ E Xn . In addition, for X an arbitrary random variable on Ω,A,P , set X ω = X + ω − X − ω , where X + ω = max X ω ,0 (the positive part of X) and (the negative part of X). Since X + ω ≥ 0 X − ω = −min X ω ,0 − and X ω ≥ 0, define E X = E X + − E X − and set E X =

Ω

X ω dP ω =

Ω

provided that not both E X + =

X + ω dP ω −

Ω

Ω

X − ω dP ω ,

X + dP and E X − =

Ω

27

X − dP are infinite. (If

both E X + ,E X − are infinite, E(X) does not exist. If only one of the terms E X + ,E X − is infinite, then E(X) itself is infinite. Thus, E(X) will be finite if both E X + ,E X − are finite.) In sum, an arbitrary random variable X on Ω,A, P is integrable if its positive and negative parts X + and X − , respectively, are both integrable or both E X + , E X − are finite. So if X is integrable, then, as defined earlier E X = E X + − E X − =

Ω

X + dP −

arbitrary event A A, it is customary to write Xdp =

Ω

A

2.3.2

Ω

X − dP =

Ω

Xdp. For an

χ A Xdp

28

Computational Considerations

Extremely useful versions of E(X) are Equations (2.5) and (2.5.1). To operationalize these expressions, let us first consider the case where X is a discrete random variable X ω x1 , x2 ,… for ω Ω and p x = P X = x is the probability mass function for X. The expectation (or mean of X) is then determined as E X =

xj p xj

=μ ,

29

j

provided that the indicated sum is absolutely convergent, that is,

j

xj p xj

converges and is finite. Clearly, (2.9) represents a weighted mean, with the sum p xj = 1. of the weights j

2.3 The Expectation of a Random Variable

In general, if g(x) is a single-valued real function of the discrete random variable X, then E g x =

291

g xj p xj j

g xj p xj is finite

E g x exists if j

Some important properties of the expectation (linear) operator are the following: a. E c = c (c a constant). b. E a ± bX = a ± bE X (a, b constants). c. E

n

X i=1 i

=

n i=1

E Xi

d. E g x ± h x = E g x ± E h x e. E a + bX

n

=

n

n

i=0

i

i n− i

ab

(g, h are functions of X). E X n− i (a, b constants).

Suppose X is a random variable defined on a probability space Ω,A,P and X2 is integrable or E X 2 < +∞. Then the variance of X is defined as the expected value of the squared deviations from the expectation and denoted as V X = E X −E X

2

= E X2 − E X

For X a discrete random variable X ω xj − E X

V X =

2

2

= σ2

2 10

x1 , x2 ,… ,ω Ω , (2.10) appears as

p xj

2 10 1

2

2 10 2

j

or xj 2 p xj − E X

V X = j

In addition, if we take the positive square root of V(X), we obtain the standard deviation of X or S X = V X = σ . Here, S(X) serves as a measure of dispersion of the xj’s around the mean of X. A few key properties of V(X) are the following: a. b. c. d.

V(c) = 0 (c a constant). V(c + X) = V(X) (c a constant). V(cX) = c2 V(X) (c a constant). If E(X2) exists, then E(X) exists and thus V(X) exists. Hence, if V(X) exists, it follows that E(X) exists.

Given a discrete random variable X, we may express the distance from the mean in terms of standard deviations by forming the standardized variable

51

52

2 Mathematical Foundations 2

Z = X − μ σ. Then it is easily shown that E(Z) = 0 and S(Z) = 1. For any value xj of X, zj = xj − μ σ indicates how many standard deviations xj is from the mean μ. If X represents a continuous random variable with X(x) = x and probability density function p(x), then the expectation (or mean) of X appears as +∞

E X =

2 11

x p x dx

−∞

(provided that the integral in (2.11) converges to a finite value). In general, if g(X) is a single-valued real function of a continuous random variable X, then +∞

E g x =

−∞

2 11 1

g x p x dx

+∞

(provided

−∞

g x p x dx exists).

To obtain the variance of a continuous random variable X, we need to calculate +∞

V X =

−∞

2

x −E X

2 12

p x dx

or V X = E X 2 −E X

2.4

2

+∞

=

−∞

x2 p x d x − E X

2

2 12 1

Moments of a Random Variable

A moment of a random variable X is defined as the expected value of some particular function of X. Specifically, moments of a random variable X are specified in terms of having either zero or E(X) as the reference point. For a discrete random variable X, the rth moment about zero is μr = E X r =

xjr p xj

2 13

j

(note that the first moment about zero is the mean of X or μ1 = E X = μ) and the rth central moment of X or the rth moment about the mean of X is μr = E X −μ

r

r

xj − μ p xj

=

2 14

j

If X is a continuous random variable with probability density function p(x), then, provided the following integrals exist, we may correspondingly define

2.4 Moments of a Random Variable +∞

μr = E X r =

−∞

xr p x d x

2 15

and +∞

r

μr = E X −μ

=

−∞

x −μ r p x d x

2 16

It is readily verified that a. the zeroth central moment of X is unity; b. the first central moment of X is zero; and c. the second central moment of X is the variance of X or μ2 = E X −μ 2 = V X . If we standardize the random variable X to obtain Z = X − μ σ, then, since E(Z) = 0, the rth central moment of Z can be expressed in terms of the rth central moment of X as X −μ σ

μr Z = E Z r = E 1 = r E X −μ σ

r

r

2 17

μ X = r r σ

Example 2.8 For the discrete probability distribution appearing in Table 2.3, it is readily determined that μ=E X =

xj p xj = 2 50; j

μ2 = E X 2 =

x2j p xj = 7 50; j

and thus μ2 = V X = μ2 −μ2 = 1 25

Table 2.3 X a discrete random variable. X

1

2

3

5

p(xj)

0.2

0.3

0.4

0.1

53

54

2 Mathematical Foundations 2

Let the probability density function for a continuous random variable X be given as 2x,0 < x < 1;

px =

0, elsewhere

Then μ=E X =

+∞ −∞

xp x d x

1

=2

2 x2 dx = x3 3 2

μ2 = E X 2 =

= 0 666;

+∞ −∞

1

=2

1 0

x2 p x d x

1 x3 dx = x4 2 0

1 0

=0 5

so that μ2 = V X = μ2 − μ2 = 0 055.

■

2.5

Multiple Random Variables

2.5.1

The Discrete Case

Given a probability space Ω,A, P , let X and Y be distinct random variables defined on this space with (X, Y): Ω R2 . For each outcome ω Ω, the bivariate random variable (X, Y) assigns the value X ω , Y ω R2 . The combination (X, Y) defines the possible outcomes of some random experiment. In this regard, if the range of (X, Y) is a discrete ordered set of points Xi ,Yj ,i = 1,…, n; j = 1,…,m in R2 , then (X, Y) is termed a discrete bivariate random variable. For convenience, we assume that both X and Y take on only a finite number of values (although assuming a countably infinite set of values is admissible). Given a random variable X : X1, X2, …, Xn (with X1 < X2 < < Xn ) and a second random variable Y : Y1, Y2, …, Ym (with Y1 < Y2 < < Ym ), let the joint probability that X = Xi and Y = Yj be denoted as P X = Xi , Y = Yj = f Xi ,Yj ,i = 1,…, n; j = 1,…, m. Here, f(Xi, Yj) depicts the probability mass at the point (Xi, Yj). (To elaborate, let Ar , As A with Ar = ω Ω X ω = Xr ,As = ω Ω Y ω = Ys . Then Ar As A with P Ar As = P X = Xr ,Y = Ys = f Xr , Ys .) In general, a function f (X,Y) that assigns a probability f(Xi, Yj) within the ranges Xi , i = 1, …, n and Yj , j = 1,…,m of the discrete random

2.5 Multiple Random Variables

variables X and Y, respectively, is called a bivariate probability mass function if a f Xi ,Yj ≥ 0 for alli, j; and 2 18

f Xi , Y j = 1

b i

j

The set of the nm events (Xi, Yj), together with their associated probability f Xi ,Yj ,i = 1,…, n; j = 1,…, m, constitutes a discrete bivariate probability distribution of the random variables X and Y. For the bivariate probability mass function f (X,Y), the joint probability that X ≤ Xr , Y ≤ Ys is provided by the bivariate cumulative distribution function F Xr ,Ys = P X ≤ Xr , Y ≤ Ys =

f Xi , Yj

2 19

i≤r j≤s

(Now, let Ar , As A with Ar = ω Ω X ω = Xr and As = ω Ω Y ω = Ys . Then Ar As A with P Ar As = P X ≤ Xr , Y ≤ Ys = F Xr , Ys .) While the bivariate probability distribution tells us something about the joint behavior of the random variables X and Y, information about these two random variables taken individually can be obtained from their marginal probabilities. In this regard, for X and Y discrete random variables, the summation of the bivariate probability mass function f (X, Y) over all Y within Yj ,j = 1,…, m yields the univariate probability mass function g(X) called the marginal probability mass function of X f Xi , Y j

g X =

2 20

j

Similarly, taking the summation of f(X, Y) over all X within Xi , i = 1, …, n produces the univariate marginal probability mass function of Y hY =

f Xi , Y j

2 21

i

Then, from (2.20), P X = Xr = g Xr =

f Xr , Y j ;

2 22

f Xi , Y s

2 23

j

and, from (2.21), P Y = Ys = h Ys = i

Once the marginal probability mass functions (2.22) and (2.23) are determined, we can also find

55

56

2 Mathematical Foundations 2

P Xa ≤ X ≤ X b =

g Xi ;

2 24

h Yj

2 25

a≤i≤b

P Yc ≤ Y ≤ Yd = c≤j≤d

Using (2.20), we can define the set of all Xi’s together with their marginal probabilities g Xi , i = 1, …, n, as the marginal probability distribution of X—it depicts the probability distribution of a single discrete random variable X when the levels of the random variable Y are ignored. Obviously, we require g Xi = 1. (The marginal probability disthat: (a) g Xi ≥ 0 for all i and (b) i tribution of Y is defined in a similar fashion.) Next, our objective is to define conditional probability distributions of the discrete random variables X and Y. To set the stage for this discussion, we first need to consider the notion of a conditional probability itself. To this end, for Ω, A,P a probability space, let events A, B A with P(B) > 0. Suppose a simple event ω Ω is randomly selected and it is determined that ω B. What is the probability that ω is also in A? To answer this question, we need the concept of a conditional probability—the probability of occurrence of one event given that another event has definitely occurred. Then the probability of A given B (denoted P(A|B) and read “the probability of A given B”) can be determined as follows: Since ω B, event B becomes the new or effective sample space, and thus our reduced or effective probability space is Ω = B,A = B C C

Ω,A, P , where

A , and P = P P B , with P Ω = 1. Then the probability

that ω A given ω B is P A B = P A B P B = P A B . To summarize, P AB =

P A B , P B > 0; P B

2 26

P A B ,P A > 0 P A

2 27

and if P(A) > 0, P BA =

From (2.26) and (2.27), we can write P A B =P A B P B =P B A P A

2 28

Multiplication law for probabilities

Events A, B A are termed independent if, on any given trial of a random experiment, the occurrence of one of them in no way affects the probability of occurrence of the other. That is, A is independent of B if an only if P A B = P A , P B > 0;

2 29

and B is independent of A if and only if P B A = P B ,P A > 0

2 30

2.5 Multiple Random Variables

Note that under the (mutual) independence of events A and B, (2.28) becomes, via (2.29) and (2.30), P A B =P A P B

3

2 31

Test for independent events

Now, back to our initial objective of defining a conditional probability distribution. Suppose X and Y are discrete random variables with bivariate probability mass function f(X,Y) and marginal probability mass functions g(X) and h(Y), respectively. Given these relationships, we may now ask “What if, on a particular trial of a random experiment, we specify the level of Y but the values of X are allowed to be determined by chance?” That is, what is the probability that X = Xi given that Y = Yj for fixed j? In a similar fashion, we may be interested in determining the probability that Y = Yj given that X = Xi for fixed i. In this regard, if (X,Y) is any point at which h Y > 0, then the conditional probability mass function of X given Y is f X, Y g XY = 2 32 hY Here, g(X|Y) is a function of X alone and, for each Xi given Y = Ys , the probability that X = Xi given Y = Ys is P X = X i Y = Y s = g Xi Y s =

f Xi , Y s , h Ys > 0, i = 1, …, n h Ys

2 33

Similarly, at any point (X, Y) at which g(X) > 0, the conditional probability mass function of Y given X (a function of Y alone) is f X, Y hY X = 2 34 g X So for any Yj given X = Xr , the probability that Y = Yj given X = Xr is P Y = Y j X = X r = h Y j Xr =

f Xr , Y j , g Xr > 0,j = 1,…,m g Xr

2 35

Note that from (2.32) and (2.34), we can solve for f (X, Y) as f X, Y = g X Y h Y = h Y X g X

2 36

Multiplication law for probability mass functions

From (2.33), we can define the set of all Xi’s together with their conditional probabilities g Xi Yj , i = 1, …, n, as the conditional probability distribution of X given Y = Yj . This distribution depicts the probability distribution of a 3 What is the distinction between events that are mutually exclusive and events that are independent? Events A and B are mutually exclusive if they cannot occur together, that is, A B = Ø so that P A B = 0. If events A and B are independent, they can occur together, that is, A B Ø so that P A B = P A P B .

57

58

2 Mathematical Foundations 2

single discrete random variable X when the level of Y is fixed at Yj. For this distribution, we must have (a) g Xi Yj ≥ 0 for all i and (b) g Xi Yj = 1. (The i conditional probability distribution of Y given X = Xi is defined in an analogous fashion.) Let X and Y be discrete random variables with bivariate probability mass function f (X,Y) and marginal probability mass functions g(X) and h(Y), respectively. Then the random variable X is independent of the random variable Y if g X Y =g X

2 37

for all X, Y values for which both of these functions are defined. Similarly, the random variable Y is independent of the random variable X if 2 38 h Y X =h Y for all X and Y at which these functions exist. Given (2.37) (or (2.38)), (2.36) renders 2 39 f X,Y = g X h Y , that is, X and Y are independent random variables if and only if their joint probability mass function can be written as the product of their individual marginal probability mass functions g(X) and h(Y), respectively. Hence, under independence, P X = Xi ,Y = Yj = P X = Xi P Y = Yj for all points (X, Y).

2 39 1

Example 2.9 Given the bivariate probability mass function appearing in Table 2.4, we can readily determine that F X2 ,Y2 = P X ≤ X2 , Y ≤ Y2 =

f Xi ,Yj i≤2 j≤2

f Xi , Y1 + f Xi , Y2

= i≤2

= f X1 , Y1 + f X1 , Y2 + f X2 , Y1 + f X2 , Y2 = 0 07 + 0 14 + 0 05 + 0 25 = 0 51; Table 2.4 Bivariate probability mass function f(X, Y). f Xi , Yj , i = 1,2 3; j = 1 2 Y

Y1

Y2

g(Xi)

X1

0.07

0.14

0.21

X2

0.05

0.25

0.30

X3

0.21

0.28

0.49

h(Yj)

0.33

0.67

1.00

X

2.5 Multiple Random Variables

P X = X2 = g X2 =

f X2 , Yj = f X2 , Y1 + f X2 ,Y2 j

= 0 05 + 0 25 = 0 30; P Y = Y1 = h Y1 =

f Xi , Y1 = f X1 , Y1 + f X2 ,Y1 + f X3 ,Y1 j

= 0 07 + 0 05 + 0 21 = 0 33; P X1 ≤ X ≤ X 2 =

g Xi = g X1 + g X2 = 0 51; 1≤i≤2

P X = X3 Y = Y 1 = g X3 Y 1 =

f X3 , Y1 0 21 = = 0 6363; and 0 33 h Y1

P Y = Y 2 X = X 1 = h Y 2 X1 =

f Y 2 , X1 0 14 = 0 6667 = 0 21 g X1

Are X and Y independent random variables? One way to answer this question is to check to see if f X2 , Y2 = g X2 h Y2 . From Table 2.4, f X2 , Y2 = 0 25 while g X2 = 0 30 and h Y2 = 0 67. Then g X2 h Y2 = 0 20 f X2 ,Y2 . Hence, X and Y are not independent random variables. ■

2.5.2

The Continuous Case

Suppose the sample space Ω R2 consists of a class of events representable by all open and closed rectangles. Then an event in Ω can be written as, say, A = X,Y a ≤ X ≤ b, c ≤ Y ≤ d , where X and Y are continuous random variables. In addition, let P be a (joint) probability measure that associates with each A Ω a number P A R . In general, a function f(X, Y) that defines the probability measure, P A = P a ≤ X ≤ b, c ≤ Y ≤ d =

b d

f x, y dy dx

2 40

a c

is called a bivariate probability density function if a f x,y ≥ 0 for all real x and y such that −∞ < x < +∞, −∞ < y < +∞ , and f x, y > 0for x, y A;and b

+∞

+∞

−∞

−∞

f x, y dy dx = 1

2 41

So if Ω is made up of all open and closed rectangles in R2 and P(A) is determined via (2.40), then the random variables X and Y follow a continuous bivariate probability distribution. (Note that the probability of an event containing a single point is always zero, for example, P X = a, Y = b = 0.)

59

60

2 Mathematical Foundations 2

Equation (2.40) defines the probability that the continuous random variables X and Y with bivariate probability density f(x, y) assumes a value within the set A = X,Y a ≤ X ≤ b, c ≤ Y ≤ d . A related function (derived from f) that yields the (joint) probability that X assumes a value less than or equal to some number t and Y takes on a value less than or equal to a number s is the bivariate cumulative distribution function for the continuous random variables X and Y or F t,s = P X ≤ t, Y ≤ s =

t

s

−∞

−∞

2 42

f x, y dy dx

F(t, s) is a continuous function of t and s and, at every point of continuity of f x, y , ∂2 F t, s ∂s ∂t = f s, t . Hence, knowing the bivariate probability density function f(x, y) enables us to determine the associated bivariate cumulative distribution function F(t, s) using (2.42). Conversely, knowing F(t, s) enables us to determine the probability density f(s, t) at each of its points of continuity by finding ∂2 F ∂s ∂t. Given the continuous random variables X and Y with bivariate probability density function f(x, y), the marginal probability density function of X is +∞

g x =

−∞

2 43

f x, y dy +∞

and is a function of x alone with (a) g x ≥ 0 and (b) the marginal probability density function of Y is

−∞

g x dx = 1. In addition,

+∞

hy =

−∞

2 44

f x, y dx

and is a function of y alone with (a) h y ≥ 0 and (b)

+∞ −∞

h y dy = 1.

Given the joint cumulative distribution function F(t, s), the marginal cumulative distribution functions of X and Y are P X ≤t =F t =

t

+∞

−∞

−∞

s

+∞

t

f x, y dy dx =

−∞

g x dx

2 45

h y dy,

2 46

and P Y ≤s =F s =

−∞

−∞

s

f x, y dx dy =

−∞

respectively, where g(x) and h(y) are determined from (2.43) and (2.44), respectively. If f(x, y) is the bivariate probability density function for the random variables X and Y and the marginal densities g(x) and h(y) are known, then we may define, for Y fixed at y, the conditional probability density function for X given Y as

2.5 Multiple Random Variables

g xy =

f x, y , h y > 0; hy

2 47

and for X fixed at x, the conditional probability density function for Y given X is h yx =

f x, y ,g x > 0 g x

2 48

Here, g(x|y) is a function of x alone with properties (a) g x y ≥ 0 and (b) +∞

−∞ +∞ −∞

g x y dx = 1; h y x is a function of y alone, and (a) h y x ≥ 0 and (b) h y x dy = 1. How may we interpret (2.47)? If Y = y but the value of X is

unknown, then the function g(x|y) gives the probability density of X along a fixed line Y = y in the X,Y-plane. (A similar interpretation can be offered for (2.48).) Once (2.47) and (2.48) are determined, we may calculate probabilities such as P a≤X ≤b y =

b

g x y dx

2 49

h y x dy

2 50

a

and P c≤Y ≤d x =

d c

Suppose X and Y are continuous random variables with bivariate probability density function f(x, y) and marginal probability densities g(x) and h(y), respectively. Then the random variable X is independent of the random variable Y if g x y =g x

2 51

for all values of X and Y for which both of these functions exist. In similar fashion, we may state that the random variable Y is independent of the random variable X if h y x =h y ,

2 52

again for all X and Y values for which both of these functions are defined. From (2.47) and (2.51) we can write f x,y = g x h y ,

2 53

that is, X and Y are independent random variables if and only if their joint probability density function f(x, y) can be written as the product of their individual marginal probability densities g(x) and h(y), respectively.

61

62

2 Mathematical Foundations 2

Given the probability density function

Example 2.10

x + y,0 < x < 1 and 0 < y < 1; f x, y = 0, elsewhere, we can readily find 05

1

P 0 333 < X < 0 5, 0 5 < y < 1 =

x + y dy dx 0 333 0 5 05

1 xy + y2 2

= 0 333

= 0 25x2 + 0 375x +∞

g x =

1

x + y dy =

−∞

0

+∞

hy =

−∞

05

dx = 05 0 333

= 0 097; 1

1 2 x + xy 2

1

x + y dx = 0

0 5x+0 375 dx 0 333

05

1 x + y dy = xy + y2 2

1

x + y dx =

1

1 =x+ ; 2 0 = 0

1 + y; 2

x+y x+y ;h y x = g xy = 1 2 +y x+ 1 2 When X = x = 2, h y 2 = 0 80 + 0 40y so that P 0 333 ≤ Y ≤ 0 5 x = 2 =

05

0 80 + 0 40y dy 0 333

= 0 80y + 0 20y2

05 0 333

= 0 16

Are X and Y independent random variables? If they are, then f x, y = g x h y . Clearly, this is not the case. ■ Example 2.11 f x,y =

Given the probability density function 4xy, 0 < x < 1 and 0 < y < 1; 0, elsewhere,

determine the associated cumulative distribution function. We thus need to find t

F t, s = 4

−∞ t

=4 0

s −∞

1 2 xy 2

t

s

xy dy dx = 4

xy dy dx 0 0

s

t

dx = 2s2 0

xdx = s2 t 2 0

2.5 Multiple Random Variables

or 0, t < 0 and s < 0; s2 t 2 , 0 ≤ t ≤ 1 and 0 ≤ s ≤ 1;

F t,s =

0, t > 1 and s > 1 Then, as expected, ∂2 F ∂s∂t = 4st. 2.5.3

■

Expectations and Moments

In what follows, it is assumed that all specified expectations exist. For X and Y discrete random variables with bivariate probability mass function f(X,Y), the rth moment of the random variable X about zero is E Xr =

Xir f Xi , Yj = i

j

Xir g Xi ;

2 54

i

and the sth moment of the random variable Y about zero is E Ys =

Yjs f Xi , Yj = i

j

Yjs h Yj

2 55

j

In addition, the rth and sth product or joint moment of X and Y about the origin (0,0) is E X rY s =

Xir Yjs f Xi , Yj i

2 56

j

and, for r and s non-negative integers, the rth and sth product or joint moment of X and Y about the mean is E X − μX

r

Y − μY

s

Xi −μX

= i

r

s

Yj − μY f Xi ,Yj ,

2 57

j

where μX = E X and μY = E Y . If X and Y are continuous random variables with joint probability density function f(x, y), then the rth moment of X about zero is +∞

E Xr =

−∞

+∞

+∞

x r f x, y dy dx =

−∞

−∞

x r g x dx;

2 58

y s h y dy

2 59

and the sth moment of Y about zero is +∞

E Ys =

−∞

+∞ −∞

+∞

y s f x, y dy dx =

−∞

In addition, the rth and sth product or joint moment of X and Y about the origin (0,0) is E X rY s =

+∞

+∞

−∞

−∞

x r y s f x, y dy dx

2 60

and, for r and s non-negative integers, the rth and sth product or joint moment of X and Y about the mean is

63

64

2 Mathematical Foundations 2

E X − μX

r

Y − μY

s

=

+∞

+∞

−∞

−∞

x− μX

r

y −μY s f x,y dy dx

2 61

Some additional considerations regarding expectations are the following: 1. If ς is a linear function of the random variables X and Y, ς = aX ± bY , where a and b are constants, then, for either the discrete or continuous case, E ς = E aX ± bY = a E X ± b E Y 2. If in (2.57) and (2.61) we set r = s = 1, then the expression E X −μX Y −μY = E XY − μX μY = COV X,Y = σ XY

2 62

is called the covariance of the random variables X and Y—it depicts the joint variability of the random variables X and Y. Note that COV aX, bY = abσ XY . While the sign of σ XY indicates the direction of the relationship between the discrete or continuous random variables X and Y (X and Y are positively related when σ XY > 0; they are negatively related when σ XY < 0), the coefficient of correlation between the random variables X and Y, ρXY =

σ XY , − 1 ≤ ρXY ≤ 1, σX σY

2 63

measures the strength and the direction of the linear relation between X and Y, where σ X and σ Y are, respectively, the standard deviations of X and Y. When ρXY = 1 (resp. ρXY = −1), we have perfect positive (resp. negative) association between X and Y. 3. If X and Y are independent discrete or continuous random variables, then σ XY = 0 and, from (2.62), E XY = μx μY If X and Y are uncorrelated or ρXY = 0, it does not necessarily follow that X and Y are independent random variables. 4. If ς = aX + bY , then the variance of ς is V ς = V aX + bY = a2 σ 2X + b2 σ 2Y + 2abσ XY ,

2 64

where σ 2X = V X and σ 2Y = V Y are the variances of the random variables X and Y, respectively. If ς = aX − bY , V ς = V aX −bY = a2 σ 2X + b2 σ 2Y − 2abσ XY

2 65

In addition, if X and Y are independent random variables, then V aX ± bY = a2 σ 2X + b2 σ 2X

2 66

5. For random variables X and Y, E XY = μX μY + σ XY

2 67

If X and Y are independent, then E XY = μX μY (as indicated in item (3)).

2.5 Multiple Random Variables

6. For random variables X and Y, 2

V XY = E XY −E XY

= E XY − μX μY + σ XY

2

= μ2Y σ 2X + μ2X σ 2Y + 2μX μY σ XY −σ 2XY + E X − μX

2

Y − μY

2

+ 2μY E X −μX

+ 2μX E X − μX Y − μY

2

Y −μY

2 68

2

If X and Y are independent random variables, V XY = μ2Y σ 2X + μ2X σ 2Y + σ 2X σ 2Y

2 68 1

7. We previous defined a Hilbert space H (Section 1.11.3) as a complete inner product space, where the inner product f , g = f x g x dx R for f ,g

H

and the norm f = f , f 1 2 . Given this setting, we now turn to the specification of the Hilbert space of random variables HR. To this end, let Ω,A, P be a probability space with event A A. Then the indicator function 1, ω A;

χA ω =

0, ω

A

is a random variable with E χ A = P A . In addition, if X is a simple random variable or n

X ω =

xi χ A i , Ai

A, i = 1,…, n,

i=1 n

x P Ai . then E X = i=1 i Next, let S be the set of simple random variables defined on Ω, A,P . With S an inner product space and for simple random variables, X, Y S, the norm of X is 1

X = X, X 2 = E X

2

1 2

;

and the inner product of X and Y on S is defined as n

n

xi χ Ai yj χ Bj

X, Y = E XY = E i=1 j=1 n

n

=

xi yj P Ai Bj i=1 j=1

for Ai ,i = 1,…, n , Bj , j = 1, …, n A. Since S might not be complete (in the sense that all Cauchy sequences of random variables defined on S must

65

66

2 Mathematical Foundations 2

converge in S), how do we make the leap from S to HR? The completion of S may be effected by augmenting S by a set of random variables in order to form HR by rendering S dense in HR. Let X and Y be discrete random variables with g(X|Y) the conditional probability mass function of X given Y and h(Y|X) the conditional probability mass function of Y given X. Then the conditional expectation of X given Y = Ys 4 is defined as E X Ys =

Xi g Xi Y s ;

2 69

i

and the conditional variance of X given Y = Ys is 2

V X Ys = E X − E X Ys

Ys = E X 2 Ys − E X Ys 2 ,

2 70

where E X 2 Ys =

Xi2 g Xi Ys i

Similarly, the conditional expectation of Y given X = Xr is E Y Xr =

Y j h Y j Xr ;

2 71

j

and the conditional variance of Y given X = Xr is 2

V Y Xr = E Y −E Y Xr

Xr = E Y 2 Xr −E Y Xr 2 ,

2 72

with E Y 2 Xr =

Yj2 h Yj Xr j

If X and Y are continuous random variables with g(x|y) and h(y|x) representing conditional probability density functions for X given Y and Y given X, respectively, then the conditional expectation of X given Y = y is +∞

E Xy =

−∞

2 73

x g x y dx;

and the conditional variance of X given Y = y is V X y = E X −E x y

2

y = E X 2 y −E X y 2 ,

2 74

where +∞

E X2 y =

−∞

x2 g x y dx

2 75

4 Appendix 2.A provides a discussion of the theoretical foundation for conditional expectations.

2.5 Multiple Random Variables

In similar fashion, the conditional expectation of Y given X = x is defined as +∞

E Y x =

−∞

2 76

y h y x dy;

and the conditional variance of Y given X = x is V Y x = E Y −E Y x

2

x = E Y 2 x − E Y x 2,

2 77

where +∞

E Y2 x =

−∞

y2 h y x dy

2 78

If X and Y are independent discrete or continuous random variables, then the preceding sets of conditional means and variances equal their unconditional counterparts, for example, under the independence of X and Y, (2.73) reduces to E X y = E X and (2.74) becomes V X y = V X . Example 2.12 Let’s generalize the univariate normal distribution presented in Example 2.7 involving the random variable X to the bivariate case where we have both X and Y as random variables. Specifically, let the random variable (X, Y) have the joint probability density function f x, y =

1 2πσ x σ y

e − 2 Q , −∞ < x, y < +∞ , 1

1 − ρ2

1 2

2 79

where Q=

1 1 − ρ2

2

x −μx σx

− 2ρ

x− μx σx

y − μy y −μy + σy σy

2

,

2 80

and where μx, μy, σ x, σ y, and ρ are all parameters, with −∞ < μx , μy < +∞,σ x > 0, σ y > 0, and ρ < 1 ((2.79) is undefined if ρ = ±1). Here, μx = E X , μy = E Y , σ 2x = V X , σ 2y = V Y , and ρ is the coefficient of correlation between X and Y. If Equations (2.79) and (2.80) hold, then X and Y are said to follow a bivariate normal distribution with joint probability density function (2.79), or X and Y are N(x, y; μx, μy, σ x, σ y, ρ). Looking to the properties of (2.79), 1. f x,y > 0 2. The probability that a point (X, Y) will lie within a region A of the x, y-plane is A =

P x,y

f x, y dydx A

3. As required, +∞

+∞

−∞

−∞

f x, y dydx = 1; and

67

68

2 Mathematical Foundations 2

4. if in (2.79) we set U = x− μx σ x , and V = y − σ y σ y , then f(x, y) can be expressed in terms of the single parameter ρ. In addition, for (X, Y) bivariate normal, the marginal probability density function of X (denoted N(x; μx, σ x)) can be obtained from (2.79) by integrating out y or +∞

g x =

−∞

f x, y dy =

1 σ x 2π

1 x−μx σx

e−2

2

, −∞ < x < +∞ ;

2 81

and the marginal probability density function of Y (denoted N(y; μy, σ y)) is obtained from (2.79) by integrating out x or +∞

1

− 12

y −μy σy

2

e , −∞ < y < +∞ 2 82 σ y 2π (Note that if the marginal distributions of the random variables X and Y are each univariate normal, this does not imply that (X, Y) is bivariate normal.) In this section 2.5.3 we noted that the random variables X and Y are independent, then they must also be uncorrelated or ρXY = 0. However, the converse of this statement does not generally apply. Interestingly, it does apply if X and Y follow a bivariate normal distribution. That is, if X and Y are bivariate normal with probability density function (2.79), then X and Y are independent random variables if and only if ρXY = ρ = 0. When ρ = 0 (X and Y are independent), (2.79) factors as f x, y = g x h y . Next, if the random variable (X, Y) is bivariate normal, then the conditional hy =

−∞

f x, y dx =

distribution of X given Y = y is univariate N x;μx + ρ σ x σ y σx

y −μy ,

1− ρ2 . Here, the conditional mean of X given Y = y is E X Y = y = y −μy , and the conditional variance of X given Y = y is

μx + ρ σ x σ y

= σ 2x

V X Y =y 1 −ρ2 . The conditional probability density function of X given Y = y is obtained from the joint and marginal densities in (2.79) and (2.81), respectively, for h y 0, as g xy =

f x, y hy

−

1

= σx

2π

e 1 −ρ2

x−μx − ρ σ x σ y 2 σ 2x 1 − ρ2

y −μy

2

, −∞ < x < +∞ 2 83

Similarly, the conditional distribution of Y given X = x is univariate N y;μy + ρ σ y σ x x −μx , σ y

1− ρ2 . Thus, the conditional mean of Y given

2.5 Multiple Random Variables

X = x is E Y X = x = μx + ρ σ y σ x x− μx , and the conditional variance of Y given X = x is V Y X = x = σ 2y 1 − ρ2 . Hence, the conditional probability density function of Y given X = x is obtained from the joint and marginal densities in (2.79) and (2.82), respectively, given g x 0, as h yx =

f x, y g x 1

=

y −μy −ρ σ y σ x x− μx 2 σ 2x 1 − ρ2

e

σy 2.5.4

−

2π

1 −ρ2

2 84

2

, −∞ < y < +∞

■

The Multivariate Discrete and Continuous Cases

In some of the preceding sections, we exclusively considered bivariate random variables (X, Y). Now, let us expand our discussion to the case of dealing with the multivariate random vector X = X1 , X2 ,…, Xn , where each Xi , i = 1, …, n, is a random variable. To this end, let Ω,A, P be a probability space and suppose that X Ω R n is a random vector. If X is discrete, then the joint probability mass function of X is defined as f x = f x1 ,…, xn = P X1 = x1 , …, Xn = xn for each x = x1 ,…, xn P X

Rn

2 85 R

R n . Then for any event A

Rn,

f x

A =

2 86

x A

If X is a continuous random vector, then the joint probability density function is a non-negative, integrable function f x = f x1 ,…,xn R n R that satisfies P X

A = … f x dx = … f x1 ,…, xn dxn …dx1 , A

2 87

A

where the limits of integration are specified so that integration occurs over all vectors X A. Given the multivariate probability mass function f(X), the joint probability that Xi ≤ Yi ,i = 1, …, n, where Y = Y1 ,…, Yn R n , is provided by the discrete multivariate cumulative distribution function F R n 0,1 , with F y1 ,…, yn = P X ≤ Y = P X1 ≤ y1 , …, Xn ≤ yn

2 88

69

70

2 Mathematical Foundations 2

In addition, for the multivariate probability density function f(x), the continu0,1 is written as ous multivariate cumulative distribution function F R n F y = P X1 ≤ y1 , …, Xn ≤ yn y1

=

−∞

…

yn −∞

f x1 ,…, xn dxn …dx1

2 89

Let Ø x = Ø x1 ,…, xn be a real-valued function defined on Ω. With Ø X a random variable, the expectation of Ø X in the discrete case is E Ø X

Ø xf x;

=

2 90

x Ω

and for X continuous, E Ø X

+∞

=

−∞

…

+∞ −∞

Ø x f x dx

2 91

As was presented earlier (Sections 2.5.1 and 2.5.2 covered, respectively, the discrete and continuous cases), we can define the discrete multivariate marginal probability mass function of a subset of coordinates of X by summing the joint probability mass function over all possible values of the remaining coordinates, for example, the marginal distribution of X1 ,…,Xk R k ,k < n, can be determined as g x1 ,…,xk =

f x

xk + 1 ,…, xn

R n− k

2 92

The continuous multivariate marginal probability mass function, again taken for a subset of coordinates of X, is obtained by integrating the joint probability density function over all possible values of the other coordinates, that is, the marginal density of X1 ,…, Xk R k , k < n, is specified as g x1 ,…, xk =

+∞

+∞

−∞

−∞

f x dxn …dxk + 1

2 93

The multivariate conditional probability mass function or probability density function of the subset X1 ,…, Xk R k , k < n, of coordinates of X, given the values of the remaining coordinates, is obtained by dividing the joint probability mass function by the marginal probability mass function (in the discrete case) or by dividing the joint probability density function by the marginal probability density function (for the continuous case). In this regard, the conditional probability mass function or probability density function of Xk + 1 ,…,Xn given X1 = x1 ,…,Xk = xk is a function of xk + 1 ,…, xn defined by f xk + 1 ,…, xn x1 , …, xk =

f x , g x1 ,…,xk > 0 g x1 ,…, xk

2 94

2.5 Multiple Random Variables

Example 2.13 Let the random variables X1, X2, …, Xn constitute the components of the n × 1 random vector X and let the means of these random variables, denoted μ1, μ2, …, μn, respectively, represent the components of the n × 1 mean vector μ. In addition, let Σ serve as the nth order variance– covariance matrix of X1, …, Xn, where X1 − μ1

2

… X n − μn X1 − μ1

X1 −μ1 X2 − μ2

… X n − μn X2 − μ2

X1 −μ1 Xn − μn

…

=E X − μ X − μ =E

Xn −μn

σ 21 σ 12 … σ 1n

σ 21

σ 21 σ 22 … σ 2n

ρ21 σ 2 σ 1

2

ρ12 σ 1 σ 2 … ρ1n σ 1 σ n σ 22

… ρ21 σ 2 σ n

=

=

σ n1 σ n2 … σ 2n

ρn1 σ n σ 1 ρn2 σ n σ 2 …

σ 2n 2 95

is a symmetric positive definite matrix and ρkl = σ kl σ k σ l , with k,l = 1, …,n. Then the joint probability density function for the multivariate normal distribution appears as f X; μ, Σ = N X; μ, Σ = 2π

−n 2

Σ

− 12

e − 2 X −μ Σ 1

−1

X −μ

, −∞ < μi < +∞, i = 1, …,n,

2 96

where a prime denotes vector (matrix) transposition. Given this expression, the following can be demonstrated: 1. The marginal probability density function of each Xi is N Xi ;μi , σ i , i = 1,…,n. 2. The conditional probability density function of Xk, given fixed values of the remaining variables, is N E Xk X1 , …, Xk −1 , Xk + 1 , …, Xn , σ 2Xk X1 , …, Xk − 1 , Xk + 1 , …, Xn 3. If R represents an nth order correlation matrix with components ρkl , k, l = 1,…, n, then (2.96) can be written as f U = 2π

− n2

σ 1 σ 2 …σ n

−1

R

− 12 − 12 U R − 1 U

e

,

2 97

71

72

2 Mathematical Foundations 2

where Ui = Xi −μi σ i , i = 1,…n. Furthermore, if ρkl = 0, k l (the jointly distributed normal random variables are uncorrelated), then R = In (the identity matrix—a matrix having ones along the main diagonal and zeros elsewhere) and f U = 2π

− n2

σ 1 σ 2 …σ n

− 1 − 12 U U

2 97 1

e

If it is also true that μi = 0 and σ i = σ for all i, then Ui = Xi σ, i = 1, …,n, and thus (2.97.1) appears as f U = 2π

− n2 − n − 12 U U

σ

2 97 2

e

In addition, if σ = 1, then U follows a multivariate standard normal distribution with probability density function f U = 2π

− n2 − 12 U U

2 97 3

e

4. Given (2.96), if the random variables X1, …, Xn are mutually independent, then f X; μ, Σ

= Π in= 1 f

Xi ; μi , σ i = Π in= 1

2π

− 12

− 12

σ i ei

− 12

Xi −μi σi

2

,

2 98

where f(Xi; μi, σ i) is univariate normal. Note that (2.98) also results if Σ is a diagonal matrix (Σ is diagonal if the normally distributed random variables Xi ,i = 1,…, n, are uncorrelated or have zero covariance). Hence, with respect to Equation (2.98), the variables Xi , i = 1, …, n, are independent if and only if Σ is a diagonal matrix. Thus, zero covariance is equivalent to independence under multivariate normality. ■

2.6

Convergence of Sequences of Random Variables

Let Xn n∞= 1 (or simply {Xn} for short) denote a sequence of random variables defined on the probability space Ω, A, P . An important question that emerges in the area of probability theory is “Does there exist a limiting random variable X to which the sequence approaches as n ∞ ?” Our objective in this section is X as n ∞ since, as we shall now to characterize the manner in which Xn see, there are many varieties of convergence criteria for sequences of random variables. Being already familiar with the concept of “pointwise convergence” of real numbers (Section 1.5), the reader may be tempted to define the convergence of {Xn} to X as n ∞ as limn ∞ Xn ω = X ω ,ω Ω. The mode of convergence just described is termed the sure convergence of {Xn} to X. However, in terms of the requirements of probability theory, this form of convergence

2.6 Convergence of Sequences of Random Variables

of random variables is “too strong.” In this regard, we shall now look to a set of “weaker” concepts pertaining to the convergence of a sequence of random variables. 2.6.1

Almost Sure Convergence

We may weaken the notion of “sure convergence” by requiring only that {Xn} converges almost surely (a.s.) or converges with probability 1 to X as n ∞. That is, let {Xn} and X be random variables defined on the same probability space Ω,A, P and let (measurable) set A = ω Ω Xn ω X as ω as n ∞ . Then {Xn} converges a.s. to X as n ∞ (written Xn X) if P as (A) = 1. In short, Xn X if P Xn X = 1. A sufficient condition for a.s. convergence is: Sufficient Condition for a.s. Convergence ∞ as P X − X ≥ ε < +∞ for all ε > 0, then X X. n n n=1

If

Alternative expressions for a.s. convergence are (a) {Xn} converges to X a.s. if P ω Ω limn ∞ Xn ω − X ω = 0 = 1; and (b) {Xn} converges to X a.s. if P ω Ω limn ∞ Xn ω X ω = 0. In addition, we may observe briefly the following: 1. The random variables {Xn} and X are all defined on the same probability space and are generally taken to be highly dependent. 2. Almost sure convergence or convergence with probability 1 is the strongest form of convergence in the context of probability theory. In this regard, it is often said that a.s. convergence implies that {Xn} “converges strongly” to X as n ∞ . 3. The definition of a.s. convergence implies that, for any ε > 0, there exists a value Nε such that Xn − X < ε for every n ≥ Nε . Hence, the probability distribution of Xn − X becomes increasingly concentrated about zero as n ∞ , that is, the values of Xn −X approach zero as n ∞ (Figure 2.9). 2.6.2

Convergence in L p ,p > 0

An alternative way to weaken the “sure convergence” concept is to require that {Xn} converges in Lp or converges in pth mean to X as n ∞. That is, for p a positive constant, {Xn} and X are random variables defined on the same probability space, and the pth absolute moments E(|Xn|p) and E(|X|p) of Xn and X, Lp respectively, exist, then {Xn} converges in Lp to X (expressed as Xn X) if Lp E Xn − X p 0 as n ∞. In short, Xn X if limn ∞ E Xn −X p = 0 for finite E(|Xn|p) and E(|X|p).

73

74

2 Mathematical Foundations 2

Xn – X

ε

n

0

–ε

Figure 2.9 P nlim∞ Xn = X = 1 or X

as

X.

For p =1, we have convergence in mean; and for p = 2, we have convergence in mean square or convergence in quadratic mean.5 In short, the sequence {Xn} ms converges to X in mean square (denoted Xn X) if limn ∞ E Xn −X 2 = 0. In fact, a necessary and sufficient condition for the mean-square limit to exist is the Cauchy Criterion for m.s. Convergence: A sequence of random variables {Xn} E Xn 2 < +∞ converges to a random variable X E X 2 < +∞ in mean square if and only if limn, m ∞ E Xn − Xm 2 = 0. We note in passing the following: ms

1. For c R, Xn c if and only if E Xn c and V Xn 0. 2. Mean-square convergence does not imply a.s. convergence; a.s. convergence does not imply mean-square convergence. HR (the Hilbert space of random variables), mean-square conver3. For Xn 0 as n ∞ . In fact, the existence of a gence is equivalent to Xn − X random variable X HR is guaranteed if {Xn} is a Cauchy sequence in HR. Lp Lq 4. For p ≥ q,Xn X implies Xn X. (So for p = 2 and q = 1, we see that convergence in mean-square implies convergence in mean.)

5 More formally, a sequence {Xn} of random variables, with E Xn < +∞, converges in L1 to a random variable X, with E X < +∞, if limn ∞ E Xn −X = 0. (If this limit is zero, then {Xn} is said to “converge strongly” to X.) A sequence {Xn} of random variables, with E Xn 2 < +∞, converges in L2 to a random variable X, with E X 2 < +∞, if limn ∞ E Xn −X implies that {Xn} “converges strongly” to X as n ∞).

2

= 0. (Mean-square convergence

2.6 Convergence of Sequences of Random Variables

5. If limn ∞ E Xn −X 2 = 0 (convergence in L2), then limn ∞ E Xn 2 = E X 2 and limn ∞ E Xn = E X (see also the discussion offered in Section 2.6.5). 2.6.3

Convergence in Probability

Suppose {Xn} and X are random variables defined on the same probability space Ω,A,P . {Xn} converges in probability or converges in measure to X if, for p 0 as n ∞ (written Xn X). In every ε > 0, P ω Ω Xn ω − X ω ≥ ε p p short, Xn X if limn ∞ P Xn − X ≥ ε = 0 for all ε > 0. Equivalently, Xn X if limn ∞ P Xn −X < ε = 1 for each ε > 0. (As this latter expression indicates, Xn −X should be arbitrarily close to zero with probability arbitrarily close to p 1. So if Xn X, then, with high probability, Xn ≈X as n ∞ .) It is important to note the following: 1. Convergence in probability means that there exists a subsequence on which p a.s. convergence occurs, that is, if Xn X, then there exists a subsequence ∞ as Xn n∞= 1 such that Xnj X. Xnj j = 1 p

2. For c R, Xn c if, for all ε > 0, limn ∞ P Xn − c ≥ ε = 0. 3. Consider now the following relationships between our first three types of convergence of a sequence of random variables. (Note that the symbol “ ” means “implies.”): Xn Xn

as ms

as

X X

Xn

p

Xn

p

X generally < X generally
0, limn ∞ P Mn − μ ≥ ε = 0. (If we require that the random variables Xn be only independent instead of i.i.d., then we need the additional restriction that V Xi = σ 2 < +∞ .) Thus, the p averages Mn converge in probability to μ as n ∞ or Mn μ. So as n becomes large, the random variables Xn become more and more concentrated about a common mean value μ; Mn is with high probability very close to μ.

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We next examine a Theorem 2.7.1 that informs us that a large sum of i.i.d. random variables, appropriately normalized, has an approximate standard normal distribution. That is, Theorem 2.7.1 Central Limit Theorem Let {Xn} be a sequence of i.i.d. random variables with E Xi = μ, 0 < V Xi = σ 2 < +∞, and Mn = S n n. If Gn(t) denotes the cumulative distribution function of Zn = n Mn − μ σ and Gn(t) represents the cumulative distribution function of the standard normal variate Z, then for any t, −∞ < t < +∞ , 1 t 2 lim Gn t = G t = e −y2 dy, n ∞ 2π −∞ that is, n Mn −μ σ has a limiting N(y;0,1) distribution. As this theorem indicates, Zn = Sn −nμ σ n = n Mn −μ σ converges in distribution to Z N z;0,1 or Zn d Z N z;0,1 —the normalized average of n i.i.d. random variables with finite variance tends in distribution to a normalized, normally distributed random variable. An alternative expression for the preceding limit is for a, b R −∞ < a < b < +∞ , 2 Sn −nμ 1 b −y lim P a < 0, where the reduced or effective probability space was denoted as

Ω, A, P , with probability measure

P = P P B . In light of this result, we can set E XB =

1 P B

XdP B

for integrable X. Given this expression, let us now look to the calculation of E(X|Y), where X, Y are random variables defined on the same probability space Ω,A, P . That is, if a simple event ω Ω is chosen and Y(ω) is determined, then, given Y(ω), we are interested in finding the value of X(ω). Suppose Y is discrete. Then we can define the expected value of X with respect to the probability measure P Y = y as E X Y =y = =

Ω

X ω dP ω Y = y

1 P Y =y

X ω dP ω Y =y

If X is P-integrable, then the preceding expression is defined for all y such that P Y = y > 0. Clearly, E X Y = y is a particularization of the random variable E(X|Y). Hence, our best estimate of X will be the average value of X over a partition of Ω. To this end, given that Y is discrete, it induces the A-partition of y χ ω is a simple random Ω,Bj = ω Y ω = Yj , j ≥ 1, where Y ω = j ≥ 1 j Bj

variable defined on Ω, A, P with Y = yj on Bj , and the Bjs are disjoint events satisfying j Bj = Ω. Let σ Bj , j ≥ 1 depict the σ-algebra generated by the partition Bj , j ≥ 1, and define E X Y ω = E X Bj if ω Bj and P Bj > 0. If P Y = yj > 0 for all j ≥ 1, set E XY ω =

j≥1

E X Y = yj χ

Y = yj

ω

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2 Mathematical Foundations 2

To summarize, given σ Bj , j ≥ 1 = σ Y , suppose a. E(X|σ(Y)) is σ(Y)-measurable; and b. for event B σ Y , E Xσ Y

ω dP ω =

B

X ω dP ω B

Then the A Y -measurable random variable E(X|Y) can be viewed as E X σ Y = E X A Y —the conditional expectation of X with respect to a σ-algebra. In general, for Ω,A, P a probability space, X an integrable random variable on Ω, and U A a sub-σ-algebra, the conditional expectation E X U is any random variable on Ω such that a. E X U is U-measurable; and E X U dP =

b. B

XdP for all events B U.6 B

Moreover, if Y is any U-measurable random variable satisfying XdP for all B U,

YdP = B

B

then Y = E X U a.s. in Ω,A, P . Any such Y is termed a version of E X U . If U is generated by a random variable Y on Ω, A, P , simply insert E(X|Y) in place of E X U . As the preceding discussion indicates, conditional expectations are not unique. But since any two conditional expectations of X with respect to U are a.s. equal as random variables on Ω, A, P , we can, for all intents and purposes, view conditional expectations as unique. In fact, from this point on, all general results involving conditional expectations should be interpreted as holding a.s. Stated more formally, if X is an integrable random variable on Ω,A, P , then for each sub-σ-algebra U A, the conditional expectation E X U exists and is unique up to U-measurable sets of probability zero. The justification for this assertion is Theorem 2.A.1. Theorem 2.A.1 Radon–Nikodỳm Theorem Let v and μ be finite measures on the measurable space Ω,A with v absolutely continuous with respect to μ (typically written v < 0, P Xt −Xs > ε 0 if t T ,t s. i. A quadratic variation process if Xt ω Ω R is continuous and the second variation of the process, X, X 2t , is defined as X, X

2 t

ω = lim Δtk

0

Xtk + 1 ω −Xtk ω

2

,

tk ≤ t

where 0 = t1 < t2 < < tn = t and Δtk = tk + 1 −tk . An additional feature is worth mentioning. Specifically, a stochastic process Xt t T is termed a Markov process if it exhibits the so-called Markov property—the change at time t is determined only by the value of the process at 1 An alternative (equivalent) statement of this theorem, specified in terms of probability measures defined on the Borel σ-algebra on R, appears in Appendix 3.A.

3.1 Stochastic Processes

time t and not by the values at any times before t. Under this property, to make predictions of the future behavior (outcome) of a system, it suffices to consider only the present state of the system and not its past performance or history. Thus, the current state of a system is critical, but how it arrived at that state is irrelevant. Thus, a Markov process “has no memory” (more on this notion in Section 3.5.2). 3.1.3

Filtrations of A

Let Ω, A, P be a probability space and, for T = N or T = 0, +∞ , let A t = σ Xs ,s T ,s ≤ t depict a σ-algebra of events defined in terms of the outcomes of a stochastic process up to time t, that is, A t represents the history of the process up to time t. Hence, the σ-algebra A t classifies all the information about the process available at time t T and, for a given event A A t , observing the process up to time t enables us to determine if that event has or has not occurred. We shall define A ∞ = σ t T A t —it is the σ-algebra generated by t T A t that encodes our information over all time points t. Clearly, At A ∞ , t T . A filtration on Ω, A, P is an increasing family F = A t t T of sub-σ-algebras of A—it enables us to assess how the process information is arranged (grouped) into a family of σ-algebras. So for s, t T , if A s and A t are σ-algebras included in A and if 0 ≤ s ≤ t < +∞ , then A s A t A. (Observe that a filtration on A is bounded above by A.) To elaborate, if A s and A t are two σ-algebras on Ω, A, P , then A s A t if and only if A t contains more information about the process than does A s ; the A s information set is a subset of the At information set. (Remember that for a discrete-time filtration, T = N; and for a continuoustime filtration, T is an interval in R.) A probability space Ω, A, P endowed with a filtration F is termed a filtered probability space and denoted Ω, A, F, P . Also, a stochastic process Xt t T on Ω, A, P is said to be adapted to the filtration F if, for each t T , Xt is At -measurable (the information at time t carried by A t determines the value of the random variable Xt). Equivalently, Xt t T is adapted to F if, for each A t . Sometimes, an adapted process is called a nonant T ,σ Xs ,s T , s ≤ t ticipating process—a process that “cannot see into the future.” Thus, Xt t T is At -adapted if and only if, for each t T , Xt is known at time t when the information carried by At is known. Given that the filtration F represents the evolution of knowledge about some random system through time, it follows that if Xt t T is an adapted process, then Xt depends only on the systems behavior prior to time t, with no prior information lost. Suppose that the expected value of a random process Xt t T is conditioned on the history of the process up to time s < t, E Xt A s . If E Xt A s is, in turn, conditioned on the process history up to time r < s, then the outcome is the same as if we had conditioned on the earlier time r to begin with. That is, the tower

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property holds or E E Xt A s A r = E Xt A r , r < s < t. This is because the Ar A s or there is less information carried by A r relative to that carried by A s . A stochastic process Xt t T is always adapted to its natural filtration X A t = σ Xs ,s ≤ t —the filtration where A t is generated by all values of Xs up to time s = t. Since A t is the smallest σ-algebra for which all variables Xs ,s ≤ t, are measurable, it follows that F X = A tX t T is the smallest filtration to which Xt t T is adapted. What is the importance of filtrations? The information carried by a stochastic process that evolves over time is modeled by filtrations. If the system is observed up to time t, then one is able to determine if an event A has occurred, that is, if A σ Xs ,s ≤ t , where σ Xs , s ≤ t is the smallest σ-algebra for which all the random variables X t1 , …, X tn ,0 ≤ t1 ≤ ≤ tn ≤ t, are measurable. In fact, if Xt t T is a stochastic process defined on Ω, A, P , then A t = σ A s ,s ≤ t is itself a filtration. Suppose Ω, A, F, P is a filtered probability space. The filtration F = A t t T is complete if the probability space Ω, A, P is complete2 and if A 0 (the collection of all subsets of A that have a zero probability) contains all the P-null sets (i.e., if A A and P(A) = 0, then A A 0 ).3 Moreover, the filtration F is said to be A s , s T ,s > t for all t T . right-continuous if A t = A t + , where A t + = 2 Obviously, A t + is also a filtration and At A t + ,t T . When the probability space Ω, A, P is complete in that A contains all P-null sets (so that the filtration F is complete), then the filtration F is said to satisfy the usual conditions if it is right-continuous and A0 contains all P-null sets. In what follows, we shall always work on a given complete probability space Ω, A, P with a filtration F = A t t T satisfying the usual conditions.4 To reflect the dynamic aspects of a filtration, the concept of progressive measurability is needed. Specifically, a continuous stochastic process Xt t ≥ 0 that is adapted to a filtration F = A t t ≥ 0 is said to be progressively measurable with respect to F if for every t T = 0, +∞ , and for all events A B R , s, ω T × Ω,Xs ω A B T × A t . While a process may be adapted but not progressively measurable, an adapted and continuous stochastic process is inherently progressively measurable. Thus, a continuous stochastic process that is adapted to a filtration F is also progressively measurable with respect to F.

2 In general, a class A of subsets of Ω is complete with respect to a measure μ if E F, F A, and μ(F) = 0 implies E A. 3 A set A Ω is termed a P-null set if there exists a set B A with A B and P(B) = 0. 4 This assumption is not all that restrictive since any filtration can be made complete and rightcontinuous. Given a filtered probability space Ω, A, F, P , we first complete the probability space Ω, A, P and then add to every t+, t T , all the P-null sets. This new filtration, called the augmentation of F, also satisfies the usual conditions. Typically, F is right-continuous from the start.

3.2 Martingales

3.2

Martingales

3.2.1

Discrete-Time Martingales

Suppose F = A t t T is a filtration on a probability space Ω, A, P . Generally speaking, a martingale on a filtered probability space Ω, A, F, P is a sequence Xt t T of integrable random variables on Ω, A, P that are adapted to the filtration F. More specifically, for Xt t∞= 1 a sequence of random variables on a probability space Ω, A, P and with F = At t∞= 1 a sequence of σ-algebras in A, the sequence Xt t T is a discrete-time martingale if a. b. c. d.

A t A t + 1 (the A t form a filtration F = A t t∞= 1 ); Xt is A t -measurable (the Xt’s are adapted to the filtration); E Xt < +∞ (the Xt’s are integrable); and E Xt + 1 A t = Xt a.s. for all t ≥ 1 (the martingale property).5

Clearly, the sequence of random variables Xt t∞= 1 is defined as a martingale relative to the sequence F = A t t∞= 1 of σ-algebras, that is, X is a martingale with respect to the A t ’s. As the martingale property indicates, the average value of Xt + 1 taken over A t is just Xt. Thus, a “fair” or “unbiased” estimate of Xt + 1 relative to A t is Xt; Xt is the best prediction the process can make given the information contained in A t up to time t. (For example, if Xt represents the fortune of a gambler after the tth play of a game of chance and if A t represents the information about the game at that time, then this property tells the gambler that the expected fortune after the next play is the same as the current fortune. Hence, a martingale Xt describes the fortune at time t of a gambler who is betting on a fair game.) Remember that the sequence Xt t∞= 1 of random variables is defined to be a martingale with respect to the filtration F = At t∞= 1 . When the filtration is not mentioned or specified, we may take as the σ-algebra the sets Gt = σ X1 ,…, Xt —Gt is the σ-algebra generated by Xt. In fact, G is the smallest σ-algebra on Ω for which Xt is measurable. Since Gt Gt + 1 (the Gt ’s form a filtration), Xt is Gt -measurable (the Xt’s are adapted to the filtration), and A t A, it follows that Gt A t so that Gt t∞= 1 is a filtration on A Xt Ω, A, P . Furthermore, if the martingale property E Xt + 1 A t = Xt holds, then E Xt + 1 Gt = E E Xt + 1 A t Gt = E Xt + 1 Gt = Xt a s. So for Gt = σ X1 ,…,Xt , we can write E Xt + 1 A t E Xt + 1 X1 , …, Xt = Xt . So with Xt t∞= 1 adapted to Gt t∞= 1 , it follows that Xt t∞= 1 is a martingale on the filtered probability space Ω, A, Gt t∞= 1 , P .

5 Sometimes, this property appears as E Xt As = Xs a.s. for all 1 ≤ s < t < +∞ . That is, if A s is our latest information set, then Xs is known and thus constitutes the best forecast of future values of Xt.

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The following few related points merit our attention: 1. Under the martingale property Xt is a version of E Xt + 1 A t and, with Xt being A t -measurable, the martingale property can be written Xt + 1 dP = A

Xt dP, A A t .

A Xt t∞= 1

2. Suppose is a sequence of independent integrable random variables on the probable space Ω, A, P with E Xt = 0 for all t. If A t = σ X1 ,…,Xt , then At

A for all t; thus, At

∞ t=1

t

is a filtration on Ω, A, P . Let Yt =

X i=1 i

t

for all t. Since Yt ≤ X and each Xi is integrable and A t -measurable, i=1 i it follows that Yt is integrable and A t -measurable. For i ≤ t,Xt + 1 is independent of both Xi and A t so that E Xt + 1 A t = E Xt + 1 = 0. Since Yt +1 = Yt + Xt +1 ,E Yt + 1 A t = E Yt + Xt +1 A t = E Yt A t + E Xt + 1 A t = Yt + 0 = Yt so that Yt t∞= 1 is a martingale adapted to the filtration A t t∞= 1 . Thus, the martingale Yt t∞+ 1 can be viewed as a representation of a sequence of independent integrable random variables on a probability space Ω, A, P . At + 1

3. Given that A t

A t + j , j ≥ 1, for event A A t ,

Xt dP = A

Xt + 1 dP = A

A

of E Xt + j A t Ω

Xt + j dP. With Xt being At -measurable, Xt is a version

=

so that E Xt + j A t = Xt a.s. For A = Ω, E Xt =

E Xt + j A t dP =

Ω

Xt + j dP = E Xt + j , j ≥ 1. Hence, if Xt

∞ t =1

Ω

Xt dP =

is a martin-

gale on a filtered probability space, then E Xt = E Xt + 1 = E Xt + 2 = . 4. Suppose X is an integrable random variable on a probability space Ω, A, P ). If and the A t ’s are nested σ-algebras in A (i.e., A t A t + 1 Xt = E Xt A t , then clearly parts (a)–(c) of the definition of a discrete-time martingale are satisfied. In addition, since E Xt + 1 A t = E E X A t + 1 A t = E X A t = Xt , it follows that Xt t∞= 1 is a discrete-time martingale adapted to the filtration At t∞= 1 . 5. The process X t ,0 ≤ t < +∞ defined by setting X 0 = X0 with X t = X0 + A1 X1 − X0 + A2 X2 −X1 + + An Xt − Xt − 1 ,t ≥ 1, is termed the martingale transform of X t t T by At t T . Clearly, this transform represents a general device for constructing new martingales from old ones, that is, the transform of a martingale is also a martingale. More formally, we have Theorem 3.2. Theorem 3.2 Martingale Transform Theorem Let Xt t T be a martingale adapted to the filtration F = A t t T with a sequence of bounded random variables that are At ,1 ≤ t < +∞

3.2 Martingales

nonanticipating with respect to F. Then the sequence of martingale transforms X t t T is itself a martingale adapted to F. If in the martingale property E Xt + 1 A t = Xt “ = ” is replaced by “ ≤ ” or “ ≥ ,” then Xt t∞= 1 is called a supermartingale or submartingale. In this regard, a stochastic process Xt t∞= 1 that is adapted to the filtration F = At t∞= 1 and such that − Xt t∞= 1 is a submartingale is termed a supermartingale. A stochastic process Xt t∞= 1 adapted to F that is simultaneously a submartingale and a supermartingale must obviously be a martingale. A concept closely allied with a martingale is that of a stopping time. Specifically, a +∞ is termed a stopping time for the filtrarandom variable τ Ω T = N tion F = A t t T if w τ w ≤ t A t for all 0 ≤ t < +∞ . That is, think of a stopping time as a random variable that posits a criterion for determining whether or not τ ≤ t has occurred on the basis of information about A t . For instance, we may view τ as a rule for stopping play when engaging in a gambling game; it is independent of the outcome of a play of a game that has yet to be undertaken. Oftentimes, we are interested in the behavior of a random process Yt t T exactly at the stopping time τ. If τ < +∞ a.s., then the stopped process Yτ ∞ is defined by setting Yτ = χ Y . We note briefly that if Xt t T is a k =0 τ=k k martingale adapted to F, then the stopped process martingale adapted to F.

Xmin t, τ

t T

is also a

3.2.1.1 Discrete-Time Martingale Convergence

We noted previously in Section 3.1 that if Xt t∞= 1 is a martingale on the probability space Ω, A, P and adapted to the filtration F = A t t∞= 1 , then Xt ω t∞= 1 is a sample path for each ω Ω. A question that naturally arises is “Does the sequence of sample paths become less erratic over time?” That is, “Does limt ∞ Xt ω = Z exist, where Z is some long-run stable value?” To answer these questions, let us first develop some conceptual material. To this end, suppose L1 Ω, A, P denotes the space of integrable random variables (see Section 1.11.3 and Equations (1.20) and (1.20.1)). Next, a collection of integrable random variables Xk k K is L1-bounded if supk K E Xk < +∞. Finally, if Xt t∞= 1 is a sequence of integrable random variables and there exists an integrable random variable Z such that limt ∞ Xt − Z = 0, then the sequence Xt t∞= 1 converges to Z in L1 Ω, A, P . Based on these considerations, we can now state Theorem 3.3. Theorem 3.3 If Xt t∞= 1 is an L1-bounded martingale on a probability space Ω, A, P , then there exists an integrable random variable Z such that limt ∞ Xt = Z a.s.

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To strengthen the conditions for convergence of a martingale sequence Xt t∞= 1 , let us consider the concept of uniform integrability. (Uniform integrability is used to prove convergence in L1.) Specifically, a sequence of integrable random variables Xt t∞= 1 on a probability space Ω, A, P is uniformly integrable if either a b

lim

m

Xt

∞

sup E Xt ; Xt > m = lim t

∞ t = 1 is bounded on

m

∞

Xt dP = 0;6 or

sup t

Xt > m

L1 Ω, A, P and for all ε > 0 there exists a δ > 0 Xt dP < ε for all t

such that, for an event A A, P A < δ implies A

33 (As an example, think of a collection of random variables Xt dominated by an integrable random variable Y, that is, Xt < Y , where E Y < +∞.) So if uniform integrability is effected, then Theorem 3.3 can be strengthened a bit, as Theorem 3.4 reveals. Theorem 3.4 Let Xt t∞= 1 denote a martingale on the probability space Ω, A, P that is adapted to the filtration A t t∞= 1 . If Xt t∞= 1 is uniformly integrable, then (1) there exists an integrable random variable Z on Ω, A, P such that Xt Z a.s. in L1 Ω, A, P as t ∞; and (2) Xm = E Z A m a.s. in Ω, A, P .7 If X is an integrable random variable, then a characteristic of the conditional expectation of X is provided by Theorem 3.6. Theorem 3.6 If Ω, A 0 , P is a probability space and X L1 , then E X A A a σ-algebra A o is uniformly integrable. More broadly, if X L1 , and the A t are arbitrary σ-algebras within A o , then the random variables E X A t are uniformly integrable. Additional equivalent convergence results for a martingale sequence Xt are the following:

∞ t=1

i. If Xt X in L1 , then Xt = X A t . ii. Xt is uniformly integrable. 6 Note that the sequence of sets Xt > m m∞= 1 decreases to Ø. Also, an m chosen large enough renders (a) to supt E Xt ≤ m + 1 < +∞. 7 The concept of uniform integrability is also useful when discussing the convergence of sequences of random variables that are not necessarily martingales, for example, Theorem 3.5 Let Xt t T be a sequence of integrable random variables with Z an integrable random variable. Then the sequence Xt t T converges to Z in L1 (i.e., limt ∞ E Xt −Z = 0 if and only if Xt t T is uniformly integrable).

3.2 Martingales

iii. iv. v. vi.

Xt converges a.s. and in L1. Xt converges in L1. There is an integrable random variable Z such that Xt = E Z A t . Suppose A t A ∞ (i.e., A t is an increasing sequence of σ-algebras and the union t∞= 1 A t generates the σ-algebra A ∞ ). Then as t ∞ , E X A t E X A ∞ a.s. and in L1.

In fact, to amplify (2): if F = A t t∞= 1 is a filtration on the probability space Ω, A, P and Z is an integrable and At -measurable random variable, then the process E Z A t t∞= 1 is a martingale with respect to F. We next offer a particularization of the dominated convergence theorem (Section 1.12.5). Specifically, we state Theorem 3.7. Theorem 3.7 Dominated Convergence Theorem for Conditional Expectations Suppose Xt t∞= 1 is a martingale on a probability space Ω, A, P . Let Xt X a.s. E X A ∞ a.s. and Xt ≤ Z for all t, with E Z < +∞. If At A ∞ , then E Xt A t In closing this portion of our discussion of martingales, we present a theorem that will be particularly useful in Chapter 4. But first a definition. The martingale Xt t ≥ 0 is said to be L2-bounded if E Xt 2 ≤ B < +∞ ,B R, for all t ≥ 1. Under this condition, Xt t ≥ 0 converges a.s. or with probability 1 as an L2 sequence. More formally, we state Theorem 3.8. Theorem 3.8 L2-bounded Martingale Convergence Theorem Suppose Xt t ≥ 0 is a martingale that is L2-bounded for all t ≥ 0. Then there exists a random variable X, with E Xt 2 < B, B R, such that P limt ∞ Xt = X = 1 and limt ∞ Xt −X 2 = 0. Next follows some convergence results for submartingales and supermartingales. We state first Theorem 3.9. Theorem 3.9 Let Xt t∞= 1 be a submartingale on the probability space Ω, A, P (Doob, 1953). If k = supt E Xt < +∞, then Xt X a.s., where X is a random variable satisfying E X < k. For Xt

∞ t=1

a submartingale, the following results are equivalent:

i. Xt is uniformly integrable. ii. Xt converges a.s. and in L1. iii. Xt converges in L1.

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A special case of Theorem 3.9 is provided by Theorem 3.10. Theorem 3.10 If Xt > 0 is a supermartingale on the probability space Ω, A, P , then, as t ∞ , Xt X a.s. and E X ≤ E Xo .

3.2.2

Continuous-Time Martingales

We now turn our attention to continuous-time martingales. Specifically, for At t ≥ 0 a filtration on the probability space Ω, A, P and Xt t ≥ 0 a collection of integrable random variables that are A t -measurable (i.e., Xt t ≥ 0 is adapted to the filtration), Xt t ≥ 0 is a continuous-time martingale if E Xt A s = Xs for all 0 ≤ s ≤ t. It is important to note that, under an appropriate conversion, a discretetime martingale can be used to generate some of the preceding results for a continuous-time martingale. For example (Dineen, 2013), suppose Z is an integrable random variable on the probability space Ω, A, P and A t t ≥ 0 is a filtration on Ω, A, P . Let E Z A t = Xt for all t. For 0 ≤ s ≤ t, E Xt A s = E E Z A t A s = E Z A s = Xs so that Xt t ≥ 0 is a martingale on Ω, A, P . Conversely, let Xt t ≥ 0 be a martingale adapted to the filtration A t t ≥ 0 and suppose there exists a strictly increasing sequence of real numbers tn n∞= 1 , with tn ∞ as n ∞, such that Xtn n∞= 1 is uniformly integrable. (Given that Xt t ≥ 0 is a continuous-time martingale, Xtn n∞= 1 is a discrete-time martingale for tn ∞ as n ∞.) By Theorem 3.4, there exists an integrable random var0 as n ∞ and E Z A tn = Xtn . If tn > t > 0 iable Z such that E Xtn Z E Z A t = E E Z A tn A t = E Xtn A t = Xt . Then it can be shown that limt ∞ E Xt − Z = 0 and that (3.3a) holds. Continuous-time martingales also admit stopping times. That is, given +∞ is a stopping time adapted to a filtration F = At t ≥ 0 , τ Ω R A t for all t ≥ 0. In addition, if F provided that the set w Ω τ w ≤ t Xt t 0, +∞ is any collection of random variables, the stopped variable Xτ on w Ω τ w < +∞ is defined by setting Xτ w = Xt w for τ w = t. We next have Theorem 3.11, an extension of Doob’s stopping time theorem in discrete time to continuous time. That is, Theorem 3.11 Doob’s Continuous-Time Stopping Theorem (Doob, 1953). Let Xt t ≥ 0 be a continuous martingale adapted to the filtration F = A t t ≥ 0 that satisfies the usual conditions (see Section 3.1.3). If τ is a stopping time for F, then the process Yt t ≥ 0 = Xmin t, τ t ≥ 0 is also a continuous martingale adapted to F.

3.2 Martingales

3.2.2.1 Continuous-Time Martingale Convergence

We briefly look to the issue of the convergence of continuous-time martingales. To facilitate this discussion we first examine a couple of requisite definitions pertaining to the sample paths of a martingale. In particular, a path Xt(ω) is said to be right-continuous if for almost all ω Ω the function Xt(ω) is rightcontinuous for t ≥ 0; it is said to be left limit if for almost all ω Ω the left limit lims t Xs ω exists and is finite for all t ≥ 0. Given these considerations, we now offer the following important convergence theorem. Specifically, we have Theorem 3.12. Theorem 3.12 Doob’s Convergence Theorem (Doob, 1953) Let F = A t t ≥ 0 be a filtration on a probability space Ω, A, P and let Xt t ≥ 0 be a martingale with respect to the filtration F whose paths are right-continuous and left limit. Then the following are equivalent: a. Xt t ≥ 0 converges in L1 when t ∞ . b. As t ∞ , Xt t ≥ 0 converges a.s. to an integrable and At -measurable random variable Z that satisfies Xt = E Z A t , t ≥ 0. c. Xt t ≥ 0 is uniformly integrable. 3.2.3

Martingale Inequalities

For every martingale Xt t T , the process Xt p , p ≥ 1, is a submartingale whenever Xt L p . (The process − Xt p , p ≥ 1, is a supermartingale if Xt L p .) For a real-valued martingale, Xt , Xt+ = max Xt , 0 and Xt− = max − Xt ,0 are submartingales. In this regard, the following martingale estimates are useful for forming approximations: 1. Discrete-Time Martingale Inequalities a If Xt

1 then P max Xk ≥ λ ≤ E Xt+ 1≤k ≤t λ

∞ t = 1 is a submartingale

holds for any λ > 0, t ≥ 1 b If Xt

∞ t = 1 is a martingale with

E max Xk 1≤k ≤t

p

E Xt ≤

p

p p −1

< +∞ and 1 < p < +∞ , then p

E Xt

p

,t ≥ 1 34

2. Continuous-Time Martingale Inequalities Suppose Xt t ≥ 0 is a stochastic process with continuous sample paths a.s.

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3 Mathematical Foundations 3

a If Xt

t ≥ 0 is a submartingale

then

1 P max Xs ≥ λ ≤ E Xt+ 0≤s≤t λ holds for allλ > 0,t ≥ 0 b If Xt

t ≥ 0 is a martingale

35 with E Xt

E max Xs 0≤s≤t

p

≤

p

p p−1

< +∞ and 1 < p < +∞ , then p

E Xt

p

,t ≥ 0

(Note that (3.4a) is a generalization of Chebyshev’s inequality.)

3.3

Path Regularity of Stochastic Processes

While the Kolmogorov theorem (see Section 3.1.1 and Appendix 3.A) addresses the issue of the existence of stochastic processes, it does not reveal anything about the paths of such processes. The Kolmogorov continuity theorem given later states that, under some rather mild conditions, we can employ processes whose paths are fairly regular. To start, we define a function f R + R as Hölder with exponent γ (or γ-Hölder) if there exist constants c,γ > 0 such that f t − f s ≤ c t − s γ for s, t R + . A function that satisfies this inequality is termed a Hölder function. Such functions are continuous. (For f a continuous 0 as s t. Hölder continuity enables us to determine function, f t − f s the rate of this convergence, that is, f t − f s 0 as s t at least as fast γ 0.) as t − s Given these considerations, we may now state Theorem 3.13. Theorem 3.13 Kolmogorov Continuity Theorem For real α,ε, c > 0, if the stochastic process Xt t 0, 1 defined on a probability

space Ω, A, P satisfies E Xt − Xs α ≤ c t −s 1 + ε for s, t 0, 1 , then there exists a modification (or version) of Xt t 0, 1 that is a continuous process and whose paths are γ-Hölder for every γ 0,ε α . As just indicated, it is possible to obtain versions of stochastic processes whose paths are highly regular. The Kolmogorov continuity theorem provides a sufficient condition that allows us to work with continuous versions of stochastic processes. But for martingales, the possibility of the existence of regular versions depends upon the regularity properties of the filtration with respect to which the martingale property holds. In this regard, let F = At t ≥ 0

3.4 Symmetric Random Walk

be a filtration on a probability space Ω, A, P and let the following assumptions hold: 1. If event A A satisfies P A = 0, then every subset of A is in A0 . 2. The filtration is right-continuous (i.e., for every t ≥ 0, A t ε > 0 A t + ε ). Then the filtered probability space Ω, A, F, P is said to satisfy the usual conditions (also see Section 3.1.3). Armed with these regularity considerations, we can now state Theorem 3.14. Theorem 3.14 Doob’s Regularization Theorem (Doob, 1953) Let Ω, A, At t ≥ 0 , P be a filtered probability space that satisfies the usual conditions and let Xt t ≥ 0 be a supermartingale with respect to the filtration F = A t t ≥ 0 . Assume the function t ↦E Xt is right-continuous. Then there exists a modified process X t t ≥ 0 of Xt t ≥ 0 with the following properties: 1. X t t ≥ 0 is adapted to the filtration F. 2. The paths of X t t ≥ 0 are locally bounded, right-continuous, and left limited. 3. X t t ≥ 0 is a supermartingale with respect to the filtration F.

3.4

Symmetric Random Walk

Let Xt t∞= 1 denote a sequence of independent random variables on the probability space Ω, A, P , where Ω = − 1 1 , A = 2Ω , and Xt assume only the value ± 1 with probability 1/2, that is, P 1 = 1 2 = P − 1 . It is easily verified that E Xt = 0, V Xt = 1. This sequence is termed a symmetric random walk since the indicated probabilities are each 1/2. Why the nomenclature “random walk?” We may consider the outcomes of this random experiment as a sequence of steps, each determined at random and independent of previous steps, taken in either the forward or backward direction, for example, if Xt = 1, a step forward is taken; and if Xt = − 1, a step backward is taken. To view the outcomes of this experiment as constituting a random process, let Ω = 2n , A be the σ-algebra generated by the sequence Xt t∞= 1 of independent random variables, and P be a probability measure8 (specified above). Then Xt t∞= 1 is defined on the probability space Ω, A, P . If we set 8 Actually, P is a product (probability) measure. That is, let Ωi , Ai , Pi , i = 1,2, be two probability spaces and let Ω = Ω1 × Ω2 and consider A 1 A 2 = σ A1 × A2 as the product σ-algebra on Ω. If we define P on the class ε = A1 × A2 Ai A i , i = 1 2 as P A1 × A2 = P1 A1 P2 A2 , then P can be extended to a unique probability measure on σ ε = σ A1 × A 2 . Then the product probability measure P = P1 P2 is the probability measure on σ A1 × A 2 .

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3 Mathematical Foundations 3 t

Yt = X ,t ≥ 1, then the sequence Yt i=1 i ∞ with Xt t = 1 , is a martingale.

∞ t = 1 , a partial sum process associated t

To verify the martingale assertion, let Yt = X , A t = σ X1 …Xt , i=1 i t ≥ 1, with Yt A t , E Yt < +∞ , and Xt + 1 be independent of A t . (Note that A t is the smallest filtration to which Yt is adapted.) Since conditional expectation is linear (see Appendix 2.A) and Yt + 1 = Yt + Xt + 1 , it follows that E Yt + 1 A t = E Yt A t + E Xt + 1 A t = Yt + E Xt + 1 = Yt so that Yt t∞= 1 is a martingale with respect to the filtration A t t∞= 1 .

3.5

Brownian Motion

3.5.1

Standard Brownian Motion

A one-dimensional standard Brownian motion (SBM) process (or Wiener process) is a stochastic process Wt t ≥ 0 on a probability space Ω, A, P with the following properties: i The SBM process begins at t0 = 0 or P W0 = 0 = 1 ii For 0 = t0 ≤ t1 ≤

≤ tk , the increments displacements

W t1 ,W t2 − W t1 , …, W tk −W tk −1 are independent random variables iii For 0 ≤ s < t, the increments Wt − Ws P Wt −Ws

B =

N 0, t −s , that is, 2 −x e 2 t − s dx

1 2π t − s

B

36 t −sN 0, 1 . where B is a Borel subset of R. Equivalently, Wt − Ws Property (3.6i) is a convention. To determine the position of a Brownian particle in one dimension, we start at t = 0, with the initial position specified as W0 = 0. Property (3.6ii) indicates that the increments W t1 ,W t2 −W t1 , …, W tk −1 −W tk −2 occurring during the time intervals t0 ,t1 , …, tk − 2 ,tk −1 , respectively, do not affect the increment W tk − W tk − 1 that obtains during the time interval tk −1 , tk , that is, the SBM process is assumed to be without memory. (For instance, the path of a pollen particle traverses in order to get to its current position does not influence its future 9

9 The SBM process just defined is the namesake of the Scottish botanist Robert Brown (1828), who first described the phenomenon of pollen particles moving erratically while suspended in water. This led Brown to conclude that the particles exhibited a “seemingly random movement.” It was later explained that the pollen particles were being impacted by the rapidly moving water molecules.

3.5 Brownian Motion

location.) Property (3.6iii) indicates that (a) Wt − Ws ,0 ≤ s < t, has a zero mean (if we think of Wt as the height above a horizontal time-axis of pollen particles at time t, then a zero mean indicates that, at time t + 1, the particle’s height is just as likely to increase as it is to decrease, with no upward or downward drift); and (b) the variance t − s of an SBM process increases with the length of the time interval [s, t] (the pollen particle moves away from its position at time s, and there is no tendency for the particle to return to that position, that is, the SBM process lacks any propensity for position reversion). Do SBM processes satisfying these properties exist? The answer is a definite yes, that is, there exists a probability space Ω, A, P and a stochastic process Wt t ≥ 0 defined on this space such that Properties (3.6i)–(3.6iii) hold. In fact, these properties lead to a consistent set of finite-dimensional distributions via the Kolmogorov existence theorem (Section 3.1.1 and Appendix 3.A). In this regard, for 0 < t1 < < tk , let μt1 , …, tk be the distribution function of S1 ,…, Sk

R k , where Si =

i

X and X1 , …, Xk j=1 j

are independent, normally dis-

tributed random variables with zero means and variances t1 ,t2 − t1 , …, tk − tk − 1 , respectively. Then it can be shown (e.g., Billingsley, 2012) that for 0 < t1 < < tk , the μt1 , …, tk satisfy the consistency conditions of the Kolmogorov existence theorem; thus, there exists a stochastic process Wt t ≥ 0 corresponding to μt1 , …, tk . In sum, if W0 = 0, then there exists on the probability space Ω, A, P a process Wt t ≥ 0 whose finite-dimensional distributions possess Properties (3.6i)–(3.6iii). The importance of an SBM process {Wt} is that it serves to represent the cumulative effect of process noise. That is, if Ws and Wt ,0 ≤ s < t, mark the position of the process at times s and t, respectively, then the increment Wt −Ws reflects pure noise over the interval [s, t]. It is important to note that a slight modification of the preceding definition of an SBM process can also be offered—one that explicitly incorporates the notion of a filtration. Specifically, let Ω, A, P be a probability space with filtration At t ≥ 0 . A one-dimensional SBM is an A t -adapted process Wt t ≥ 0 with the following properties: i P W0 = 0 = 1 ii For 0 ≤ s < t, the incrementWt − Ws is independent of A s

37

iii For 0 ≤ s < t, the increment Wt −Ws is N 0, t −s So given the filtration A t t ≥ 0 for 0 ≤ s < t, there is at least as much information available at time t as there is at time s. Thus, A s A t or information accumulates. Also, the information available at time t is sufficient to evaluate Wt at time t is A t -measurable. Also, finally, for 0 ≤ t < u, the future increment Wu − Wt is independent of At —any increment in the SBM process subsequent to time t is independent of the information existing at that time.

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3 Mathematical Foundations 3

While the filtration At t ≥ 0 is a key element of this alternative specification of the properties (3.7) of an SBM process, we can readily incorporate a specialized type of filtration into our discussion of SBM even if (3.6) holds. To this end, let A tw = σ Ws ,0 ≤ s < t represent the σ-algebra generated by Ws 0 ≤ s < t so that A tw

t ≥0

is the natural filtration generated by Wt

t ≥ 0.

Here, Wt

under (3.6) is an SBM process with respect to the natural filtration A tw

t≥0 t≥0

.

(If A t A t , t ≥ 0, and Wt − Ws is independent of A s , 0 ≤ s < t, then Wt t ≥ 0 is an SBM process with respect to the filtration A t t ≥ 0 .) Suppose Wt t ≥ 0 is an SBM process defined on a probability space Ω, A, P and let Ω, A, P denote the completion of Ω, A, P . Thus, Wt t ≥ 0 is an SBM process on the complete probability space Ω, A, P . For N the collection of P-null sets (see Section 3.1.3), let A t = σ A tw N ,t ≥ 0, depict the augmentaw

tion with respect to P of

Atw

t ≥0

. Since

At

t ≥0

is a filtration on

Ω, A, P satisfying the usual conditions (Section 3.1.3), it follows that Wt t ≥ 0 is an SBM process on Ω, A, P with respect to A t t ≥ 0 . So given an SBM process defined on a probability space Ω, A, P , one can construct a complete probability space with a filtration satisfying the usual conditions. In this regard, unless otherwise stated, we can assume that we have an SBM process defined on a complete probability space Ω, A, P with a filtration A t t ≥ 0 satisfying the usual conditions. Looking to some of the salient features of SBM processes we have the following: 1. The increments of the SBM process are stationary, that is, the distribution of Wt − Ws , 0 ≤ s < t, depends only upon the difference t − s. (By virtue of Properties (3.6i and iii), the distribution of Wt − Ws is characterized by noting that Wt N 0, t .) 2. For 0 ≤ s < t, it is readily determined that E Wt = 0, E Wt 2 = V Wt = t (SBM is a quadratic variation process with quadratic variation equal to t), and, via property (3.6ii), E Ws Wt = E Ws Wt − Ws + E Ws 2 = E Ws E Wt − Ws + E Ws 2 = s. In general, E Ws Wt = min s, t . 3. If we represent the value of Wt at ω Ω by t ↦W t, w , then the sample path functions of SBM processes can be denoted as W , w . For each w, W(0,w) = 0 and, a.s., t ↦W t, w is continuous in t. More specifically, for every ω Ω, the sample path t ↦ W t, w is Hölder-continuous for each 0 < γ < 1 2 ; it is nowhere Hölder-continuous for any γ > 1 2 . 4. If the SBM process Wt t ≥ 0 is measurable, then each sample path function W , w is A-measurable.

3.5 Brownian Motion

5. SBM process sample paths W , w , ω Ω, are highly irregular and, a.s., are of unbounded variation.10 6. Sample paths of SBM processes are, a.s., nowhere differentiable (see Appendix 3.B). 7. Symmetry property: the process − Wt t ≥ 0 is an SBM. 8. Scaling property: for every real c > 0, the SBM process Wct t ≥ 0 follows the cWt t ≥ 0 . same probability law as the SBM process 9. Time-inversion property: a.s., limt 0 Wt t = 0, that is, the SBM process tW1 t t ≥ 0 follows the same probability law as the process Wt t ≥ 0 . 10. For any h > 0, the process Wt + h − Wt t ≥ 0 is an SBM. 11. If Wt t ≥ 0 is an SBM process, then Wt N 0, t and, for real numbers a and b, a ≤ b, the probability that the sample path assumes values between a and b at time t is P a ≤ Wt ≤ b =

1 2πt

b a

and for the function f R E f Wt

=

1 2πt

− x2

e 2t dx; R,

+∞ −∞

− x2

f Wt e 2t dx

(Note that if f = χ a, b x , where χ a, b x =

1, x

a, b ;

0, x

a, b ,

then the preceding expression obtains.) 12. An SBM process is a continuous-time martingale. That is, if Wt t ≥ 0 is an SBM process and At = σ Ws ,0 ≤ s < t , then, for Wt = Wt − Ws + Ws ,E Wt A s = E Wt − Ws A s + E Ws A s = E Wt − Ws + Ws = 0 + Ws (since Wt − Ws is independent of A s and Ws is A s -measurable) and E Wt – Ws 2 A s = t – s a.s. since the increment Wt − Ws is independent of the past and normally distributed with mean 0. Conversely, let Wt t ≥ 0 be a stochastic process with A t t ≥ 0 an increasing family of σ-algebras such that Wt is A t -measurable and the preceding two equalities hold a.s. for 0 ≤ s < t. Then Wt t ≥ 0 is an SBM process. In fact, since an 10 A right-continuous function f sup

k j=1

variation.

0, t

R is a function of bounded variation if

f tj −f tj −1 < +∞. If the supremum is infinite, then f is said to be of unbounded

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3 Mathematical Foundations 3

SBM process Wt t ≥ 0 has independent increments, it is actually a square integrable martingale. The preceding discussion on SBM processes has focused on what we called standard BM (SBM). A more general BM process has us modify (3.6iii) to the following: for 0 ≤ s < t, Wt − Ws is normally distributed with E Wt −Ws = μ t −s and E Wt −Ws 2 = σ 2 t −s ,σ 2 0, where μ (called the drift coefficient) and σ 2 (the variance coefficient) are real constants. If μ = 0 and σ 2 = 1, we have a normalized BM process. In fact, for any BM process Wt t ≥ 0 with drift μ and variance σ 2, i. Wt t ≥ 0 is a BM process with drift −μ and variance σ 2; ii. for constants a, b > 0, aWbt t ≥ 0 is a BM process with drift abμ and variance a2bσ 2; and iii. Wt −Ws − μ t −s σ is a normalized BM process. We close this section by noting that a d-dimensional random process Wt = Wt1 ,…, Wtd t ≥ 0 is a d-dimensional SBM process if every Wti t ≥ 0 is a one-dimensional SBM process and the Wti t ≥ 0 are independent (or the σ-algebras σ Wti , t ≥ 0 are independent), i = 1,…,d. In this regard, if Wt = Wt1 ,…, Wtd t ≥ 0 is a d-dimensional SBM process, then for i,j = 1, …, d, i. E Wt – Ws A s = 0, 0 ≤ s < t < +∞; ii. E Wti Wsj = min t, s δij ,1 ≤ i, j ≤ d, where δij =

1,i

j;

0,i

j

is the Dirac delta function. iii. E

Wti −Wsi

j

Wt − Wsj

= t − s δij ,0 ≤ s < t.

In addition, a d-dimensional SBM process is a continuous martingale. That is, let Xt = Xt1 ,…, Xtd t ≥ 0 be a d-dimensional martingale with respect to the filtration A t t ≥ 0 , with X0 = 0 a.s., and having joint quadratic variations X i ,X j t = tδij ,1 ≤ i, j ≤ d. Then Xt = Xt1 ,…, Xtd t ≥ 0 is a d-dimensional SBM process with respect to A t t ≥ 0 . 3.5.2

BM as a Markov Process

Let Xt t ≥ 0 be a stochastic process defined on a probability space Ω, A, P . The σ-algebra A t = σ Xs ,0 ≤ s < t , t ≥ 0, is the history of the process up to and including time t. (At records the information available from our observation of Xs for all times 0 ≤ s < t.) In addition, a real-valued, A t -adapted stochastic

3.5 Brownian Motion

process Xt t ≥ 0 is called a Markov process if the Markov property P Xt B A s = P Xt B Xs holds a.s. for all 0 ≤ s ≤ t < +∞ and all Borel sets B R. So given Xs, one can predict the probabilities of future values Xt just as well as if you knew the entire history of the process prior to time s. The process only knows Xs and is not aware of how it got there so that the future depends on the past only through the present, that is, once the present is known, the past and future are independent. Looked at in another fashion, Xt t ≥ 0 is a Markov process if for any 0 ≤ s < t, the conditional distribution of Xt + s A t is the same as the conditional distribution of Xt + s Xt, that is, P Xt + s ≤ y A t = P X t + s ≤ y X t a s The transition probability of the Markov process is a function P(x, s; B, t) defined on 0 ≤ s ≤ t < +∞, x R, with the following properties: 1. For every 0 ≤ s ≤ t < +∞ , P Xs , s; B, t = P Xt B Xs . 2. For every 0 ≤ s ≤ t < +∞ and x R, P x, s; , t is a probability measure on the family of Borel sets B. 3. For every 0 ≤ s ≤ t < +∞ and B B, P , s; B, t is Borel measurable. 4. For every 0 ≤ s ≤ r ≤ t < +∞, x R, and B B, P x, s; B, t =

P y, r ; B, t P x, s;dy, r R Chapman–Kolmogorov equation

(That is, a single-step transition probability can be expressed in terms of a combination of two-step transition probabilities with respect to an arbitrary intermediate time r.) In terms of the preceding transition probability, the Markov property becomes P Xt

B A s = P Xs , s; B, t ;

and thus we can write P Xt

B Xs = x = P x, s; B, t ,

the probability that the process will be in set B at time t given that the process was in state x at time s ≤ t. A Markov process Xt t ≥ 0 is termed homogeneous (with respect to t) if its transition probability P(x, s; B, t) is stationary, that is, P x,s + u; B, t + u = P x, s; B, t , 0 ≤ s ≤ t < +∞, x R, and B B. In this circumstance, the transition probability is a function of x, B, and the difference t − s since P x, s; A, t = P x,0; B, t − s . Under this observation, we can simply write P x, 0; B, t = P x; B, t .

105

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3 Mathematical Foundations 3

The process Xt t ≥ 0 is said to be a strong Markov process if the following strong Markov property holds: for any bounded Borel-measurable function ξ R R, any finite A t stopping time τ, and t ≥ 0, E ξ X τ + t A τ = E ξ X τ + t Xτ In terms of the transition probability, the strong Markov property can be written as P Xτ + t

B A τ = P Xτ , τ; B, τ + t

While a strong Markov process is a Markov process, the converse of this statement does not generally hold. To relate the notion of a Markov process to our current discussion, we note the following: if Wt t ≥ 0 is an SBM process, then Wt t ≥ 0 is a Markov process with stationary transition probability P Wt

B Ws =

1 2π t −s

e

− x −Ws 2 2 t −s

dx

B

a.s., 0 ≤ s ≤ t < +∞ , for Borel sets B R. Moreover, this process is homogeneous with respect to its state space (since W0 = 0) and time and, via Theorem 3.13 (the Kolmogorov continuity theorem) can be chosen so that it has continuous sample paths with probability 1. It is also true that {Wt} is a strong Markov process. 3.5.3

Constructing BM

3.5.3.1 BM Constructed from N(0, 1) Random Variables

Let us first consider the construction of an SBM process from countably many standard normal random variables (Ciesielski, 1961; Evans, 2013; Lévy, 1948; McKean, 1969). This construction will verify that an SBM process actually exists. (Section 3.5.3.2 offers a construction of an SBM process based upon the notion of the limiting behavior of a collection of symmetric random walks.) It was determined earlier in Section 1.11.3 that, for functions f t , g t H = L2 0, 1 , the norm and inner product were defined, respectively, as 1 2

f =

f t dt

1 2

and

0 1

f,g =

f t g t dt 0

In addition, a countable set of functions Φ = Ø 1 , Ø 2 ,… was defined as orthonormal if Ø i = 1 for all i; and Ø i , Ø j = 0, i j Moreover, it can be demonstrated that Ø i is a complete orthonormal basis for H.

3.5 Brownian Motion

If {X1, X2, …} is a sequence of independent, identically distributed N(0, 1) random variable defined on a probability space Ω, A, P , then for n = 1, 2 …, define n

t

Wtn =

Xi i=1

Ø i s ds

38

0

For each t ≥ 0, Wtn is a Cauchy sequence in L2 Ω, A, P whose limit Wt is a N(0, t) random variable. This being the case, the remaining issue of Wt having continuous sample paths will now be addressed. While (3.8) holds for an arbitrary complete orthonormal basis, let us now be a bit more specific. To this end, let us define the Haar functions h0 , hj, n , j = 1 2, …,2 n−1 ; n = 1 2, … as h0 t = 1 and with k = 2j − 1, 2 n− 1

2

−2 n −1

hj, n =

, k − 1 2n ≤ t ≤ k 2n ; 2

, k 2n < t ≤ k + 1 2n ;

39

0, elsewhere (Figure 3.2a). These Haar functions constitute a complete orthonormal basis in L2[0, 1]. Moreover, when the Ø i in (3.8) are Haar functions, it follows that Wtn Wt uniformly a.s. In this regard, let us define the Schauder functions {S0, Sn,j} as the indefinite integrals of the Haar functions or t

S n, j =

3 10

hj, n s ds 0

Figure 3.2 (a) The Haar function hj,n and (b) the Schauder function Sn,j(t).

(a) 2(n – 1)/2 t k– 1 2n

k+ 1 2n

k 2n

–2(n – 1)/2

(b) 2–(n + 1)/2 t k– 1 2n

k 2n

k+ 1 2n

107

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3 Mathematical Foundations 3

so that S0 t = t, and S n, j t =

2 n− 1

2

t − k −1 2n , k −1 2n ≤ t ≤ k 2n ;

2 n− 1

2

k + 1 2n − t , k 2n ≤ t ≤ k + 1 2n

3 11

0, elsewhere (Figure 3.2b). Next, for X0 ,Xn, j , n = 1 2,…; j = 1 2,…,2 n− 1 a collection of independent N (0, 1) random variables defined on the probability space Ω, A, P , and for t 0, 1 , N = 1, 2, …, and ω Ω, define (via (3.8)–(3.11)) N

Wtn ω = X0 S0 +

Yn t, ω ,

3 12

n=1

where 2 n −1

Yn t, ω =

Xn, j ω Sn, j t j=1

Now, for each N and ω Ω, the sample path function t ↦ W N t, ω is continuous. The justification that WtN ω has a limiting continuous-path process Wt is provided by Theorem 3.15. Theorem 3.15 The sequence {WtN ω } defined by (3.12) converges uniformly a.s. as N ∞ to a stochastic process Wt with continuous sample paths. Moreover, Wt is an SBM for t 0, 1 . The gist of the preceding discussion is that, in general, we can construct an SBM process on a probability space on which there exists countably many independent standard normal random variables. That is, suppose we have countably many independent N(0, 1) random variables Xn n∞= 1 defined on the probability space Ω, A, P . Then there exists a one-dimensional SBM process W(t, ω) defined for t ≥ 0, ω Ω. 3.5.3.2 BM as the Limit of Symmetric Random Walks

What sort of random behavior can be used to generate (model) an SBM process? Our approach to framing the answer to this question, and to verify that SBM processes exists, is to look to the Donsker’s invariance principle which states that an SBM process may be constructed as the limit of appropriately rescaled random walks. To see this, let Xi i ≥ 1 be a sequence of independent random variables defined on the probability space Ω, A, P and which satisfy P Xi = 1 = P Xi = −1 = 1 2. In addition, for n ≥ 1, let Yn =

n

X i=1 i

(our

Appendix 3.A Kolmogorov Existence Theorem: Another Look

position after n trials or steps); and for n ≥ 0, let A n = σ Y0 ,Y1 ,…,Yn . Then the sequence Yn n ≥ 0 , Y0 = 0, is a symmetric random walk on the set of integers. Next, the sequence of processes Ytn t 0, 1 , n N, where Ytn = n

t−

k k +1 k k +1 − t YK , ≤ t ≤ , Yk + 1 + n n n n

is a stochastic process that represents a piecewise continuous linear interpolation of the rescaled discrete sequence Yn n ≥ 0 . Given this construction, we can now conclude our argument with Theorem 3.16. Theorem 3.16

The sequence

Ytn

(i.e., weakly) to an SBM process Wt 3.5.4

t 0, 1 t 0, 1

, n N, converges in distribution (Donsker, 1951).

White Noise Process

The time derivative of W t ,W t = dW t dt = ξ t , is termed (one-dimensional) white noise. As was noted earlier (see characteristic (5) of SBM processes), W t does not exist in the ordinary sense; it is a generalized or idealized derivative. What then are the properties of a white noise process ξ(t)? If Xt t ≥ 0 is a Gaussian stochastic process with E Xt2 < +∞, t ≥ 0, the covariance function of Xt t ≥ 0 is defined as r t, s = E Xt Xs , where t, s ≥ 0. If r(t, s) = h(t − s) for some real-valued function h R R and if E Xt = E Xs for all t, s ≥ 0, then Xt is called stationary in the wide sense. Then a white noise process ξ(t) is one that is Gaussian as well as wide-sense stationary, with h = δ0 , where δ0 is the Dirac point mass11 concentrated at t = 0.

Appendix 3.A Kolmogorov Existence Theorem: Another Look A stochastic process Xt t T is usually described in terms of its family of finitedimensional distributions. In this regard, for each n-tuple (t1, …, tn) of distinct elements of T, let the associated distribution of the random vector (X(t1), …, X(tn)) over Rn be denoted as μt1 , …, tn , with μt1 , …, tn A = P X t1 , …, X tn

A , A Rn

11 For Ω, A a measurable space with x Ω, the function δx A δx A =

1,x A; 0,x A

3A1

0, 1 , defined for set A A by

is termed the Dirac point mass function at point x.

109

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3 Mathematical Foundations 3

Then μt1 , …, tn is a probability measure on Rn and is termed a finite-dimensional distribution of the process Xt t T . We require that the family of probability measures or finite-dimensional distributions induced by Xt t T satisfy the following two consistency conditions; that is, if t1 ,…, tn T ,B R is the Borel σ-algebra on R, and if {π(1), …, π(n)} is a permutation of the numbers 1, …, n, then a μt1 , … , tn A 1 ×

× A n = μ tπ 1 , … , tπ n A π

b μt1 , … , tn A 1 ×

× An−1 × R = μt1 , …, tn A1 ×

1

×

, Ai

B R , i = 1, …,n;

× An− 1 , Ai

B R , i = 1, …, n

× Aπ

n

3A2 A1 × × An and (Note that (3.A.2a) holds because X t1 , …, X tn Aπ 1 × × Aπ n are the same events.) X tπ 1 ,…,X tπ n The probability measures (3.A.1) generated by a process such as Xt t T must satisfy (3.A.2). Conversely, the Kolmogorov existence theorem states that if a given system of measures satisfies the consistency requirements (3.A.2), then there exists a stochastic process with these finite-dimensional distributions. In this regard, we have Theorem 3.A.1. Theorem 3.A.1 Kolmogorov Existence Theorem Suppose for every t1 , …, tn R n we have a probability measure or system of distributions μt1 , …, tn on R n and that these measures satisfy the consistency conditions (3.A.2). Then there exists on some probability space Ω, A, P a stochastic process Xt t T having μt1 , …, tn as its finite-dimensional distributions. Hence, this theorem offers a justification for the existence of a stochastic process Xt t T having a family μt1 , …, tn n ≥ 1, ti T of probability measures as its finite-dimensional distributions.

Appendix 3.B

Nondifferentiability of BM

Our objective herein is to examine the proposition that, for any time t, almost all trajectories of BM are not differentiable. To this end, consider the interval from t to t + h. For this interval, let us express the difference quotient for the BM process Wt t ≥ 0 as Yh =

W t + h −W t h

Here, Yh represents a normally distributed random variable with

3B1

Appendix 3.B Nondifferentiability of BM

1 E Yh = E W t + h −W t = 0; h 1 1 1 V Yh = 2 V W t + h − W t = 2 h = h h h Thus, the standard deviation of Yh is V Yh = 1 h . From (3.6iii), we know that we can write Yh as 1 h Z, Z N 0, 1 , and thus, for constant k > 0, P Yh > k = P

Z >k h

Taking k to be arbitrarily large, as h 0, P Z h > k 1 and thus +∞ (in distribution) so that the rate of change in Wt at time t Yh = Z h is infinite or the derivative of Wt with respect to t at time t does not exist. In addition, with t also arbitrary, the BM path is nowhere differentiable—one cannot determine at any time t the immediate (local) direction of the BM path.

111

113

4 Mathematical Foundations 4 Stochastic Integrals, Itô’s Integral, Itô’s Formula, and Martingale Representation

4.1

Introduction

Suppose we have an ordinary differential equation (ODE) depicting the growth in, say, population (N) over time or dN t = a t N t , N 0 = N0 , dt

41

where the relative growth rate is dN t dt N t = a t . If a(t) is deterministic or a t = a = constant, then (1.4) has the solution N t = N0 e at

1

42

However, what if a(t) is subject to random fluctuations that, possibly, can be attributed to extraneous or chance factors such as the weather, location, and the general environment? In this circumstance, we can rewrite a(t) as a t =r t +j t

random error 43

= r t + j t noise, where r(t) and j(t) are given functions of t and the (one-dimensional) noise term follows a “known probability law.” In view of (4.3), Equation (4.1) can be rewritten as the stochastic differential equation (SDE) dN t = r t N t + j t N t noise, N 0 = N0 dt

411

1 Rewrite (1.4) as dN t N t = a dt. Then lnN t = at + lnC, where ln C is the constant of integration, and N t = Ce at . For t = 0,N 0 = C so that (4.2) obtains.

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

114

4 Mathematical Foundations 4

Under integration, this expression becomes t

N t = N0 +

t

r s N s ds + 0

j s N s noise ds

44

0

But how do we integrate t

j s N s noise ds?

45

0

To answer this question, we must find a stochastic process that can be used to represent noise. An obvious choice is the white noise term W t = dW t dt—the time derivative of the Brownian motion (BM) process Wt t ≥ 0 (see Section 3.5.4). In this regard, a substitution of W t dt = dW t = noise dt into (4.5) yields t

t

j s N s noise ds = 0

j s N s dW s

451

0

What are the key issues surrounding the calculation of the integral given in (4.5.1)? For one thing, (4.5.1) is not an ordinary integral since BM is nowhere differential almost surely (a.s.) or with probability 1. For another, BM is a stochastic process whose paths are a.s. of infinite variation for almost every ω Ω. To circumvent these difficulties, we shall look to a specialized mathematical apparatus that directly exploits the random nature of BM. Specifically, we shall look to the Itô stochastic integral (Itô, 1946) defined with respect to BM.

4.2

Stochastic Integration: The Itô Integral

Let us now define the class of admissible functions or processes to which the Itô integral will be applied (Allen, 2010; Evans, 2013; Friedman, 2006; Mao, 2007; Øksendal, 2013). That is, our objective is to obtain a solution of the stochastic integral t

f s, ω dW s, ω , ω Ω, or, written more succinctly, of

0 t

f s dW s

46

0

To this end, suppose Ω, A, P is a complete probability space with a filtration F = At t ≥ 0 satisfying the usual conditions (see Section 3.1.3). In addition, let Wt t ≥ 0 be a one-dimensional BM process defined on the filtered probability space Ω, A, F, P that is adapted to the filtration F. For a,b R, with 0 ≤ a < b < +∞, let the interval [a, b] be endowed with its Borel σ-algebra and

4.2 Stochastic Integration: The Itô Integral

let M2 a, b denote the space of all real-valued processes f = f t the product space Ω × a, b such that

t a, b

on

i process f is measurable with respect to the product σ-algebra on Ω × a, b 2 a, b , the random variable f t is A t -measurable

ii for each t

i e , the stochastic process f is adapted to F; and iii

f

2 2

b

a, b = E

f t

2

47

dt < +∞

a

Under the norm (4.7iii), M2 a, b is complete and, for processes f , h M2 a, b , these processes are equivalent if f −h 22 a, b = 0. R f satisfies 4 7 , which (The set of processes M2 a, b = f Ω × a, b displays a Hilbert space structure, is termed the natural domain of the Itô integral.) Given that f t M2 a, b , how should (4.6) be defined? A brief sketch of our approach to providing an answer to this question proceeds in three steps. For an integrand f M 2 a, b , let the Itô stochastic integral be denoted as b

I f =

f t dW t

L2 P , with P a probability measure.

a

Step 1. If ϕ = ϕ t

t

a, b

M20 a, b

is an elementary or step process

a, b is the subspace of elementary functions in M 2 a, b ), define the stochastic integral of ϕ as (M20

I ϕ =

b

ϕ t dW t =

a

k −1

ei W ti + 1 −W ti

i=0

Step 2. The Itô isometry on M20 a, b holds, that is, for all elementary functions ϕ M20 a, b , ϕ 2M20 a, b = I ϕ 2L2 P = E I ϕ 2 or, for

L2 P is an isometry and is ϕ M20 a, b , the mapping I: M 20 a, b continuous. Step 3. For every elementary function ϕ M 20 a, b , there exists a sequence of elementary processes ϕn M20 a, b such that f − ϕn

2 M 20 a, b

0 as n

∞

2 One way to determine measurability is: if f (t) is adapted and t ↦f t is a.s. right- (or left-) continuous in t, then f is measurable.

115

116

4 Mathematical Foundations 4

Then define b

I f = lim

n ∞ a

ϕn dW t in L2 P ,

that is, since each f M20 a, b can be approximated by a step process, the Itô stochastic integral is the L2-limit of integrals of elementary functions or b

I( f ) is the limit of

ϕ t dW t as ϕ

f

a

M20 a, b .

We now fill in some of the details. We may view the notion of an elementary or step process in the following fashion. A real-valued stochastic process ϕ t t a, b is an elementary (step) process if there exists a partition a = t0 < t1 < < tk = b of a, b , with mesh size max ti + 1 − ti , and bounded random variables ei , 0 ≤ i ≤ k − 1, such that ei is 0 ≤ i ≤ k −1

A ti -measurable, E e2i < +∞, and ϕt =

k −1

ei χ ti , ti + 1 t ,

48

i=0

where χ ti , ti + 1 t = 1 if t ti , ti + 1 and 0 otherwise, and with the family of all M2 a, b . In this regard, for an elemensuch processes denoted as M20 a, b 2 tary processes ϕ t M 0 a, b , define the random variable I ϕ =

b

ϕ t dW t =

a

k −1

e i W ti + 1 − W ti

49

i=0

as the Itô stochastic integral of ϕ with respect to the BM process Wt b

Here,

t ≥ 0.

ϕ t dW t is A b -measurable, ei is A ti -measurable, and W ti + 1 − W ti

a

is independent of A ti , i = 0 1, …, k − 1. Moreover, (4.9) is an element of L2(P) since, if ϕ M20 a, b (i.e., Ø is elementary and bounded), then it is true that b

a E

ϕ t dW t

= 0; and

b

2

a

b E

ϕ t dW t

3 b

=E

a

ϕt

2

4 10

dt

a the Itˆo isometry

3 In general, think of an isometry as involving a function from one metric space to another metric space that displays the property that the distance between two points in the first space is equal to the distance between the image points in the second space. We thus have a function that preserves a metric or preserves length (e.g., preserves norms).

4.2 Stochastic Integration: The Itô Integral

Moreover, if both ϕ1 , ϕ2 M20 a, b and b

M20 a, b , with c1 , c2

c1 ϕ1 t + c2 ϕ2 t dW t = c1

a

b a

R, then c1 ϕ1 + c2 ϕ2

ϕ1 t dW t + c2

b a

ϕ2 t dW t dt 4 11

Given (4.10b) and (4.11), we can now broaden the definition of the integral based upon elementary processes to also include processes in M2 a, b . The justification for this extension is Theorem 4.1, an important approximation result: Theorem 4.1 For any f M2 a, b , there exists a sequence ϕn t elementary processes such that b

lim E

n ∞

a

f t −ϕn t

2

dt = 0

∞ n=1

of

4 12

As promised, we can now define the Itô stochastic integral for a process f M2 a, b . To this end, via (4.12) (4.11), and (4.10b), b

lim E

n, m ∞

a

ϕn t dW t − b

lim E

n, m ∞

a

n, m ∞

a

ϕn t dW t

= 2

b

b

a

2

ϕm t dW t

ϕn t − ϕm t dW t

lim E

Hence,

b

a

ϕn t − ϕm t

2

=

dt = 0

is a Cauchy sequence in L2(P) so that its limit exists

in L2(P). In addition, we will define this limit as the Itô stochastic integral. To summarize, the Itô stochastic integral of f with respect to W t t ≥ 0 is b

b

f t dW t = lim a

n ∞ a

ϕn t dW t inL2 P ,

4 13

where {ϕn} is a sequence of elementary processes satisfying (4.12). (Note that the limit in (4.13) does not depend upon the choice of {ϕn} so long as (4.12) holds.)

117

118

4 Mathematical Foundations 4

Given that f ,g M 2 a, b following properties: b

1

Measurability

and α, β

R, this stochastic integral has the

f t dW t is A b -measurable

a b

2 E

b

f t dW t

= 0 and E

a

f t dW t A a = 0

a 2

b

3 E

b

=E

f t dW t

dt and

a 2

b

Aa

f t dW t

E

2

f t

a

b

=E

2

f t

a

dW t A a

a b

=

2

E f t

A a dt

a b

4

a

f t dW t + β

a

b

5 E

b

αf t + βg t dW t = α

Linearity

b

f t dW t a

b

g t dW t a

b

=E

g t dW t

f t g t dW t

a

a

6 If Z is a real − valued bounded A b -measurable random variable then Zf

M2 a, b

b

b

Zf t dW t = Z a

7

f t dW t a

Gaussian if the integral f is deterministic i e , f is independent of ω Ω , then the It oˆ stochastic integral

b

b

f t dW t is N 0, a

b

8

c

f t dW t =

Additivity a

b

f t dW t + a

a, b ,

f t dW t = A

f t dW t for a < c < b c

b

9 For a set A

f t 2 dt

a

a

f t χ A dW t 4 14

4.2 Stochastic Integration: The Itô Integral

M2 0, T , the indefinite Itô stochastic integral as

We next define, for f t

I t =

f s dW s , t

0, T ,

4 15

0

where I 0 = 0 and I t

is A t -adapted and has the following martingale

t 0, T 2

property. That is, if f M 0, T , then I t t 0, T is a square integrable martingale with respect to the filtration F = At t ≥ 0 so that, via (4.14.2), I t t 0, T has the martingale property t

E I t As = E I s As + E

f r dW r A s = I s ,

s

0 ≤ s < t ≤ T . Specifically, Doob’s martingale property (3.5b) holds or 2

t

≤ 4E

f s dW s

E max

0≤t≤T

0

T

f s

2

ds

0

Moreover, I t t 0, T has a continuous version a.s. That is, for f M2 0, T , there exists a t-continuous version of (4.15) in that there exists a t-continuous stochastic process {Jt} on Ω, A, P with continuous sample paths such that t

P Jt =

= 1, t

fdW

0, T . (From this point on, when we speak of an indef-

0

inite Itô integral (4.15), we mean its t-continuous version.) In sum, for f M2 0, T , the indefinite Itô integral I t t 0, T is a square-integrable continuous martingale with quadratic variation. t

I t ,I t

t

=

2

f s

0, T

ds, t

0

Given that the Itô stochastic integral (4.15) is a one-dimensional stochastic t

process, we write Xt = 0 ≤ s ≤ t ≤ T . In addition,

f s dW s , t 0

0, T ;

b. COV Xt ,Xs = E Xt Xs =

E f u

min s, t

2

f u dW u ,

2

du, 0 ≤ s ≤ t ≤ T ;

0

t

=

t s

a. E Xt = 0 and V X t = E Xt2 , t

c. E Xt

0, T . Then Xt −Xs =

E f u 0

2

du, t

0, T ; and

t

d. the process Xt =

f s dW s , t 0

0, T , has orthogonal increments, that is,

for 0 ≤ r ≤ s ≤ t ≤ T , E Xu − Xt Xs − Xr = 0. (Note that although the Itô

119

120

4 Mathematical Foundations 4

stochastic integral has orthogonal, and thus uncorrelated, increments, these increments are not, in general, independent. However, if f is independent of ω, then property (4.14.7) obtains. In this instance, the uncorrelatedness of a set of normally distributed random variables implies their independence so that Xt is a random process with independent increments. This said, we can thus conclude that a normal process Xt that has E Xt = 0 and X 0 = 0 has independent increments in the interval and thus can be represented by an Itô stochastic integral of the form (4.15).) It is important to note that, in our subsequent discussions, all (three) of the following notations are equivalent and will be used interchangeably: t

t

fdW = 0

4.3

t

f s dW s = 0

f s, ω d s, ω

0

One-Dimensional Itô Formula

Let Wt t ≥ 0 , W0 = 0, be a one-dimensional BM on a probability space Ω, A, P that is adapted to the filtration F = A t t ≥ 0 . What is the solution of the integral t

W s dW s ? Under ordinary (Riemann) integration, 0 t

1 W s dW s = W t 2 0

2

However, as will be determined later, the desired integral is actually t

1 1 W s dW s = W t 2 − t, 2 2 0 that is, the extra term − 1 2 t demonstrates that the Itô stochastic integral cannot be evaluated as one would an ordinary integral. To correctly evaluate Itô stochastic integrals, we shall use what is called the Itô chain rule or Itô formula. To this end, let Wt t ≥ 0 be a one-dimensional BM defined on a complete probability space Ω, A, P and adapted to the filtration F = A t t ≥ 0 . In addition, let f t L1R + and g t L2R + . Then a one-dimensional continuous adapted process Xt t ≥ 0 of the form t

X t = X0 +

t

f s ds + 0

g s dW s

4 16

0

is called an Itô process with stochastic differential dX t = f t dt + g t dW t

4 17

4.3 One-Dimensional Itô Formula

Next, suppose V = V X, t :R × R + R is twice continuously differentiable in X and once in t. Then under (4.16) and (4.17), Y t = V X t , t is also an Itô process with stochastic differential 1 dY t = Vt X t , t + VX X t , t f t + VXX X t , t g t 2

2

dt 4 18

+ VX X t , t g t dW t a s , where Vt = ∂V ∂t, VX = ∂V ∂X, and VXX = ∂2 V ∂X 2 . This expression is commonly known as the one-dimensional Itô chain rule or Itô formula.4 Moreover, (X(t), t) is the argument of Vt, VX , and VXX while for all times t ≥ 0, (4.18) is interpreted as Y t − Y 0 = V X t , t −V X 0 ,0 t

1 Vt X s , s + VX X s , s f s + VXX X s , s g s 2

= 0

2

ds

t

VX X s , s g s dW a s

+ 0

4 19 A convenient computational device for evaluating certain Itô integrals is Theorem 4.2. Theorem 4.2 Integration-by-Parts Formula Let f (s) depend only on s (w must not be an argument of f ) with f continuous and of bounded variation on [0, t]. Then t

f s dW s = f t W t −

0

t

4 20

W s df s 0

One final point is in order. Suppose Xt t ≥ 0 is an Itô process satisfying dX t = fdt + gdW t and let both f and g be deterministic functions of t, with X 0 = X0 = constant, that is, t

Xt = X0 +

t

f s ds + 0

Then Xt

t≥0

g s dW s 0

is a t

N E X t ,V X t

= N X0 +

t

f s ds, 0

g s 2 ds

0

process with independent increments.

4 A rationalization of Itô’s formula is provided by Appendix 4.A.

121

122

4 Mathematical Foundations 4 t

Let us calculate

Example 4.1

W s dW s . Two approaches will be offered: 0

(1) direct use of Itô’s formula and (2) a modification thereof. 1. From experience with classical theory, one might surmise that the solution should include a term of the form (1/2)W(t)2. Hence, we shall apply Itô’s formula to Y = V X t , t = 1 2 X t 2 or, for X t = W t , to V = 1 2 W t 2 , that is, since VX = W t , VXX = 1, and (4.17) becomes dX t = 0 dt + 1 dW t , (4.18) yields d

W t 2

2

1 Vt + VX f t + VXX g t 2 =0 =0

=

2

dt + VX g t dW t =W t

=1 2

=

1 dt + W t dW t 2

Thus, the solution in differential form is W t 2

d

2

1 = dt + W t dW t 2

In integral form, the solution is, from (4.19), W t 2

2

1 = t+ 2

t

W s dW s or 0

t

W s dW s = 0

W t 2

2

1 − t 2

2. Since V(X(t), t) might involve only X t = W t and not explicitly include a “t” term (the present case), let us examine the resulting differential and integral forms of the solution by supposing that V = V X t is twice differentiable with a continuous second derivative. Then for X t = W t , a second-order Taylor expansion of V = V W t renders Itô’s formula in differential form or 1 dV W t = V W t dW t + V W t dW t 2 2 4 21 1 = V W t dW t + V W t dt 2 via Table 4.A.1. The integral form of Itô’s formula then appears as t

V W t =V W 0 +

V W s dW s + 0

1 2

t

V W s ds 0

4 22

4.3 One-Dimensional Itô Formula

So given V = V W t = 1 2 W t 2 , (4.21) yields W t 2

d

2

= W t dW t +

1 1 dt 2

= W t dW t +

1 dt; 2

and from (4.22) W t 2

2

t

=0+

W s dW s + 0

t

W t 2

W s dW s = 0

2

1 − t 2

1 2

t

ds or 0

■

t

Example 4.2

Find

s dW s

From classical analysis, we might guess that

0

the solution should have a term of the form tW(t). In this regard, set Y = V X t ,t = tX t or, for X t = W t , V = tW t so that (4.17) becomes dX t = 0 dt + 1 dW t and Itô’s formula (4.18) yields d tW t =

Vt =W t

1 + VX f t + VXX g t 2 =0 =0

2

dt + VX g t dW t =t

Hence, the solution in differential form is d tW t = W t dt + t dW t ; and the solution in integral form is t

t

W s ds +

tW t =

s dW s or

0 t

0 t

s dW s = tW t −

0

W s ds

■

0 t

Example 4.3

e W s dW s . For X t = W t , (4.17) becomes

Evaluate 0

dX t = 0 dt + 1 dW t . Then set V X t = V W t = e W t (since deterministic theory supports the solution having a term of the form eW(t)). Then from (4.21), the differential form of the solution is d eW

t

1 = e W t dW t + e W t dt; 2

123

124

4 Mathematical Foundations 4

and the integral form is t

eW

t

e W s dW s +

= 0

t

e W s dW s = e W

t

t

1 2

e W s ds + V W 0 or 0

−1−

0 t

Find

Example 4.4

1 2

t

e W s ds

■

0

e μs + σW s dW s . Set X t = W t . Then from (4.17),

0

dX t = 0 dt + 1 dW t , and thus V = V X t ,t = V W t ,t = e μt + σW t . Then by Itô’s formula (4.18), 1 dV = Vt + VX f t + VXX g t 2 = μV

=0

2

dt + VX g t dW t

1 2 σ V 2

=

= σV

so that the solution in differential form is 1 dV = μV + σ 2 V 2

dt + σVdW t

1 = μ + σ 2 Vdt + σVdW t 2 The solution in integral form is, for V W 0 , 0 = 1, e μt + σW

t

=1+ μ+

σ2 2

t

e μs + σW s ds + σ

0

e μs + σW s dW s or

0

t

1 e μs + σW s dW s = e μt + σW σ 0

t

t

1 μ σ − − + σ σ 2

t

e μs + σW s ds

0

(Note that if μ = 0 and σ = 1, then the result of Example 4.3 emerges.) t

Example 4.5

Determine

■

W s 2 dW s . Classical analysis provides a hint

0

that the solution should contain a term of the form (1/3)W(t)3. Hence, we shall apply Itô’s formula to Y = V X t = 1 3 X t 3 or, for X t = W t , to V = 1 3 W t 3 . In this regard, since V = W t 2 and V = 2W t , Itô’s formula (4.21) yields the solution in differential form d

1 W t 3

3

= W t 2 dW t + W t dt

4.3 One-Dimensional Itô Formula

or, in integral form,

1 W t 3= 3

t

W s 2 dW s +

0

t

1 W s 2 dW s = W t 3 − 3 0

t

W s ds or 0 t

W s ds

■

0

For the convenience of the reader, Table 4.1 of some common stochastic integrals and their solutions now follows.

Table 4.1 Common stochastic integrals (for real a < b; t > 0). t

1.

dW s = W t 0 t

2. 0 b

3.

1 1 W s dW s = W t 2 − t 2 2 cdW s = c W b −W a ,c a constant

a b

4.

W s dW s = a t

5.

1 W b 2 −W a 2

sdW s = tW t −

0

W s ds 0 t

1 W s 2 dW s = W t 3 − 3 0

0

t

t

e W s dW s = e W

7.

t

−1−

0

1 2

W s ds e W s ds 0

t

W s e W s dW s = 1 + W t e W

8.

t

−e W

0 t

9.

sW s dW s = 0 t

10. 0 t

11.

1 b −a 2

−

t

t

6.

2

t t 1 W t 2− − 2 2 2

t

t

−

1 2

t

eW

s

1 + W s dW s

0

W s 2 ds

0

1 W s 2 −s dW s = W t 3 −tW t 3 e − 2 + W s dW s = e − 2 + W s

t

t

−1

0 t

12. 0

sinW s dW s = 1−cos W t −

1 2

t

cos W s ds 0

(Continued overleaf )

125

126

4 Mathematical Foundations 4

Table 4.1 (Continued) t

13.

cos W s dW s = sin W s + 0

1 2

t

sin W s ds 0

t

14.

dW s

E

=0

0 t

15.

E

W s dW s

=0

W s dW s

=

0 t

16.

4.4

V 0

t2 2

Martingale Representation Theorem

It was mentioned earlier in Section 4.2 that the Itô integral is a martingale with respect to A t . The converse of this statement is also true—any A t -martingale Xt t ≥ 0 with E Xt2 < +∞ can be written as a stochastic integral. In what follows, we shall argue that any square-integrable random variable that is measurable with respect to a BM is representable as a stochastic integral in terms of this BM. In particular, any martingale that is adapted to a BM filtration can be written as a stochastic integral. To set the stage for this discussion, suppose Wt t ≥ 0 is a BM. If f L2 P , t

then the random variable V =

f s dW s ,t 0

t

and, by the Itô isometry, E V 2 = E

0, T , is A t -measurable

f s 2 ds < +∞ so that V

L2 P .

0

Theorem 4.3, the Itô representation theorem, informs us that any f can be represented in this fashion. That is,

L2 P

Theorem 4.3 Itô Representation Theorem Let Wt t ≥ 0 be a BM. For every function f L2 P and adapted to the filtration F = A t t ≥ 0 , there is a unique stochastic process Ut t ≥ 0 such t

that E

U s 2 ds < +∞.

0

The importance of this theorem is that it enables us to obtain the following characterization of the square-integrable martingales adapted to the filtration F. Specifically, Theorem 4.4 describes the martingale representation theorem.

4.5 Multidimensional Itô Formula

Theorem 4.4 Martingale Representation Theorem Let Wt t ≥ 0 be a BM with Xt t ≥ 0 a square-integrable martingale adapted to the filtration F = A t t ≥ 0 . Then there is a unique stochastic process Ut t ≥ 0 such that, for every t ≥ 0, E

t

U s 2 ds < +∞.

0

In this regard, any square-integrable martingale adapted to the filtration F admits a continuous version.

4.5

Multidimensional Itô Formula

Let us now extend the one-dimensional Itô formula (Section 4.3) to the multidimensional case. To this end, suppose W t t ≥ 0 = W1 t ,…, Wm t ' (a prime “'” denotes matrix transposition) is an m-dimensional BM defined on the complete probability space Ω, A, P and adapted to the filtration F = A t t ≥ 0 , where the components Wr(t) and Ws(t) are independent if r s. In addition, let f t = f1 t , …, fd t ' L1R + R d and g t = gij dxm L2R + R dxm . Then a Rd-valued continuous adapted process X t t ≥ 0 = X1 t ,…, Xd t ' of the form X t = X0 +

t

f s ds +

0

t

g s dW s

4 23

0

is termed a d-dimensional Itô process with stochastic differential dX t = f t dt + g t dW t ,

4 24

where X1 t

f1 t

X t =

, f Xd t

g1m

, g t =

=

d×1

d×1

g11

,

d×m

fd t

gd1

gdm

dW1 t and dW t = m×1

dWm t Here, the ith component of (4.23) is t m

t

X i t = Xi 0 +

fi s ds + 0

0 j=1

gij s dWj s ,i = 1, …,d;

4 23 1

127

128

4 Mathematical Foundations 4

and the ith component of (4.24) is m

dXi t = fi t dt +

gij t dWj t , i = 1,…, d

4 24 1

j=1

R is twice continuously differenNow, suppose V = V X t , t :R d × R + tiable in X and once in t. Then under (4.23) and (4.24), Y t = V X t ,t is also an Itô process with stochastic differential 1 dY t = Vt X t , t + VX X t , t f t + tr g t VXX X t ,t g t dt 2 + VX X t , t g t dW t a s , 4 25

or d

dY t = Vt +

VXi fi t + i=1

d

1 d d m VX X gik t gjk t dt 2 i=1 j=1 k =1 i j

4 25 1

m

VXi gij t dWj t a s , 5

+ i=1 j=1

where Vt , VXi , and VXi Xj i = 1, …, d; j = 1,…, m.

are all evaluated at the point

X t ,t ,

Example 4.6 Suppose d = 1 and m = 3. Also, X t = X1 t , f t = f1 = 0, with a, b,c R, W t t ≥ 0 = W1 t ,…, g t = g11 t , …, g13 t = a, b, c , W3 t , and Y = V X t , t = V X1 , t = 2X12 t . Then from (4.24.1), dX1 t = f1 dt + g11 dW1 t + g12 dW2 t + g13 dW3 t = adW1 t + bdW2 t + cdW3 t and, from (4.25.1), dY t = Vt + VX1 f1 t +

1 3 VX X g1k t 2k =1 1 1

dt

m

+

VX1 g1j t dWj t j=1

1 = Vt + VX1 f1 t + VX1 X1 g11 t g11 t + g12 t g12 t + g13 t g13 t 2 + VX1 g11 t dW1 t + g12 t dW2 t + g13 t dW3 t = 2 a2 + b2 + c2 dt + 4X1 adW1 t + bdW2 t + cdW3 t 5 A rationalization of this expression is provided by Appendix 4.B.

■

dt

Appendix 4.A Itô’s Formula

Appendix 4.A

Itô’s Formula

Let Wt t ≥ 0 be a one-dimensional continuous BM defined on the filtered probability space Ω, A, F, P that is adapted to the filtration F = A t t ≥ 0 . Suppose [t, t + dt] is an infinitesimal time interval, where both dt and dt are positive and dt i , i = 2 3, …, is 0. For dWt = Wt + dt −Wt , we know that dWt

1 2

N E dWt , V dWt = N 0,dt

so that dWt is a differential of size (dt)1/2. Moreover, 2

a E dWt b V dWt

2

= dt; and

4A1

= dt 2 = 0 = constant

so that dWt 2 = dt, that is, under (4.A.1b), (dWt)2 must equal its mean value of dt. In a similar vein, a E dt dWt = dt E dW = 0; and

4A2

b V dt dWt = dt 2 V dWt = 0 = constant

so that dt dWt = 0 (since (4.A.2b) implies that dt dWt must equal its mean value of 0). To summarize, dWt

2

= dt, dt 2 = 0,dt dWt = dWt dt = 0

4A3

An aid to remembering these results is provided by the Itô multiplication table (Table 4.A.1). The second-order Taylor expansion of V(X(t), t) generally appears as dV X, t = Vt dt + VX dX +

1 VXX dX 2

2

+ 2VXt dXdt + Vtt dt

2

= Vt dt + VX fdt + gdW +

1 VXX fdt + gdW 2

2

+ 2VXt fdt + gdW dt + Vtt dt

2

4A4 Then from the assumptions on V(X, t) and Table 4.A.1, Equation (4.A.4) simplifies to 1 dV X, t = Vt + VX f + VXX g 2 dt + VX g dW 2 or Equation (4.18).

4A5

129

130

4 Mathematical Foundations 4

Table 4.A.1 Itô multiplication table. •

dt

dWt

dt

0

0

dWt

0

dt

Note that Itô’s chain rule (4.A.5) reduces to the chain rule of ordinary calculus in the deterministic case where g ≡ 0. The stochastic version of the chain rule has an additional term involving VXX. Table 4.A.1 can also be used to develop Itô’s product formula. To this end, suppose Xt and Yt are one-dimensional continuous adapted processes of the form (4.16), that is, t

Xt = X 0 +

t

a s ds + 0 t

Yt = Y0 +

b s dW s , 0 t

α s ds +

0

β s dW s ,

0

and let V x, y = xy. Then, via Itô’s formula, d Xt Yt = dV Xt , Yt = Vx dXt + Vy dYt 1 + Vxx dXt 2

2

1 + Vxy dXt dYt + Vyy dYt 2

2

4A6

= Yt dXt + Xt dYt + dXt dYt , the Itô’s product formula. (Note that each of the partial derivatives in (4.A.6) is evaluated at the point (Xt, Yt).) From this expression, one can readily obtain the general integration by parts formula t

Xt Yt = X0 Y0 +

t

Xs dYs +

dXs dYs

0

0

0

t

t

t

= X0 Y0 +

Ys dXs + 0

Appendix 4.B

t

Ys dXs +

Xs dYs + 0

4A7 b s β s ds

0

Multidimensional Itô Formula

Our objective here is to provide some guidance for the development of Equation (4.25.1). We shall first collect some of our major results from Section 4.5. To this end, set Y t = V X t , t . Let us write the ith component of dX t = f t dt + g t dW t

4B1

Appendix 4.B Multidimensional Itô Formula

Table 4.B.1 Itô multiplication table i

j .

•

dt

dWi

dWj

dt

0

0

0

dWi

0

dt

0

dWj

0

0

dt

as m

dXi t = fi t dt +

gij t dWj t , i = 1,…, d

4B2

j=1

Next, the second-order Taylor expansion of Y(t) can be written as d

dY t = Vt dt +

VXi dXi + j=1

1 d 2 i=1

m

VXi Xj dXi dXj

4B3

j=1

(here, all partial derivatives are evaluated at the point (X(t), t), and a substitution of (4.B.2) into (4.B.3) and simplifying (by “brute force”) yields Equation (4.25.1). An aid for executing the said simplification is the following set of relationships among the various differentials: dt 2 = 0; dWi dt = dt dWi = 0, i = 1, …, d; dWj dt = dt dWj = 0, j = 1,…, m; and dWi dWj = δij dt, where δij =

1, i = j; 0, i

j

These results are summarized in the Itô multiplication table (Table 4.B.1).

131

133

5 Stochastic Differential Equations 5.1

Introduction

Our objective in this chapter is to examine the solution to a stochastic differential equation (SDE) of the form dX t = f X t , t dt + g X t , t dW t , t

t0 ,T , T > 0,

51

with initial condition X t0 = X0 , where X0 is taken to be a random variable independent of W t −W t0 and E X0 2 < +∞. Here, f and g are termed the drift and diffusion coefficient functions, respectively. In particular, we can view f as measuring short-term growth and g as depicting short-term variability. (If g X t ,t ≡ 0 in (5.1), then the resulting expression is simply the ordinary differential equation (ODE) dX t dt = f X t ,t ,t t0 ,T .) Looking to specifics, we shall endeavor to answer questions such as the following: Under what conditions does a solution exist? If a solution does exist, is it unique? How, and under what circumstances, can we determine an exact solution? If an exact solution is found, in what sense is it stable? To address these issues, let Ω, A, P be a complete probability space with filtration F = A t t ≥ 0 satisfying the usual conditions (see Section 3.1.3). In addition, let Wt t ≥ 0 be a one-dimensional Brownian motion (BM) defined on the filtered probability space Ω,A, F, P . For 0 ≤ t0 < T < +∞, let X0 be a A t0 -measurable random variable such that E(|X0|2) is finite. Also, suppose R and g R × t0 , T R are each Borel-measurable functions. f R × t0 ,T Then the solution to the SDE (5.1) in integral form is t

X t = X0 +

t

f X s , s ds + t0

g X s , s dW s ,t

t0 ,T

52

t0

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

134

5 Stochastic Differential Equations

More to the point, a one-dimensional stochastic process X t a solution of (5.1) if

t t0 , T

is termed

i. process {X(t)} is continuous and A t -adapted; ii. process f X t , t L1 t0 , T and process g X t ,t L2 t0 ,T ; and iii. Equation (5.2) holds for every t t0 , T almost surely (a.s). Moreover, a solution X t

t t0 , T

is indistinguishable from X t

5.2

is unique if any other solution X t

t t0 , T

or P X t = X t , t

t t0 , T

t0 ,T = 1.

Existence and Uniqueness of Solutions

Under what circumstances will a unique solution to (5.1) exist? The answer to this question is provided by Theorem 5.1 (Allen, 2010; Arnold, 1974; Evans, 2013; Gard, 1988; Mao, 2007; Øksendal, 2013; Steele, 2010). Theorem 5.1 Existence and Uniqueness Theorem If the coefficient functions f, g of SDE (5.1) satisfy the conditions f x, t −f y, t + g x, t − g y, t ≤ k x− y

53

uniform Lipschitz condition

for some constant k and all t f x, t

2

+ g x, t

2

t0 , T , T > 0; and

≤ k2 1 + x 2

54

linear growth condition

for some constant k2 and all t t0 , T , T > 0, then there exists a continuously adapted solution Xt of (5.1) such that X t = X0 and supt0 ≤ t ≤ T E Xt 2 < +∞ (i.e., Xt is uniformly bounded in M 2 t0 , T ). Moreover, if Xt and X t are both continuous M2 t0 , T bounded solutions of (5.1), then P

sup Xt −X t = 0 = 1,

0≤t≤T

55

and thus the solution Xt is unique. Here, the uniform Lipschitz condition (5.3) guarantees that the functions f and g do not change faster than a linear function of X relative to changes in X, thus implying the Lipschitz continuity of f , t and g , t for all t t0 ,T . Condition (5.4) serves to bound f and g uniformly with respect to t t0 ,T ; it allows at most

5.2 Existence and Uniqueness of Solutions

linear increases in these functions with respect to X. This restriction on the growth of f and g guarantees that, a.s., the solution Xt does not “explode” (tend to +∞) in the interval [t0, T] no matter what X0 happens to be. The solution of (5.1) just described is termed a strong solution in that it has a strongly unique sample path and because the filtered probability space, the BM, and the coefficient functions f and g are all specified in advance. However, if only f and g are specified in advance and the pair of processes X t ,W t are defined on a suitably filtered probability space, then, provided (5.1) holds, X t is called a weak solution. If two weak solutions determined under these conditions are indistinguishable, then path uniqueness holds for (5.1). In addition, two solutions to (5.1) (either strong or weak) are termed weakly unique if they have the same finite-dimensional probability distribution (even though they have different sample paths). While a strong solution to (5.1) is also a weak solution, the converse is not generally true. The uniform Lipschitz condition (5.3) and the linear growth condition (5.4) tend to be fairly restrictive requirements for the existence and uniqueness of solutions to the SDE (5.1). To overcome this characteristic, let us broaden the class of functions that can serve as the coefficient functions f and g in (5.1). To this end, let us relax the uniform Lipschitz condition to a local one: for every integer n ≥ 1, there exists a positive constant kn such that, for all x, y R, t t0 , T , T > 0, and with max x , y ≤ n, f x, t −f y, t + g x, t − g y, t ≤ kn x− y

531

local Lipschitz condition

Now, if the linear growth condition (5.4) holds but the uniform Lipschitz condition (5.3) is replaced by the local Lipschitz condition (5.3.1), then there exists a unique solution Xt to (5.1) with Xt M2 t0 , T . (Note that (5.3.1) will be in force when f and g are continuously differentiable in x on R × t0 , T .) Similarly, let us replace the linear growth condition by a monotone condition: suppose there exists a positive constant k such that, for all X t ,t R × t0 ,T ,T > 0, xf x, t +

1 g x, t 2

2

≤k 1+ x 2

541

monotone condition

If the local Lipschitz condition (5.3.1) holds but the linear growth restriction (5.4) is replaced by the monotone condition (5.4.1), then there exists a unique solution Xt to (5.1) with Xt M 2 t0 , T . The role of the monotone condition is that it ensures the existence of a solution on the entirety of the interval [t0, T]. (Note that if the monotone condition (5.4.1) holds, then the linear growth restriction (5.4) may or may not hold. However, if (5.4) is satisfied, then (5.4.1) is satisfied as well.)

135

136

5 Stochastic Differential Equations

A couple of final points are in order. First, suppose that the coefficient functions f and g are defined on R × t0 , T and the assumptions of Theorem 5.1 hold on every finite subinterval t0 , T t0 , +∞ . Then the SDE dX t = f X t ,t dt + g X t , t dW t , t t0 , +∞ , X t0 = X0 , has a unique solution defined on the entirety of t0 , +∞ . In this circumstance, Xt is called a global solution to this SDE. Next, suppose we have an autonomous SDE of the form dX t = f X t dt + g X t dW t , t

t0 , T , T > 0,

56

with X t0 = X0 (here both f and g are independent of t so that f X t ,t ≡ f X t , g X t , t ≡ g X t . If X0 is a random variable independent of W t −W t0 , then, provided that the assumptions underlying Theorem 5.1 are in effect, this SDE has a unique, continuous global solution Xt on t0 , +∞ such that X t0 = X0 given that the Lipschitz condition f x −f y + g x −g y ≤ k x−y

532

Lipschitz condition

is satisfied; and, for a fixed y value, the linear growth condition f x

2

+ g x

2

≤ k2 1 + x 2

542

linear growth condition

holds.

5.3

Linear SDEs

In the SDE (5.1) (repeated here as dX t = f X t ,t dt + g X t ,t dW t , t t0 ,T ,T > 0), the coefficient functions f and g are typically nonlinear in form. When nonlinearity does hold, approximation methods are used to generate a solution Xt to (5.1) since, in general, nonlinear SDEs do not have explicit or exact solutions. However, it is possible to determine an explicit solution to a linear SDE, that is, a more thorough treatment of SDEs can be undertaken when the coefficient functions f and g are linear functions of X (and especially when the coefficient functions are independent of X). More specifically, the SDE (5.1) is linear if the functions f and g are each linear functions of X on R × t0 , T , that is, if f X t ,t = a t + A t X t ; g X t ,t = b t + B t X t ,

57

where a(t), b(t), A(t), and B(t) are all mappings from [t0, T] to R. Then, under (5.7), our linear SDE appears as

5.3 Linear SDEs

dX t = a t + A t X t dt + b t + B t X t dW t , 58 t

t0 , T , T > 0,X t0 = X0

Moreover, again assuming that X t0 = X0 , a linear SDE is said to be homogeneous if a t = b t ≡ 0, t t0 , T , or dX t = A t X t dt + B t X t dW t , t

t0 ,T ,T > 0;

59

and it is termed linear in the narrow sense if B t ≡ 0 or dX t = a t + A t X t dt + b t dW t ,t

t0 ,T , T > 0

5 10

What can be said about the existence and uniqueness of a solution to a linear SDE? To answer this, suppose that for the linear SDE (5.8), the functions a(t), A(t), b(t), and B(t) are all measurable and bounded on t0 , T ,T > 0, that is, sup

t0 ≤ t ≤ T

at + At + bt + Bt

< +∞

Then f and g in (5.7) satisfy the hypotheses of the existence and uniqueness Theorem 5.1. Hence, (5.8) has a unique, continuous solution Xt over [t0, T] provided E X0 2 < +∞ and X0 is independent of W t −W t0 , t ≥ t0 . If these conditions hold in every subset of t0 , +∞ , then Xt is a unique, continuous global solution to (5.8). If the functions a(t), A(t), b(t), and B(t) are all independent of t, then (5.8) becomes an autonomous linear SDE or dX t = a + AX t dt + b + BX t dW t ,t

t0 ,T ,t > 0,

5 11

with a, A, b, and B real constants. For this type of SDE, a unique, continuous global solution always exists. 5.3.1

Strong Solutions to Linear SDEs

Consider the stochastic process defined by Yt = V Xt ,t (Arnold, 1974; Gard, 1988; Steele, 2010). This process can also be written as a function of the BM process Yt = h Wt , t (e.g., Xt may be a smooth function of Wt and t). In this regard, if we are looking for a solution to (5.1) of the form Xt = h Wt ,t , then, to determine any such solution, we need only apply Itô’s formula to this expression by (1) writing Xt = h Wt , t in terms of a second-order Taylor expansion and (2) invoking the products displayed within the Itô multiplication table (Table 4.A.1). To this end, let

137

138

5 Stochastic Differential Equations

1 hWW dWt 2 1 = ht dt + hW dWt + hWW dt 2

dXt = ht dt + hW dWt +

2

+ 2htW dWt dt + htt dt

2

5 12

1 = ht + hWW dt + hW dWt 2 But since h Wt , t = V Xt , t , (5.12) can be rewritten (via the chain rule) as 1 dXt = ht Wt , t + hXX Wt , t 2

dt + hX Wt ,t dWt

5 13

So to solve (5.1), we must match the coefficients on dt and dWt in (5.13) to those on the respective terms in (5.1). That is, we need to find a function h(x, t) that satisfies the two equations 1 f h x, t , t = ht x, t + hxx x, t , g h x, t , t = hx x, t 2

5 14

It^ o coefficient matching

This matching procedure will be termed Itô coefficient matching. Example 5.1 (Geometric BM) Suppose we choose to solve the geometric BM SDE dXt = μXt dt + σXt dWt , t

0, +∞ , X 0 = X0 ,

5 15

where −∞ < μ < +∞ measures short-term growth (drift) and σ > 0 gauges shortterm variability. More specifically, in structure, Xt exhibits both short-term growth and short-term variability in proportion to the level of the process. To find a function h(x, t) that satisfies (5.14), set 1 a μh x,t = ht x, t + hxx x, t , 2 b σh x, t = hx x, t

5 16

Then solving the partial differential equation system (5.16) involves the following three steps: 1. Integrate (5.16b) partially with respect to x to obtain h(x, t) up to an additive (arbitrary) function of integration j(t). 2. Substitute the function h(x, t) into (5.16b) in order to determine j(t). 3. The final solution is of the form Xt = h Wt , t + c, where c is a constant that is determined from the initial condition X0 on Xt.

5.3 Linear SDEs

From (5.16b), hx h = σ, ln h = σx + j t , and thus h x, t = e σx + j t , j t arbitrary

5 17

Then a substitution of (5.17) into (5.16a) gives 1 μe σx + j t = j t e σx + j t + σ 2 e σx + j t 2 or j t = μ− 1 2 σ 2 . Hence, j t = μt + 1 2 σ 2 t + ln c,c = constant, and thus a particular solution to (5.15) is Xt = X0 e σWt +

μ − 12 σ 2 t

5 18

Geometric BM process

It is important to note that Xt reflects the fact that the log of the process is BM so that, at each point in time, the distribution of the process is log-normal. Looking to the long-term dynamics of (5.18), we may conclude the following: +∞ as t +∞. i. If μ > 1 2 σ 2 , then Xt ii. If μ < 1 2 σ 2 , then Xt 0 as t +∞. iii. If μ = 1 2 σ 2 , then Xt fluctuates between arbitrarily large and arbitrarily small values as t +∞ ■ How do we know if (5.15) has been solved correctly? A direct check is the following. Let X = k y, t = X0 e Y , where Y = μY − 1 2 σ 2 t + σWt . By Itô’s formula 2

(4.18), dX = X0 e Y dY + 1 2 X0 e Y dY dY , where dY = μ − 1 2 σ dt + σdWt and dY dY = σ 2 dt (see Table 4.A.1). Then dX = X0 e Y

1 1 μ− σ 2 dt + σdWt + X0 e Y σ 2 dt 2 2

= X0 e Y μdt + X0 e Y σdWt = μXdt + σXdWt A specific class of SDEs that are particularly well suited for solution by Itô coefficient matching are those that may be characterized as exact. (A review of how to handle exact ODEs is provided by Appendix D.) A test for exactness appears as Theorem 5.2. Theorem 5.2 Test for Exactness If the SDE (5.1) (or the SDE dXt = f Wt , t + g Wt ,t dWt ) is exact, then the coefficient functions f(x, t) and g(x, t) satisfy the condition 1 fx = gt + gxx 2

Exactness Criterion

139

140

5 Stochastic Differential Equations

To see this, let us note that if the SDE is exact, then there is a function h(x, t) such that system (5.14) holds. Differentiating the first equation of this pair with respect to x yields fx =

∂ 1∂ ht + hxx ∂x 2 ∂x

Then substituting g = hx renders 1∂ gx since hxt = htx 2 ∂x 1 = gt + gxx 2

fx = hxt +

For instance, is the SDE dXt = e t 1 + 2Wt 2 dt + 1 + 4e t Wt dWt ,X0 = 0, exact? Here, f x, t = e t 1 + 2x2 , g x, t = 1 + 4e t x Since fx = 4xe t and gt + 12 gxx = 4e t x + 12 0 = 4e t x are equal, we can legitimately conclude that this SDE is exact and thus may be solved via Itô coefficient matching. Example 5.2 It is instructive to note that the SDE (5.15) specified in Example 5.1 can (alternatively) be solved by utilizing Equation (4.18) (Itô’s formula) directly. That is, if we rewrite (5.15) as dXt = μdt + σdWt , Xt then we see that the term ln Xt should appear in the solution. In this regard, set V Xt ,t = ln Xt ,f = μXt , and g = σXt in (4.18) so that dV = 0 + μXt

1 1 1 + σ 2 Xt2 Xt 2 Xt 2

dt + σXt

or 1 d ln Xt = μ − σ 2 dt + σdWt , 2 t

1 d ln Xu = μ − σ 2 2 0

t 0

du + σ

t

dWu , and 0

1 ln Xt −ln X0 = μ − σ 2 t + σ Wt − W0 2 Then, for W0 = 0,

1 dWt Xt

5.3 Linear SDEs

ln

Xt 1 = μ − σ 2 t + σWt 2 X0

Thus, Xt = X0 e

μ − 12 σ 2 t + σWt

More generally, if dXt = u t Xt dt + v t Xt dWt , X 0 = X0 ,

5 19

then Itô’s formula (4.18) can be applied to V Xt ,t = ln Xt (with f = u t Xt , g = v t Xt ) to yield the particular solution t

Xt = X0 exp 0

Example 5.3

1 us − vs 2

2

t

ds +

v s dWs

■

5 20

0

Use the method of Itô coefficient matching to solve the SDE

1 dXt = − e − 2Xt dt + e −Xt dWt , t 2

0, +∞ , X 0 = X0

5 21

Our objective is to find a function h(x, t) that satisfies (5.14). 1 1 a − e − 2h x, t = ht x, t + hxx x, t ; 2 2 − h x, t b e = hx x, t

5 22

From (5.22b), x + j t = e h or h = ln x + j t A substitution of this expression for h into (5.22a) gives 1 j t 1 1 − e −2 ln x + j t = − 2 x+j t 2 x+j t

2

Upon simplifying this latter result, we obtain j t x + j t = 0 or j t = 0. Clearly, j(t) = constant = c. Hence, h x, t = ln x + j t = ln x + c or Xt = ln Wt + c With W0 = 0, X0 = ln c or c = e X0 . Hence, a particular solution to (5.12) is Xt = ln Wt + e X0

5 23

141

142

5 Stochastic Differential Equations

Given that e X0 is constant, after a finite random time, Xt +∞ . More concretely, suppose the SDE in (5.21) holds for t t0 , T . Then Xt = ln W t − W t0 + e X0

5 23 1

so that this expression “explodes” when T = inf t W t − W t0 = −e X0 . Example 5.4

■

To solve the SDE

dXt = Xt dt + dWt , t ≥ 0,X 0 = X0 ,

5 24

let us set V Xt ,t = e − t Xt and apply Itô’s formula (4.18) to this expression (here f = Xt ,g = 1). d e −t Xt = −e − t Xt + Xt e −t +

1 0 2

dt + e −t dWt

= e −t dWt Then t

t

d e − s Xs =

0

e − s dWs or

0 t

e − t Xt − X0 =

e −s dWs 0

so that a particular solution to (5.24) is t

Xt = X 0 e t +

e t − s dWs

5 25

0

More generally, if dXt = μXt dt + σ dWt , X 0 = X0 ,

5 26

with α and σ constants, set V Xt , t = e − μt X t and apply Itô’s formula (4.18). The resulting particular solution has the form Xt = X0 e μt +

t

e μ t −s dWs

5 27

0

(For solution details and a special case of (5.26), see Example 5.6.)

■

Example 5.5 Let’s take a look at a slightly different type of SDE—one with coefficients that are not constants but that are deterministic functions indexed by time. Specifically, dXt = at + bt X t dt + σ t dWt , X 0 = X0 ,

5 28

5.3 Linear SDEs

where at, bt, and σ t are all continuous, deterministic functions of t. If we rewrite (5.28) as dXt − bt X t dt = at dt + σ t dWt

5 29

and concentrate on the left-hand side of this expression, we see that t

d dt

t

bu du

−

Xt e

dXt e dt

=

0

t

d Xt e

0

− X t bt e

bu du

− 0

or

t

bu du

−

t

bu du

−

bu du

−

=e

0

0

dXt − Xt bt dt

1

5 30

t

bu du

−

Hence, (5.30) implies that e

0

t

bu du

−

d Xt e

is an integrating factor for (5.29) so that

t

t

bu du

−

= at e

0

0

dt + σ t e

bu du

− 0

5 31

dWt

Since the left-hand side of (5.31) integrates to t

d Xs e

t

bu du

−

t

bu du

−

= Xt e

0

0

− X0 ,

0

it follows that a particular solution to (5.28) is t

Xt = X0 e

t

bu du 0

+

as e 0

t

bu du

t s

t

ds +

σs e

bu du s

dWs

2

5 32

0

x

1 Let f (t) be continuous, with a ≤ t ≤ b, and define F x = f t dt. Then F(x) is differentiable with a derivative F x = f x 2 To evaluate (simplify) expressions such as these, remember that a

i.

f t dt = 0 a b

ii.

b

f t dt = a

f t dt −

0

a

f t dt; and 0

a

iii. if a < b, then b

f t dt = −

b

f t dt. a

143

144

5 Stochastic Differential Equations

If bt is replaced by b = constant in Equation (5.28), then the preceding solution becomes t

Xt = X0 e bt +

e b t −s as ds + σ s dWs

■

5 32 1

0

For a more general discussion on integrating factors, see Appendix 5.B. (Also included therein is the technique of variation of parameters for solving certain SDEs.) Example 5.6

(Ornstein–Uhlenbeck (OU) processes)

To obtain a solution to the OU SDE dXt = − αXt dt + σ dWt , t ≥ 0,X 0 = X0 ,

5 33

with α and σ positive constants, let us apply Itô’s formula (4.18) to V Xt t = e αt Xt here f = − αXt , g = σ : d e αt Xt = αe αt Xt −αXt e αt +

1 0 2

dt + σe αt dWt or

d e αt Xt = σe αt dWt Then t

d e αs Xs = σ

0

e αt Xt −X0 = σ

t

e αs dWs or

0 t

e αs dWs

0

Hence, a particular solution to (5.23) has the form Xt = X0 e −αt + σ

t

e −α t − s dWs 0 Ornstein −Uhlenbeck process mean = 0

5 34

Equation (5.33) has a very interesting property. Specifically, the drift term − α Xt dt is negative (resp. positive) when Xt > 0 (resp. Xt < 0). Given that the OU process always experiences random fluctuations, Xt is stationary, that is, it is drawn back to its mean of zero whenever it drifts away from zero. Hence, the OU process is said to experience mean reversion. Moreover, since the short-term variability term g Xt , t = σ = constant, Xt tends to fluctuate robustly even when it is near its average value of zero. In this regard, Xt crosses the zero

5.3 Linear SDEs

level rather frequently. A slightly more general mean-reverting OU process involves modeling fluctuations that occur about a nonzero mean (or equilibrium) value μ 0 . The OU-type SDE that captures such behavior appears as dXt = − α Xt −μ dt + σdWt , t ≥ 0, X 0 = X0 ,

5 35

with α, μ, and σ all constant. A particular solution to (5.35) is Xt = X0 e − αt + μ 1 − e −αt + σ

t

e −α t − s dWs

■

5 36

0 Ornstein −Uhlenbeck process mean = μ

The aforementioned mean-reversion characteristic can be applied to many varieties of stochastic processes. For instance, consider the SDE for the geometric mean-reversion process (Tvedt, 1995) dXt = κ α− ln Xt Xt dt + σ Xt dWt , X 0 = X0 > 0, with κ, α, σ, and X0 all positive constants. The solution to this expression is Xt = exp e − κt ln X0 + α −σ 2 2κ 1 −e − κt + σe − κt

t

e κs dWs

0

Up to this point in our discussion about generating solutions to linear SDEs, we considered two (related) approaches: Itô coefficient matching; and the application of Itô’s formula (4.18) proper. Other possibilities exist, for example, we may also express solutions in, for example, what is called product form (see Appendix 5.A). Let us, at this point, summarize some of our results pertaining to the solution of linear SDEs (Table 5.1). The principal structures are included therein. Example 5.7 (Brownian bridge processes) Think of the Brownian bridge process as a BM process, defined over t 0,1 , that returns to t = 0 at time t = 1, that is, we have a “bridge” between the origin at time t = 0 and at time t = 1. Let us write the Brownian bridge SDE as dXt = −

Xt dt + dWt , X 0 = X0 = 0 1−t

5 37

To solve this SDE, we need only look to Table 5.1. Equation (5.37) appears to be of the linear narrow sense (LNS) form with at ≡ 0, bt = − 1 1 − t and ct ≡ c. Then

145

146

5 Stochastic Differential Equations

Table 5.1 Taxonomy of SDEs and their solutions. STRUCTURES (G) General form: dXt = f Xt , t dt + g Xt ,t dWt , 0 ≤ t ≤ T . (L) Linear: f Xt , t = at + bt Xt ;g Xt , t = ct + et Xt so that dXt = at + bt Xt dt + ct + et Xt dWt . (LH) Homogeneous: at = ct ≡ 0 so that dXt = bt Xt dt + et Xt dWt . (LNS) Narrow sense: et ≡ 0 so that dXt = at + bt Xt dt + ct dWt . (A) Autonomous: f Xt , t ≡ f Xt ;g Xt , t ≡ g Xt so that dXt = f Xt dt + g Xt dWt . (AL) Linear: f Xt = a + bXt ;g Xt = c + eXt so that dXt = a + bXt dt + c + eXt dWt . (ALH) Homogeneous: a = c ≡ 0 so that dXt = bXt dt + eXt dWt . (ALNS) Narrow sense: e ≡ 0 so that dXt = a + bXt dt + c dWt . SOLUTIONS (L) See Equation (5.A.20). t

(LH) Xt = X0 exp 0

1 bs − e2s ds + 2

t

(LNS) Xt = e

t

es dWs . 0

s

bs ds 0

X0 +

e

s

bu du

−

t

0

as ds +

0 t 1 (AL) Xt = exp e dWs + b− e2 2 0

bu du

−

t

e

0

cs dWs .

0 t

ds × X0 + a −ec

0

0

where s 1 Φs−1 = exp −e dwu − b− e2 2 0

s

du . 0

(ALH) See Equation (5.A.9). t

t

0

0

t

(ALNS) Xt = X0 e bt + a e b t −s ds + c e b t −s dWs .

t

Φs−1 ds + c Φs−1 dWs , 0

5.3 Linear SDEs t

−

Xt = e

1 ds 1 0 −s 0+0+

s

t

1 du 1 −u e 0 dWs

0 t

= e ln 1−t

e −ln 1 −s dWs

0 t

1 dWs , t 0,1 1 0 −s Brownian bridge process

= 1−t

5 38

A more general version of (5.37), for a and b constants, is dXt =

b − Xt dt + dWt , t 1 −t

0,1 , X 0 = X0 = a

5 39

The solution to this expression is Xt = a 1 −t + bt + 1 − t

5.3.2

t

dWs ,t 0 1−s

0,1

■

5 40

Properties of Solutions

Section 5.3.1 addressed the issue of determining strong solutions to linear SDEs. Let us now examine some of the properties of such solutions. In particular, we shall consider items such as boundedness, continuity, the specification of exact moments of a solution, and the Markov property of solutions of SDEs. For the SDE (5.1) or dXt = f Xt , t dt + g Xt , t dWt , t

t0 , T , T > 0,

51

where f and g satisfy the assumptions posited in Section 5.1, the solution Xt, with X t0 = X0 , is stochastically bounded if for each ε > 0 there exists a γ ε = γ ε t0 , X0 > 0 such that inf P Xt ≤ γ ε > 1 −ε

5 41

t t0 , T

If in (5.41) γ ε depends only on X0, then Xt is said to be uniformly stochastically bounded. Inequality (5.4.1) implies that Xt t t0 , T does not exhibit any tendency for large fluctuations in this process to beget even larger ones, that is, the average frequency of large fluctuations does not increase over time or large fluctuations in X are observed infrequently (Gard, 1988). A sufficient condition for a process X t t t 0 , T to be stochastically bounded is that for some p > 0, the pth-order moment E|Xt|P is bounded on the interval [t0, T] (more on this condition shortly).

147

148

5 Stochastic Differential Equations

Next, suppose f and g satisfy conditions (5.3) and (5.4) of Theorem 5.1 and Xt t t0 , T is a solution of (5.1). Then Xt is continuous on [t0, T] if there is a constant c > 0 such that Xt −Xr 2 ≤ c t −r , t0 ≤ r, t ≤ T ,

5 42

where we can write t

Xt −Xr =

t

f Xs , s ds +

g Xs , s dWs

r

r

Now, in the preceding sufficient condition for stochastic boundedness set p = 2. Then there is an M > 0 such that E Xs 2 ≤ M,0 ≤ s ≤ T , and thus it can be shown that (5.42) holds if we take c = 2k T + 1 1 + M . We now turn to the topic of moment inequalities and the calculation of pthorder moments E|Xt|p of the solution of the SDE (5.1), with [t0, T], T < +∞ , and X t0 = X0 . We begin with Existence of Moments (Arnold, 1974)

Theorem 5.3

Suppose that the assumptions of the existence and uniqueness theorem (Theorem 5.1) are satisfied and that E X0 2p < +∞ , where p a positive integer. Then for Xt a solution to the SDE (5.1) on [t0, T], T < +∞, i. E Xt

2p

≤ 1 + E X0

ii. E Xt − X0

2p

2p

e c t −t0 ; and

≤ D 1 + E X0

t − t0 p e c t −t0 ,

2p

5 43

where c = 2p 2p + 1 k 2 and D are constants (dependent upon p, k, and T −t0 ). Furthermore, a. E Xt − Xs 2p ≤ c1 t − s p , t0 ≤ s, t ≤ T , c1 constant. For E X0 2 < +∞ and p = 1, limt s Xt −Xs 2 = 0. b. Let p ≥ 2, X t0 = X0 L p Ω; R , and assume that the linear growth condition (5.4) is satisfied. Then the pth-order moment p−2 2

E Xt p ≤ 2

1 + E X0

p

e pα t −t0 , t

t0 , T ,

where α = k + k p− 1 2. c. The solution Xt of the linear SDE (5.8) has, for all t moment E|Xt|p if and only if E X0 p < +∞. d. For the autonomous linear homogeneous SDE,

5 44

2

t0 , T , a pth-order

dXt = bXt dt + eXt dWt , X t0 = X0 , E Xt p = E X0 p exp p b − e2 2 t − t0 +

p2 2 e t − t0 2

5 45

5.3 Linear SDEs

e. If X

= N μ, σ 2 , then, for every p > 0,

N E X ,V X E eX

p

= e pμ + p

2 σ2

2

5 46

This is because +∞

1 2πσ

e px −

−∞

x −μ

2 2 σ2

σ 2

2 2

dx = e pμ + p

(For p = 1, obviously E e x = e μ + e

2

2

.)

Continuing with our moment inequalities, we have M2 0,T , and E

Suppose p ≥ 2,g t

Theorem 5.4

p

T

g s

ds < +∞. Then

0 p

t

g s dW s

E 0

p 2

p p −1 ≤ 2

T

p −2 2

p

T

E

g s

ds

0

For p = 2, 2

t

E

T

=E

g s dW s

2

g s

0

ds

0

(Itô’s isometry). Example 5.8

For the Brownian bridge process (5.38) or t

1 dWs , t 0 1− s

Xt = 1 −t

0,1 ,

find E(Xt) and Vt(X).3 3 As an aid (review) for solving the current and the next few example problems, we offer some useful expectation/stochastic integral results as follows: t

1. E Wt = 0,V Wt = t,

dWs

2

t

=

ds

0 t

2. E

t

0 t

0

4. E

0 t

0 t

2

E Ws 2 ds =

0 2

t 0

s2 ds 0

α2

7. For t

0,T , E

e 0

sds 0

6. E e αWt = e 2 t , E e − Wt = e t 2 . In general, E e g T

t

t

sdWs =

0

Xs2 ds. t

0

sdWs = 0,E

5. E

0

Ws dWs =

Ws dWs = 0,E

ds.

t

Xs dWs =

0 t

g s 0

2

t

Xs dWs = 0, E

2

t

g s dWs = E

g s dWs = 0, E

3. E

0 2

−Wt

Wt

T

dt =

e 0

t 2

dt = 2 e

T 2

= eE g −1 .

Wt

2

2

.

149

150

5 Stochastic Differential Equations t

1 dWs = 0 1 0 −s

E Xt = 1− t E

2

1 dWs 1 0 −s

2

t

= E 1 −t = 1 −t

2

= 1 −t

2

2

= E Xt

V Xt = E Xt − E X t

t

1 ds 1 0 −s 1 − 1 = t 1 −t 1 −t

■

For the geometric BM process (5.18) or

Example 5.9

Xt = X0 exp

μ−

σ2 t + σWt , X0 arbitrary, 2

find E(Xt) and Vt(X). For p > 0 and E X0 E Xt p = E

X0 p exp p μ −

p

0, let us start by finding

σ2 t + pσWt 2

= exp p μ −

1 − p σ2 t E 2

= exp p μ −

1 − p σ2 t E X0 2

Then, for p = 1, E Xt = e μt E X0 = X0 e μt ; and for p = 2,

E Xt 2 = e

2

μ+

σ2 2

= X02 e2μt eσ

t

E X0 2

2

t

Then V Xt = E Xt 2 − E Xt = X02 e2μt eσ t −1 2

2

■

X0 p exp − p

p2 σ 2 t + pσWt 2

5.3 Linear SDEs

Example 5.10

For the OU process (5.34) or

Xt = X0 e −αt + σ

t

e −α t − s dWs ,

0

determine E(Xt) and Vt(X). t

E Xt = E X0 e −αt + σ e −αt

e −αs dWs

0 t

= E X0 e −αt + σ e −αt E

e αs dWs = E X0 e − αt

0

V Xt = E Xt − E Xt

2

t

= E X0 e −αt + σ e −αt

e αs dWs − e −αt E X0

2

0

= E e −αt X0 − E X0 + σ e −αt

t

e αs dWs

2

0

= e −2αt E X0 − E X0

2

+ σ 2 e − 2αt E

t

e αs dWs

2

0

= e −2αt V X0 + σ 2 e − 2αt

t

e2αs ds 0

= e −2αt V X0 +

σ2 1 − e − 2αt 2α

Looking at the structure of the equation for Xt , we can see that, for arbitrary X0, t

and σ

lim e −αt X0 = 0a s +∞

t

e − α t −s dWs is N 0, σ 2 2α 1 −e − 2αt . Hence, the distribution of

0

Xt N 0, σ 2 2α as t +∞ with X0 arbitrary. If X0 is normally distributed or constant, then Xt is also normally distributed, that is, if X0 N 0,σ 2 2α , then Xt N 0, σ 2 2α . ■ Example 5.11

Consider item no. 6 of footnote 3

E e αWt = e α

2

2t

. It is

instructive to verify this equality using Itô’s formula (4.18). Given the SDE dXt = f Xt , t dt + g Xt , t dWt , set f Xt ,t = 0,g Xt , t = 1, Xt = Wt , so that dXt = dWt , and choose V Xt ,t = e αWt . From Itô’s formula, dV = 0 + 0 + Vt = V0 +

α2 2

t 0

1 1 α2 e αWt dt + α e αWt dWt , 2 t

e αWt ds + α e αWt dWt , 0

151

152

5 Stochastic Differential Equations

and thus E Vt = E V0 + t

(E t

α2 2

t

E Vs ds 0

e αWt dWt = 0 since, for any nonanticipatory random variable Vt,

0

Vs dWs = 0). Then

E 0

dE Vt α2 = E Vt 2 dt (via the fundamental theorem). Hence, dE Vt α2 = dt, 2 E Vt ln E V t =

α2 t, 2

and thus E Vt = e α 5.3.3

2

2t

= E e αWt . ■

Solutions to SDEs as Markov Processes

We now turn to the examination of solutions to SDEs as Markov processes. In particular, the solution Xt of the SDE (5.1) dXt = f Xt ,t dt + g Xt ,t dWt , t ≥ 0 is a Markov process, with the probability law for any such process completely determined by the initial distribution P0 B = P X 0

B ,B B

5 47

(where the initial value X 0 = X0 is an arbitrary random variable), and the transition probability P x,s; B, t = P Xt

B Xs = x ,

5 48

where 0 ≤ s ≤ t ≤ T , and B B. For x and s fixed, the transition probability P x, s; B, t = P Xt x, s

B

5 48 1

is exactly the distribution of the solution Xt = Xt x, s of the integral equation t

Xt = x +

t

f Xr , r dr + s

g Xr , r dWr , s ≤ t

s

Stated more formally, we have Theorem 5.5.

5 49

5.3 Linear SDEs

Theorem 5.5 Let Xt be a solution of the SDE (5.1) or (Arnold, 1974; Gard, 1988; Mao, 2007) dXt = f Xt , t dt + g Xt , t dWt , t ≥ 0, with the coefficient functions f and g satisfying the conditions of the existence and uniqueness theorem (Theorem 5.1). Then Xt is a Markov process whose initial and transition probabilities are given by (5.47) and (5.48.1), respectively, and where Xt = Xt x, s is a solution to (5.49). (It is important to note that while the distribution of X0 determines the initial probability of the process Xt, the transition probabilities do not depend on X0 but are completely specified by the coefficient function f and g.) In order for the solution to exhibit the strong Markov property, the conditions/assumptions of Theorem 5.5 must be strengthened somewhat to yield Theorem 5.5a: Theorem 5.5a Let Xt be a solution of SDE (5.1) and assume that the coefficient functions f, g are uniformly Lipschitz continuous and satisfy the linear growth condition (conditions (5.3) and (5.4), respectively, of Theorem 5.1) for all x, y R and t ≥ 0 (Mao, 2007). Then Xt is a strong Markov process. A Markov process is said to be homogeneous if the transition probabilities are stationary, that is, P x, s + u;B, t + u = P x, s; B, t ,0 ≤ u ≤ T − t, is identically satisfied. Moreover, for fixed x and B, the transition probability is a function of t − s only or P x, s; B, t = P t − s, x, B . Suppose a process in differential form is characterized by an autonomous SDE, that is, the coefficient functions f Xt , t ≡ f Xt and g Xt ,t ≡ g Xt are independent of t on [0, T] so that dXt = f Xt dt + g Xt dWt , t ≥ 0

5 50

Also, let f (Xt), g(Xt) satisfy the conditions of the existence and uniqueness theorem (Theorem 5.1). Then the solution Xt of (5.50) is a homogeneous Markov process with stationary transition probabilities. In addition, if f (Xt), g(Xt) are uniformly Lipschitz continuous, with the linear growth condition satisfied, then the solution Xt to (5.50) is a homogeneous strong Markov process. To fully appreciate the stochastic nature of SDEs, we now assert that a solution of an SDE can be viewed as a specialized type of stochastic process called an Itô diffusion. More formally, a one-dimensional Markov process Xt t t0, T having continuous sample paths with probability 1 is termed a diffusion process if its transition probability P(x, s; B, t) is smooth in the sense that it satisfies the following conditions. For every s t0 , T , x R, and ε > 0,

153

154

5 Stochastic Differential Equations

a limt

s

1 t −s

y− x > ε

b limt

s

1 t −s

y− x < ε

c limt

s

1 t −s

y −x < ε

p x, s; t, y dy = 0; y − x p x, s; t, y dy = a x, s ; and

5 51

y −x 2 p x, s; t, y dy = b x,s 2 ,

where p(x, s; t, y) is the transition density; a(x, s) is the drift coefficient function; and b(x, s) is the diffusion coefficient function of the diffusion process. Here, a(x, s) may be viewed as the instantaneous rate of change in the mean of the process given that Xs = x; and b(x, s)2 can be thought of as the instantaneous rate of change of the squared fluctuations of the process given that Xs = x. The importance of (5.51a) is that it ensures that the diffusion process will not exhibit instantaneous jumps over time. To summarize, a diffusion process is a Markov process that has its probability law determined by the drift and diffusion coefficients. Such coefficients represent the conditional infinitesimal mean and variance of the process, respectively. We noted earlier that the solutions Xt of SDEs such as (5.1) are Markov processes; such solutions can also be thought of as representing arbitrary diffusion processes that, essentially, are conversions of standard Wiener processes. In fact, the standard Wiener process is a one-dimensional diffusion process with drift coefficient a x, s ≡ 0 and diffusion coefficient b x,s ≡ 1. We close with Theorem 5.6. Theorem 5.6 Given the SDE (5.1) or dXt = f Xt , t dt + g Xt ,t dWt , if the functions f and g satisfy the conditions of the existence and uniqueness theorem (Theorem 5.1), then any solution Xt is a diffusion process on the interval [0,T] with drift coefficient f(Xt, t) and diffusion coefficient g(Xt, t)2.4

5.4

SDEs and Stability

One of the difficulties associated with SDEs is that it may be impossible to find a solution that appears in a convenient and/or explicit form. Hence, it is 4 Conversely, suppose Yt is a diffusion process defined on some interval with a(y, t) and b(y, t) serving as drift and diffusion coefficients, respectively. Under what conditions will Yt satisfy an SDE? If the functions a and b satisfy the conditions of Theorem 5.1 and if {Wt} is any standard Wiener process, then the solution Xt of the SDE dXt = a Xt , t dt + b Xt , t dWt ,X t0 = Y t0 shares the same probability law as Yt so that {Xt} and {Yt} are equivalent processes (although their sample paths may not coincide with probability 1). (See Gard (1988, pp. 89–92), for a stronger version of this converse question.)

5.4 SDEs and Stability

important to focus on what qualitative information can be elicited about the solutions of SDEs without actually solving them. In this regard, in what follows we shall consider the notion of the stability of a solution. Simply stated, we shall consider the question of whether small changes in the initial conditions (state) or parameters of a dynamic system lead to small changes (stability) or to large changes (instability) in the solution. That is, the sample paths of a stable dynamic system that are “close” to each other at a particular point in time should remain so at all subsequent time points. Our point of reference for studying the stability of solutions to SDEs is the deterministic stability theory developed by A. M. Lyapunov (1892) for ODEs (see also Bucy (1965) and Hahn (1967)). This will be our “jumping off point” for the assessment of stability for solutions to SDEs. Lyapunov’s direct (second) method for determining stability will be applied to an ODE of the form dXt = f Xt , t dt, X t0 = X0 , t ≥ t0 ,

5 52

where Xt R. To investigate this technique, let us first develop some basic definitions. Suppose that for every initial value X0, there exists a unique global solution Xt(t0, X0) defined on t0 , +∞ and that f Xt , is continuous. In addition, assume that f 0,t = 0 for all t ≥ t0 . Then (5.52) is said to have a trivial or equilibrium solution Xt ≡ 0 corresponding to the initial value X0 = 0. This equilibrium is stable if for every ε > 0 there exists a δε, t0 > 0 such that Xt t0 ,X0 < ε for all t ≥ t0 whenever X0 < δε, t0 . Thus, small changes in X0 do not produce large changes in the solution for t ≥ t0 . Otherwise, the solution Xt ≡ 0 is said to be unstable. The equilibrium solution is termed asymptotically stable if it is stable and there exists a δt0 such that limt +∞ Xt t0 ,X0 = 0 whenever X0 < δt0 . If δ does not depend on t0, then stability is characterized as uniform. In addition, if the preceding limit holds for all X0, then Xt ≡ 0 is termed globally asymptotically stable (or asymptotically stable in the large). To cover the basics of Lyapunov stability for ODEs, we note first that a continuous scalar-valued nondecreasing function v(x) defined on a suitably R is restricted neighborhood of the zero point Nh = x x < h, h > 0 (Lyapunov) positive definite if v 0 = 0 and v x > 0, x Nh , for all x 0. More broadly, a continuous function v(x, t) defined on Nh × t0 , +∞ is positive definite if v 0,t ≡ 0 and there exists a positive definite function w(x) such that v x, t ≥ w x for t ≥ t0 . A function v is negative definite if −v is positive definite. A continuous non-negative function v(x, t) is termed decrescent (i.e., it has an arbitrarily small upper bound) if there exists a positive definite function u(x) such that v x,t ≤ u x for t ≥ t0 . (If the positive definite function v(x) is independent of t, then it is decrescent.) The function v(x, t) is radially unbounded if lim x +∞ inf t ≥ t0 v x, t +∞.

155

156

5 Stochastic Differential Equations

Now, suppose Xt is a solution to (5.51) and v(Xt, t) is a positive definite function having continuous first partial derivatives with respect to t and Xt. In addition, let vt = v Xt , t depict a function of t whose derivative, via (5.52), is dVt ∂v dXt ∂v ∂v ∂v = + = f Xt , t + dt ∂Xt dt ∂t ∂Xt ∂t

5 53

If dVt dt ≤ 0, then Xt varies in a fashion that would not cause Vt to increase, that is, the distance between Xt and the trivial or equilibrium solution, as measured by v(Xt, t), does not increase. If dVt dt < 0, then Vt will decrease to zero and thus the distance from Xt to the trivial solution concomitantly diminishes or Xt 0. The preceding discussion serves as the foundation of the Lyapunov method and leads to the following (sufficient) condition for the stability of an equilibrium solution to an ODE. Specifically, we have Theorem 5.7. Theorem 5.7 i. If there exists a positive definite function Vt = v Xt , t with continuous first partial derivatives such that the derivative of Vt along the solutions to (Lyapunov, 1892) dXt = f Xt , t dt, t ≥ 0,f 0, t ≡ 0,

5 54

satisfies dVt ∂v ∂v = f Xt , t + ≤ 0 ∂t dt ∂Xt

5 55

for all Xt ,t Nh × t0 , +∞ , then the trivial or equilibrium solution to (5.52) is stable. ii. If there exists a positive definite decrescent function Vt = v Xt ,t with continuous first partial derivatives such that dVt/dt taken along the solutions of (5.54) is negative definite for Xt , t Nh × t0 , +∞ , then the trivial or equilibrium solution to (5.52) is asymptotically stable. A function v(Xt, t) that satisfies the stability conditions (i) and (ii) of Theorem 5.5a is termed a Lyapunov function corresponding to the ODE (5.52). For some ODEs, a Lyapunov function may be difficult to find. However, if an appropriate Lyapunov function can be readily determined, then the Lyapunov method can be an extremely valuable tool for determining if an equilibrium point is stable. Example 5.12 The ODE dXt dt = f Xt , t = −Xt −Xt2 , Xt 0 = X0 = 0,t t0 , +∞ ,Xt R, has an equilibrium point at Xt = 0. Demonstrate that v Xt = Xt serves as a Lyapunov function for this ODE. Here, ∂v f Xt ,t = 1 − Xt − Xt2 = −Xt −Xt2 ≤ − v Xt ∂Xt

5.4 SDEs and Stability

Then along the trajectories of (5.54), d v Xt t0 , X0 , t ≤ −v Xt t0 , X0 , t ; dt and thus dv Xt t0 ,X0 , t ≤ −v Xt t0 , X0 , t dt or v Xt t0 ,X0 , t ≤ v X0 e −

t −t0

for all X0 and t ≥ t0 .5 Hence, the trivial point Xt = 0 is uniformly globally asymptotic stable. ■ When we attempt to transfer the principles of Lyapunov stability for deterministic ODEs to their stochastic counterparts, certain compelling questions present themselves. 1. How should stochastic stability be defined? 2. How should Lyapunov functions be defined, and what properties should a stochastic Lyapunov function possess? 3. What should the condition dVt dt ≤ 0 be replaced by in order for us to make stability assessments? To address these questions, let us first introduce a set of operational assumptions. Our investigation of the stability of solutions to SDEs will focus on Equation (5.1) or dXt = f Xt , t dt + g Xt , t dWt , t ≥ t0 ,

51

where (a) the assumptions of the existence and uniqueness theorem (Theorem 5.1) are satisfied; (b) the initial value X t0 = X0 R is taken to be, with probability 1, a constant; and (c) for any X0 independent of Wt ,t ≥ t0 , Equation (5.1) has a unique global solution Xt(t0, X0) with continuous sample paths and finite moments; and d f 0,t = 0 and g 0, t = 0 for t ≥ t0 so that the unique solution Xt ≡ 0 corresponds to the initial value X t0 = X0 = 0. Mirroring the deterministic case treated earlier, this solution will also be termed the trivial or equilibrium solution, but now to SDE (5.1). For Xt the solution to (5.1), let Vt = v Xt , t be a positive definite function defined on Nh × R + , Nh = Xt Xt < h, h > 0 , and which is continuously twice differentiable in Xt and once in t. Then the operator L associated with (5.1) is

5 Since dv v = −dt, lnv = −t + lnC, where C is a constant of integration. Then v = Ce − t and, at t0 , X0 , C = v X0 e t0 so that the solution readily follows.

157

158

5 Stochastic Differential Equations

∂ ∂ 1 ∂2 + f Xt , t + g Xt , t 2 ; ∂t ∂Xt 2 ∂Xt 2

L=

5 56

and if L acts on the function Vt = v Xt , t , then Lv Xt ,t =

∂v ∂v 1 ∂2 v + f Xt , t + g Xt , t 2 ∂t ∂Xt 2 ∂Xt 2

Note that, via Itô’s formula (4.18), if Xt dv Xt ,t = Lv Xt , t dt + g Xt , t

5 57

Nh , then

∂v dWt ∂Xt

5 58

In the presence of the random term dWt in (5.58) (since g(Xt, t) 0) a stable solution Xt to (5.1) requires that, on the average, E dVt ≤ 0. But since E dVt = E Lv Xt , t dt , this condition is satisfied if Lv Xt ,t ≤ 0 for all t ≥ 0. Clearly, the inequality dVt dt ≤ 0 required for stability in the deterministic case (with g Xt , t ≡ 0) is replaced by Lv Xt , t ≤ 0 in order to make stochastic stability statements. Thus, the function Vt = v Xt , t serves as the stochastic Lyapunov function corresponding to SDE (5.1). Armed with these considerations, our discussion of stability of solutions of SDEs must now be framed in terms of probability, that is, suppose assumptions (a)–(d) are satisfied. Then the trivial or equilibrium solution Xt ≡ 0 is termed stochastically stable or stable in probability if for every ε > 0, lim P

X0

0

sup t

Xt t0 , X0 ≥ ε = 0;

5 59

t0 , +∞

otherwise, the said solution is stochastically unstable. The equilibrium solution is stochastically asymptotically stable if it is stochastically stable and lim P

X0

0

t

lim Xt t0 , X0 = 0 = 1 +∞

5 60

The equilibrium solution is globally stochastically asymptotically stable or stochastically asymptotic stable in the large if it is stochastically stable and, for all X0 R, P

t

lim Xt t0 , X0 = 0 = 1 +∞

5 61

To extend the Lyapunov approach for determining the stability of ODEs to that of SDEs, let us consider the general stochastic stability theorem (Theorem 5.8).

Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988)

Theorem 5.8 i. Suppose assumptions (a)–(d) are in effect and that there exists a positive definite stochastic Lyapunov function v(Xt, t) defined on Nh × t0 , +∞ that is everywhere continuously twice differentiable in Xt and once in t. In addition, Lv Xt ,t ≤ 0, t ≥ t0 ,0 ≤ Xt Nh , where L is given by (5.56) ( or Lv(Xt, t) is determined by (5.57)). Then the trivial or equilibrium solution of (5.1) is stochastically stable. ii. If, in addition, v(Xt, t) is decrescent and Lv(Xt, t) is negative definite, then the equilibrium solution is stochastically asymptotically stable. iii. If the assumptions of (ii) hold for a radially unbounded function v(Xt, t) defined everywhere on t0 , +∞ × R, then the equilibrium solution is globally stochastically asymptotically stable (Has’minskiy, 1980). Example 5.13 Suppose an SDE has the form dXt = μXt dt + σXt dWt , t ≥ 0. If we select Vt = v Xt ,t = Xt2 as the stochastic Lyapunov function, then 1 Lv = 0 + μXt 2Xt + σ 2 Xt2 2 2 1 = 2 μ + σ 2 vt ≤ 0 2 if μ + 1 2 σ 2 < 0. If this inequality holds, then the trivial solution Xt = 0 is stochastically asymptotically stable in the large. ■

Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988) 5.A.1

Linear Homogeneous Variety

Suppose we have a linear SDE of the form dXt = bt Xt dt + et Xt dWt , X 0 = X0

5A1

Our objective is to find a solution to (5.A.1) in product form or X t = X1 t X2 t ,

5A2

where a dX1 t = et X1 t dWt , X1 0 = X0 ; b dX2 t = At dt + Bt dWt , X2 0 = 1,

5A3

159

160

5 Stochastic Differential Equations

where the process coefficients At, Bt are, as yet, undetermined. From the product rule (4.A.6) and Itô’s multiplication table (Table 4.A.1), dX = d X1 X2 = X1 dX2 + X2 dX1 + dX1 dX2 = X1 dX2 + X2 dX1 + et X1 dWt At dt + Bt dWt

5A4

= X1 dX2 + X2 dX1 + et X1 Bt dt = X1 dX2 + X2 et X1 dWt + et X1 Bt dt = et X dWt + X1 dX2 + et Bt dt Next, select At and Bt in (5.A.3b) so that dX2 + et Bt dt = bt X2 dt

5A5

(Under this equality, (5.A.4) becomes dX = et XdWt + bt Xdt, which is consistent with (5.A.1).) Given (5.A.5), for (5.A.3b) to hold, set Bt ≡ 0 and take At = bt X2 so that (5.A.3b) becomes dX2 = bt X2 dt, X2 0 = 1

5A6 t

With this expression nonrandom, upon integrating we obtain ln X2 =

bs ds or 0

t

X2 = e

bs ds

5A7

0

To solve (5.A.3a) or dX1 t = et X 1 t dWt , X1 0 = X0 , let us employ Itô’s formula (4.18). To this end, we have, for V X1 t , t = ln X1 t and g = et X1 t , 1 dV = 0 + 0 − e2t X1 t 2

1

2

X1 t

dt + et X1 t

2

1 X1 t

dWt

or 1 d ln X1 t = − e2t dt + et dWt , 2 t 0

dln X1 t = −

1 2

t

t

0

e2s ds +

ln X1 t −ln X1 0 = − t

X1 t = X0 exp 0

1 2

es dWs , 0

t 0

5A8

t

e2s ds +

es dWs −

1 2

es dWs , 0

t 0

e2s ds

Appendix 5.A Solutions of Linear SDEs in Product Form (Evans, 2013; Gard, 1988)

Upon combining (5.A.7) and (5.A.8), our product form solution to (5.A.1) is X t = X 1 t X2 t t

t

bs ds +

= X0 exp 0

5.A.2

es dWs −

0

1 2

t 0

e2s ds

5A9

Linear Variety

More generally, suppose our linear SDE now has the form dXt = at + bt Xt dt + ct + et Xt dWt , X 0 = X0

5 A 10

Again a solution of the product form X t = X 1 t X2 t

5 A 11

is sought, where a dX1 t = bt X1 t dt + et X1 t dWt , X 0 = 1; b dX2 t = At dt + Bt dWt , X2 0 = X0 ,

5 A 12

and again the process coefficients At, Bt are, at this point, unknown. Again invoking the product rule (4.A.6) and Table 4.A.1, dX = d X1 X2 = X1 dX2 + X2 dX1 + dX1 dX2 = X1 dX2 + X2 dX1 + bt X1 dt + et X1 dWt At dt + Bt dWt = X1 dX2 + X2 dX1 + et X1 Bt dt = X1 At dt + Bt dWt + X2 bt X1 dt + et X1 dWt + et X1 Bt dt = X1 At dt + Bt dWt + bt Xdt + et XdWt + et X1 Bt dt 5 A 13 Now, select At, Bt in (5.A.12b) so that X1 At dt + Bt dWt + et X1 Bt dt = at dt + ct dW

5 A 14

(If this equality holds, then (5.A.13) becomes dX = at dt + et dWt + bt Xdt + et XdWt = at + bt X dt + ct + et X dWt , which is consistent with (5.A.10).) So given (5.A.14), for this expression to hold, set At = at − et ct X1−1 ; Bt = ct X1−1

5 A 15

161

162

5 Stochastic Differential Equations

We determined earlier that an expression such as (5.A.12a) integrates to t

t

X1 t = exp

1 bs − e2s ds 2

es dWs + 0

0

5 A 16

(e.g., use Example 5.1 as a guide). Given (5.A.15), (5.A.12b) integrates to t

a s − e s c s X1 s

X2 t = X0 +

−1

t

ds +

0

cs X1 s

−1

dWs

5 A 17

0

Combining (5.A.16) and (5.A.17) renders the product form solution to (5.A.10) X t = X1 t X2 t t

t

1 bs − e2s ds 2

es dWs +

= exp 0

0 t

× X0 +

as −es cs X1 s

−1

5 A 18 t

ds +

c s X1 s

0

−1

dWs ,

0

where X1 s

−1

= exp −

s

eu dWu −

0

s 0

1 bu − e2u du 2

5 A 19

A more conventional representation of this solution appears as X t =Φ t

t

X0 + 0

as −es cs Φs−1 ds +

t 0

cs Φs−1 dWs ,

5 A 20

where t

Φ t = exp

t

es dWs + 0

Φs− 1 = exp −

0 s 0

eu dWu −

s 0

1 bs − e2s ds ; 2

5 A 21

1 bu − e2u du 2

5 A 22

Appendix 5.B Integrating Factors and Variation of Parameters In this section, we shall consider two additional techniques that can be utilized to solve SDEs, namely the use of integrating factors and the method of variation of parameters.

Appendix 5.B Integrating Factors and Variation of Parameters

5.B.1

Integrating Factors

Consider a class of SDEs of the form dXt = f Xt , t dt + c t Xt dWt , X 0 = X0 ,

5B1

where f R × R T and C R R are given continuous deterministic functions. Given (5.B.1), let the integrating factor be expressed as ρt = exp −

t

c s dWs + 0

1 2

t

c s 2 ds

0

Then d ρt Xt = ρt f Xt , t dt

5B2

represents an exact SDE. To see this, suppose Yt = ρt Xt . Then Xt = ρt−1 Yt , and thus dYt = d ρt Xt = ρt f Xt , t dt

5B3

= ρt f ρt− 1 Yt , t dt

Here, (5.B.3) represents a deterministic differential equation in Yt as a function of t, that is, as (5.B.2) reveals, multiplying both sides by the integrating factor, we obtain a deterministic differential equation. For example, let us solve dXt =

1 dt + αXt dWt , X 0 = X0 , α a constant Xt

Clearly, this is an SDE of the form (5.1) with f Xt ,t = 1 Xt , g Xt ,t = αXt , and, c t = α. Hence, ρ t = exp −

t

c s dWs + 0

1 2

t

c s 2 ds

0

1 = exp −dWt + α2 t , 2 1 Xt = ρt−1 Yt = exp αWt − α2 t Yt , 2 dYt = ρt f ρt− 1 Yt , t dt 1 = exp −αWt + α2 t 2

1 ,and exp αWt − 1 2 α2 t Yt

1 Yt dYt = exp 2 − αWt + α2 t 2

dt

163

164

5 Stochastic Differential Equations

Then t

t

Yt dYt = 0

exp − 2αWt + α2 t dt,

0 t

Yt2 = Y02 + 2 exp − 2αWt + α2 t dt, 0

1 2

t

Yt = Y02 + 2 exp − 2αWt + α2 t dt , 0

ρ t Xt =

t

Y02

1 2

+ 2 exp −2αWt + α t dt , 2

0

and thus 1 Xt = exp αWt − α2 t 2

t

X02

1 2

+ 2 exp − 2αWt + α t dt , 2

0

where Y0 = ρ0 X0 = X0 . 5.B.2

Variation of Parameters

Suppose our objective is to obtain a solution to the geometric BM SDE (5.15) or to dXt = μXt dt + σXt dWt , with μ and σ constant. The general procedure that we shall follow in order to accomplish this task has two basic steps. 1. Transform the SDE into an equivalent integral form and integrate directly and arrive at an artificial solution with c representing an arbitrary constant of integration. (The resulting solution is termed artificial since Xt does not satisfy the original SDE.) For instance, from the preceding equation let us write dXt = μdt + σdWt , Xt dXt = dt + σ dWt , Xt ln Xt = μt + σWt + c, so that the artificial solution is Xt = e μt + σWt + c

Appendix 5.B Integrating Factors and Variation of Parameters

2. Replace c by the function c(t) (the variation of the parameter c with t), Xt = e μt + σWt + c t , and apply Itô’s formula (4.18) to this expression so as to obtain dXt = Xt μ + c t +

σ2 dt + σXt dWt 2

Substituting the last term from the original SDE gives c t +

σ2 dt = 0 2

or c t =−

σ2 t + c1 , c1 a constant 2

Then Xt = e μt + σWt + c t =e

μt + σWt −

σ2 t + c1 2

For t = 0,X0 = e c1 so that our final solution is 2

Xt = X0 e

μ − σ2 t + σWt

165

167

6 Stochastic Population Growth Models 6.1

Introduction

It is a well-known fact that all biological populations exhibit some form of stochastic behavior and that environmental noise should thus be an integral component of any dynamic population model. Such noise is generally introduced as a stochastic (Wiener) process and, in particular, as a Brownian motion (BM) process. In general, a dynamic model of a population will have deterministic and stochastic components that operate simultaneously. Deterministic elements serve to make the level of the response variable predictable from the initial conditions, while stochastic elements are typically attributable to the following sources: 1. Demographic stochasticity—refers to random factors that might impact personal reproduction, mortality, etc., and that are thus taken to be individual specific. Since this variety of stochasticity operates independently among population elements, it tends to “average itself out” in larger populations; its impact is mostly an issue for small populations. Under demographic stochasticity, random factors affect the individual birth and death processes. 2. Environmental stochasticity—refers to the effect on local and global populations (both large and small) of factors such as the weather, major accidents/ catastrophes, epidemics, natural disasters, crop failures, and international (political, financial, and societal) dislocations. It is assumed that these factors operate independently over the long run. For environmental stochasticity, random fluctuations in the environment affect the entire population. 3. Mensuration stochasticity—refers to the notion that measurement error resulting from determining, say, the size of a population at various points in time is a legitimate source of uncertainty that may result in biased assessments of some phenomenon. 4. Informational stochasticity—refers to the possibility that critical errors of omission or specifications may be made when trying to model reality, either

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

168

6 Stochastic Population Growth Models

through structural deficiencies or through inappropriate/unrealistic modeling assumptions. The preceding discussion has indicated that it is imperative to explicitly take into account random phenomena in our modeling activities. This, ostensibly, will enable us to enhance the accuracy of our estimates/predictions of population growth (rates) and sustainability. This need for greater accuracy has prompted the use of stochastic process models, and these types of models can be solved via stochastic differential equations (SDEs), that is, the sample paths of the process can be described by an SDE. The fundamental deterministic population growth model offered in Section 6.2 will serve as our starting point for developing population growth models that, for the most part, can be solved (in closed form) using SDEs: models that incorporate random effects attributed to, say, environmental fluctuations.

6.2

A Deterministic Population Growth Model

Let us express the basic continuous-time population growth model as dN t = rN t , r = constant, 61 dt that is, the instantaneous absolute growth rate is proportional to the existing population level at time t, or the instantaneous relative growth rate of the population is dN t dt = r = constant N t

62

For a single-species biological population with no migration, we may view r in the following fashion. Given that dN t = births − deaths, 63 dt if the births and deaths are each taken to be proportional to N(t), then births = bN(t) and deaths = dN(t), where b is the constant per capita birth rate and d is the constant per capita death rate. Then (6.3) becomes dN t = bN t − dN t = b− d N t dt = rN t ,

64

where r = b − d is the average per capita number of offspring less the average per capita number of deaths per unit time. Hence, r will be termed the constant net birth rate or the constant per capita growth rate of the population. The solution of the ordinary differential equation or ODE (6.4) is N t = N0 e rt , N 0 = N0

65

6.3 A Stochastic Population Growth Model

So if r > 0, then b > d, and thus the population grows exponentially. If r < 0, then b < d, and thus the population declines exponentially. Our introductory discussion (Section 6.1) has indicated that, for instance, environmental variability will affect a biological population in a random fashion. To model environmental effects, one possibility is to explicitly include additional variables (e.g., food supply, rainfall, and average temperature) into, say, Equation (6.4). In a nonstatic environment, the per capita birth and death rates (b and d, respectively) could be expressed as functions of the aforementioned additional environmental variables. So as these supplemental variables change, then so do the per capita birth and death rates. However, the inclusion of additional explanatory variables into the expressions for b and d may unduly complicate the model. As an alternative, let us approximate the effect of environmental variability on population growth by varying the per capita birth and death rates in a random fashion. To this end, let us hypothesize that changes in the environment precipitate random changes in the population’s per capita birth and death rates, or changes in the net birth rate r. Moreover, changes in r are deemed independent of changes due to demographic, mensuration, and information stochasticity. Section 6.3 introduces, in a very rudimentary fashion, how random effects are readily incorporated into an elementary deterministic population growth model. More complicated stochastic models then follow.

6.3

A Stochastic Population Growth Model

Let us return to the population growth model offered in Section 4.1. Given that N(t) depicts population at time t, we posited that dN t = a t N t , N 0 = N0 , t ≥ 0, dt

66

where a(t) is the relative population growth rate. Then, based upon the arguments underlying Equations (4.3)–(4.5.1), we can write (6.6) as dN t = r t − α t noise N t , N 0 = N0 , t ≥ 0 dt

67

If r(t) = r = constant, α t = α = constant, and noise = dW t dt, with W 0 = W0 = 0, then (6.7) becomes dN t dW t = r+α dt dt

N t , N 0 = N0 , t ≥ 0,

68

or dN t = rN t dt + αN t dW t , N 0 = N0 , t ≥ 0

681

169

170

6 Stochastic Population Growth Models

Clearly, this SDE (which is autonomous, linear, and homogeneous) is of the form (5.15) and thus has a particular solution mirroring Equation (5.18) or N t = N0 e

r − 12 α2 t + αWt

,

69

a geometric BM process. (Note that the solution to (6.8.1) is also provided by Equation (5.A.9).) As far as the long-term dynamic behavior of (6.9) is concerned, 1 +∞ as t +∞; i if r > α2 , then N t 2 1 0 as t +∞; and ii if r < α2 , then N t 2 1 iii if r = α2 , then N t fluctuates between arbitrarily large 2 +∞ and arbitrarily small values as t

6 10

Suppose N0 is independent of Wt, then via Example 5.9, a E N t = N0 e rt ; b V N t = N0 e2rt eα t − 1 2

6 11

6.4 Deterministic and Stochastic Logistic Growth Models If the per capita rate of growth of a population depends upon population density, then the population cannot expand without limit. If a population is far from its growth limit, then it can possibly grow exponentially. However, when nearing its growth limit, the population growth rate declines significantly and reaches zero when that limit has been attained (Verhulst, 1838). (If the limit is exceeded, the population growth rate becomes negative so that the size of the population declines.) Verhulst proposed a long-run adjustment to exponential growth via a self-regulating mechanism (it compensates for overcrowding) that becomes operational when population becomes “too large” (in the sense that the population size approaches a “limiting population” that is determined by the carrying capacity of the environment). To model this phenomenon, let us modify r by multiplying it by an “overcrowding term” so that we obtain a density-dependent population growth equation, that is, using (6.2), set dN t dt N t = r 1− N t K

= r n t , N 0 = N0 ,

6 12

6.4 Deterministic and Stochastic Logistic Growth Models

r (n(t)) a. if N(t) < K, then b. if N(t) = K, then c. if N(t) > K, then 0

dN(t) dt N(t) dN(t) dt N(t) dN(t) dt N(t)

> 0; = 0; and < 0.

N(t)

K

Figure 6.1 Per capita rate of growth is density dependent.

so that the instantaneous relative growth rate is a decreasing function of N t or r n t = r 1 − N t K . As Figure 6.1 reveals, (6.12) implies a constant linear decrease in r as population size increases, where r is the constant maximum population growth rate when N(t) = 0, and K represents the limiting size or carrying capacity of the population.1 The solution to the ODE (6.12) or to dN t = r 1 −

N t K

6 12 1

N t dt,

is termed the (deterministic) logistic population growth function and appears as N t =

KN0 , N 0 = N0 , t ≥ 0 N0 + K −N0 e −rt

6 13

Given this expression, it is readily seen that i. for 0 < N0 < K, populations increases asymptotically to K, that is, population follows a logistic curve; ii. for N0 > K , population decreases asymptotically to K; iii. for N0 = K , the population level N(t) = K for all t; and iv. for N0 = 0, N t = 0 for all t. Looking to the specification of the stochastic version of the logistic population growth model, let us augment (6.12.1) with the multiplicative noise term 1 How might K be determined? One way to rationalize the choice for K is to assume that births = b1 −b2 N t and deaths = d1 + d2 N t , where b1, b2, d1, and d2 are positive scalars. Then, dN t = births− deaths N t = b1 −d1 − b2 + d2 N t N t dt = b1 −d1

1−

N t b1 −d1 b2 + d2

where r = b1 − d1 > 0 and K = b1 −d1

N t = r 1− b2 + d2 .

N t K

N t ,

171

172

6 Stochastic Population Growth Models

αN(t)dW(t). (Note: additive noise appears as αdW(t).) For convenience, we shall work with the resulting equation in the form dNt = rNt 1 −

Nt dt + αNt dWt , N 0 = N0 , t ≥ 0 2 K

6 14

Turning to Itô’s equation (4.18) for solving the SDE dXt = f Xt , t dt + g Xt , t dWt , t ≥ 0, or 1 dV X t , t = Vt X t , t + f X t , t VX + g X t , t 2 VXX dt 2 + g X t , t VX dW t , let V Nt , t = Nt− 1 , VN = − Nt− 2 , VNN = 2Nt− 3 , f = rNt 1 − = r Nt − dV = 0 + r Nt −

Nt K

Nt 2 , and g = αNt K

Nt2 K

1 −Nt−2 + α2 Nt2 2Nt−3 2

dt

+ αNt − Nt− 2 dWt , d Nt−1 = r −Nt−1 +

1 + α2 Nt− 1 dt − αNt−1 dWt K

6 18

To linearize (6.18), set Nt− 1 = yt so that r + −r + α2 yt dt − αyt dWt dyt = 6 19 K From Equations (5.A.20)–(5.A.22), (6.19) becomes, for a = r K , b = − r + α2 , c = 0, and e = − α, t r Φs− 1 ds , 6 20 yt = Φ t y0 + K 0 where Φ t = exp

t

− α dWs +

0

t 0

1 − r + α2 ds , 2

6 21

2 An alternative way of stochasticizing the deterministic logistic model is to write dNt dt Nt

= a t 1 − NKt , N 0 = N0 , t ≥ 0,

(6.15)

and replace a(t) in this expression by r + α dWt dt (see (6.6)–(6.8)) so that we obtain dNt dt Nt

t = r + α dW 1− NKt , N 0 = N0 , t ≥ 0, dt

(6.16)

or dNt = Nt 1 − NKt rdt + αdWt , N 0 = N0 , t ≥ 0.

(6.17)

6.4 Deterministic and Stochastic Logistic Growth Models

Φs− 1 = exp −

s

− α dWu −

0

s

1 − r + α2 du 2

0

6 22

Evaluating (6.21) yields 1 Φ t = exp − αWt + −r + α2 t , 2

6 21 1

and (6.22) gives 1 Φs− 1 = exp αWs + r − α2 s 2

6 22 1

Then a substitution of (6.21.1) and (6.22.1) into (6.20) renders 1 yt = exp −αWt − r − α2 t 2 × y0 +

r K

t

1 r − α2 s + αWs ds 2

exp 0

or exp r − 1 2 α2 t + αWt

Nt = N0− 1

t

,

6 23

exp r − 1 2 α s + αWs ds 2

+ r K 0

the stochastic logistic population growth equation, where y0 = N0−1 . (See Appendix 6.A for a discussion of both the deterministic and stochastic logistic population growth equations with an Allee effect.) Since the stochastic logistic model is widely used in areas such as biology, engineering, and so on, a useful exercise is to determine the mean and variance of the process (6.23). To this end, we first rewrite this expression as Nt =

K exp r − 1 2 α2 t + αWt t

K N0 + r

6 23 1

exp r − 1 2 α2 s + αWs ds

0

Then employing the expectation rules presented in footnote 3 of Chapter 5, we can write t

t

0

0

E exp αWs ds = exp E a.

t

= exp b. E exp r − 12 α2 t + αWt c. from (6.22.1),

0

αWs 2

2

α2 s ds; 2 = e rt ; and

ds

173

174

6 Stochastic Population Growth Models t

Φs− 1 ds

E exp r 0

t

1 r − α2 s + αWs ds 2

E exp r 0 t 0 t

exp r

=

1 r − α2 s + αWs ds = 2

E

exp r

=

rs ds = e rt − 1

0

Given these results, it follows, from (6.23.1), that E Nt =

K e −rt

K N0 +

e rt −1

=

K 1 + K N0 − 1 e −rt

Next, since V Nt = E Nt2 −E Nt 2 , we first obtain E Nt2 as K 2 e 2r + α t K N0 + e rt − 1 2

E Nt2 =

2

Then V Nt = K 2 e2rt

K + e rt − 1 N0

−2

eα t − 1 2

6.5 Deterministic and Stochastic Generalized Logistic Growth Models We may generalize the deterministic logistic population growth law (Richards, 1959) by considering the modified growth equation dNt Nt = βNt 1 − dt N∞

r

6 24

or dNt = βNt + ηNtr + 1 , N 0 = N0 , t ≥ 0, dt

6 24 1

where N∞ represents an upper horizontal asymptote (the saturation level) when t +∞ . As (6.24) indicates, the instantaneous absolute population growth rate is proportional to Nt times a feedback term. A glance back at Equation (6.24.1) reveals an ODE (Bernoulli’s equation) that integrates to the Richards growth function or the deterministic generalized logistic equation

6.5 Deterministic and Stochastic Generalized Logistic Growth Models

Nt = N∞ 1 +

−r

N0 N∞

− 1 e −βrt

− 1r

, N 0 = N0 , t ≥ 0

6 25

To obtain a stochastic generalized logistic growth equation, let us insert the multiplicative noise term αNtdWt into (6.24) so as to determine the SDE as follows: dNt = βNt 1 −

r

Nt N∞

dt + αNt dWt

6 26

Next, if we divide both sides of this equation by N∞ and set Yt = Nt N∞ , then (6.26) becomes Nt Nt Nt r Nt d =β 1− dt + α dWt N∞ N∞ N∞ N∞ or dYt = βYt 1 −Yt r dt + αYt dWt

6 27

To solve this autonomous, nonlinear SDE, we shall employ the reduction method (see Appendix 6.B for details). To determine an appropriate transformation function Zt = U Yt , let us find, from (6.B.11), A Yt =

f Yt 1 dg Yt − , g Yt 2 dYt

where f Yt = βYt 1 − Ytr and g Yt = αYt . Then A Yt =

β 1 1 − Ytr − α α 2

0

so that dA Yt β = − r Ytr − 1 α dYt In addition, with g Yt

dA Yt = − rβYtr dYt

and d dA Yt g Yt dYt dYt

= −r 2 βYtr − 1 ,

it follows from (6.B.14) that b2 = − rα so that (6.B.13) is satisfied. Then from (6.B.15a) and (6.B.9), U Yt = Ce − rαB

Yt

= Ce

−r ln

Yt Y0

= Yt− r

175

176

6 Stochastic Population Growth Models

(given that C and Y0 are arbitrary) is our transformation function for the reduction process. To determine the coefficients of (6.B.6), we look to (6.B.7). From (6.B.7a), 1 fUY + g 2 UYY = βYt 1 − Ytr 2 =

1 − rYt−r − 1 + α2 Yt 2 r r + 1 Yt−r −2 2

1 2 α r r + 1 − βr Zt + βr, 2

where Zt = Yt− r . Similarly, from (6.B.7b), gUy = −αrYt− r = − αrZt Hence, the reduced form of Equation (6.27) is the autonomous linear SDE dZt =

1 2 α r r + 1 − βr Zt + βr dt − αrZt dWt 2

6 28

To obtain the solution to (6.28), let us look to the general form offered for the solution to a linear SDE of the type AL dZt = a + bZt dt + c + eZt dWt (See Table 5.1.) Specifically, from Equation (5.A.20), Zt = Φt Z0 + a− ec

t 0

t

Φs−1 ds + c Φs− 1 dWs ,

6 29

0

where t 1 Φt = exp e dWs + b − e2 2 0

t

ds ;

6 30

0

s 1 Φs− 1 = exp − e dWu − b − e2 2 0

s

du

6 31

0

In terms of (6.28), 1 a = βr, b = α2 r r + 1 − βr, c = 0, and e = −αr 2 Evaluating (6.30) and (6.31) yields, respectively, Φt = exp −αrWt +

1 2 α r − βr t and 2

6 30 1

Φt− 1 = exp αrWs −

1 2 α r −βr s 2

6 31 1

Hence, (6.29) becomes

6.6 Deterministic and Stochastic Gompertz Growth Models t

Zt = Φt Z0 + βr

0

Φs−1 ds

6 29 1

Then substituting (6.30.1) and (6.31.1) into (6.29.1) gives 1 2 α − β t − αWt Zt = exp r 2 × Z0 + βr

t

6 29 2

1 − α2 + β s + dWs ds 2

exp r 0

−1 r

Since Zt = Yt−r , an application of the reverse transformation Yt = Zt (6.29.2) yields −1 r Yt = Φt

Y0−r

+ βr

t 0

Φs− 1 ds

to

− 1r

6 29 3

Since Nt = N∞ Yt , (6.29.3) becomes −1 r

Nt = N∞ Φt

N0 N∞

−r

+ βr

t 0

Φs−1 ds

− 1r

,

6 32

the stochastic generalized logistic population growth equation.

6.6 Deterministic and Stochastic Gompertz Growth Models Suppose we posit that the instantaneous rate of growth of the population Nt at time t is proportional to lnN∞ − lnNt or dNt dt = β ln N∞ − ln Nt 6 33 Nt (here we have a linear relationship between the instantaneous growth rate and ln Nt). The solution to this ODE is the deterministic Gompertz (1825) population growth model Nt = N∞ exp − α exp − βt −βt

= N∞ e −αe , N 0 = N0 , t ≥ 0,

6 34

where α = ln N∞ N0 .3 3 Note that from (6.34), we can obtain = αβe − βt so that dNt dt Nt

ln

dNt dt Nt

= ln αβ −βt.

We thus have a linear relationship between the logarithm of the instantaneous growth rate and t.

177

178

6 Stochastic Population Growth Models

To obtain the stochasticized version of the deterministic Gompertz model, let us rewrite (6.33). Set Xt = Nt N∞ . Then dXt 1 dNt = , dt N∞ dt and thus (6.33) becomes dNt dt Nt N∞ =β ln N∞ N∞ Nt or dXt = βXt ln Xt− 1 = −βXt ln Xt dt

6 35

Next, if we insert the multiplicative noise term kXtdWt into (6.35), we obtain the autonomous nonlinear SDE. dXt = − βXt ln Xt dt + kXt dWt ,

6 36

where f Xt = −βXt ln Xt and g Xt = kXt . To solve (6.36), let us apply the reduction method developed in Appendix 6.A. To this end, it can be shown that the requisite transformation function has the form Zt = U Xt = ln Xt .4 Turning to (6.A.7) we have, 1 1 1 fUX + g 2 UXX = −β lnXt − k 2 = −βZt − k 2 , gUX = k, 2 2 2 and thus the reduced form of Equation (6.36) is 1 dZt = − k 2 − βZt dt + kdWt , 2

6 37

an autonomous linear SDE. Looking to the general form for the solution to a linear SDE (see Table 5.1), we have AL dZt = a + bZt dt + c + eZt dWt In this regard, from Equation (5.A.20), Zt = Φt Z0 + a− ec

t 0

t

Φs−1 ds + c Φs− 1 dWs ,

6 38

0

where t 1 Φt = exp e dWs + b − e2 2 0

t

ds ;

6 39

0

4 From (6.A.11), A Xt = − βk lnXt − 2k1 . Then g dA dXt = −β and, thus, from (6.A.14), b2 = 0. Then (6.A.15b) yields U Xt = b1 B Xt + C = b1 k lnXt , where b1 = k and C = 0.

6.7 Deterministic and Stochastic Negative Exponential Growth Models s 1 Φs− 1 = exp − e dWu − b − e2 2 0

s

6 40

du 0

In terms of Equation (6.37), 1 a = − k 2 , b = − β, c = k, and e = 0 2 Evaluating (6.39) and (6.40) yields, respectively, Φt = exp − βt and Φs− 1 = exp βs Hence, (6.38) becomes t

t

0

0

Zt = Φt Z0 + a Φs−1 ds + c Φs−1 dWs = e −βt Z0 −

1 2 βt k e −1 + k 2β

t

e βs dWs

0

or ln Xt = e βt Z0 −

1 2 βt k e −1 + k 2β

t

e βs dWs

0

Hence, Xt = exp e − βt ln X0 −

1 2 βt k e −1 + k 2β

t

e βs dWs

,

6 41

0

the stochastic Gompertz growth function.

6.7 Deterministic and Stochastic Negative Exponential Growth Models Let the time profile of the values of a variable N reflect the operation of the deterministic negative exponential model Nt = N∞ 1 − e −βt , t ≥ 0,

6 42

where the parameters N∞ and β are both positive. Here, N∞ is the limit to growth parameter limt +∞ Nt = N∞ , while β is the growth rate parameter. For t = 0, the initial value of N is N0 = 0. It is readily shown that dNt dt N∞ =β −1 , Nt Nt

6 43

179

180

6 Stochastic Population Growth Models

that is, the instantaneous rate of growth in Nt at time t approaches zero as t +∞ and that this growth rate is proportional to the feedback term N∞ Nt − 1. To determine the stochastic version of the deterministic negative exponential model, let us write (6.43) as dNt = β N∞ −Nt dt

6 43 1

Then upon inserting a multiplicative noise term of the form αNtdWt into (6.43.1), we obtain the autonomous linear SDE dNt = βN∞ −βNt dt + αNt dWt

6 44

A glance at Table 5.1 reveals that for a general linear SDE of the form AL dNt = a + bNt dt + c + eNt dWt , the requisite solution (see Equation (5.A.20)) appears as t

Nt = Φt N0 + a− ec

0

t

Φs−1 ds + c Φs− 1 dWs ,

6 45

0

where t 1 Φt = exp e dWs + b − e2 2 0

t

ds ;

6 46

0

s 1 Φs− 1 = exp − e dWu − b − e2 2 0

s

du

6 47

0

In terms of Equation (6.44), a = βN∞ , b = −β, c = 0, and e = α Evaluating (6.46) and (6.47) yields, respectively, 1 Φt = exp αWt − β + α2 t and 2 1 Φs− 1 = exp − αWs + β + α2 s 2 Hence, (6.45) becomes 1 Nt = exp αWt − β + α2 t 2 ×

N0 + βN∞

t

exp 0

1 β + α2 s −αWs ds , 2

the stochastic negative exponential growth equation.

6 48

6.8 Deterministic and Stochastic Linear Growth Models

6.8 Deterministic and Stochastic Linear Growth Models Suppose the growth process is described by the deterministic linear growth function Nt = N0 + βt, t ≥ 0,

6 49

where dNt = β = constant dt or dNt = βdt

6 50

If we insert the multiplicative noise term αNtdWt into (6.50), then we obtain the autonomous linear SDE dNt = βdt + αNt dWt

6 51

Looking to Table 5.1, we see that the general SDE AL dNt = a + bNt dt + c + eNt dWt has the solution (using Equation (5.A.20)) Nt = Φt N0 + a − ec

t 0

t

Φs−1 ds + c Φs−1 dWs ,

6 52

0

where t 1 Φt = exp e dWs + b − e2 2 0

t

ds ;

s 1 Φs− 1 = exp − e dWu − b − e2 2 0

s

du 0

In terms of Equation (6.51), a = β, b = c = 0, and e = α Evaluating (6.53) and (6.54) yields, respectively, 1 Φt = exp αWt − α2 t and 2 1 Φs− 1 = exp − αWs + α2 s 2 Hence, (6.52) becomes

6 53

0

6 54

181

182

6 Stochastic Population Growth Models

1 Nt = exp αWt − α2 t 2 t

1 × N0 + β exp α2 s − αWs ds , 2 0

6 55

the stochastic linear growth equation with multiplicative noise. Note that if the additive noise term αdWt is inserted into (6.50), we obtain the autonomous linear SDE in the narrow sense. dNt = βdt + αdWt

6 56

That is, from Table 5.1, we obtain the (ALNS) equation dNt = adt + cdWt with solution t

t

Nt = X0 + a ds + c dWs 0

6 57

0

In terms of Equation (6.56), a = β, b = e = 0, and c = α Then clearly, (6.57) renders N t = X0 + β

t

t

ds + α dWs

0

0

6 58

= X0 + βt + αWt , the stochastic linear growth equation with additive noise.

6.9 Stochastic Square-Root Growth Model with Mean Reversion Although the mean-reverting square-root SDE (Cox et al., 1985; Feller, 1951a, 1951b) dXt = α − βXt dt + σ

Xt dWt , X 0 = X0 > 0,

6 59

was introduced by Feller (1951a, 1951b) in the context of population growth,5 it has, of late, come to be known as the Cox, Ingersol, and Ross (CIR) SDE model for describing short-run interest rate dynamics. Here, α (the speed of

5 Suppose the random variable Xt depicts population size (or gene frequency). Then, as noted by Feller, for the growth of large populations, a continuous approximation is appropriate, and this leads to diffusion-type processes, that is, for a growing population, the process in its later stages converges to a diffusion process mirroring simple growth. Hence, we should expect, in the limit, diffusion equations with drift, and thus normal distributions for the deviation of Xt from some equilibrium point.

6.9 Stochastic Square-Root Growth Model with Mean Reversion

adjustment), β, and σ are all taken to be strictly positive. Under the restriction 2α > σ 2 for all t a.s., the CIR process (the solution to (6.59)) is well defined and strictly positive; otherwise, it is nonnegative. Moreover, the drift coefficient α −βXt ensures mean reversion of Xt toward its long-term level α/β, and σ Xt is the diffusion coefficient describing volatility. Since the diffusion coefficient is singular at the origin, it follows that the initial nonnegative Xt value can never turn negative for increasing t. (This is appealing when the Xt process is used to describe interest rate behavior, or since interest rates must always remain and fluctuate around a long-term trend.) To solve the SDE given by (6.59), let us first rewrite it as dXt + βXt dt = α dt + σ

6 59 1

Xt dWt

βt

For e , an integrating factor, the preceding expression becomes d e βt Xt = αe βt dt + σe βt

Xt dWt ,

and thus t

t

t

0

0

d e βt Xt = α e βt dt + σ

0

e βu

Xu dWu

Then it is readily verified that Xt =

α − βt α +e + σe −βt X0 − β β

t

e βu

Xu dWu ,

6 60

0

the stochastic mean-reverting square-root growth equation or, as it is now commonly called, the CIR growth equation. Since the CIR model has been widely used to explore the behavior of shortterm interest rates, it is important to consider some of its salient features. In particular, we shall examine its mean and variance. To this end, since dWu N 0, du , it follows from (6.60) that E Xt =

α α + e −βt X0 − ; β β 2

V Xt = E Xt − E Xt =E

σe − βt

t

e βu

2

Xu dWu

0

= σ 2 e −2βt E

t

e βu

2

Xu dWu

0

= σ 2 e −βt

t

e2βu E Xu du 0

6 61

183

184

6 Stochastic Population Growth Models

via Itô’s isometry. Then, upon substituting E Xu =

α α + e −βu X0 − β β

into (6.61) and integrating, we obtain V Xt = X0

σ 2 e −βt − e2βt β

+

ασ 2 1 − e −2βt 2β2

6 61 1

An extension of (6.59) has been offered by Chan, Karolyi, Longstaff, and Sanders or CKLS (1992) and appears as dXt = α − βXt dt + σXtγ dWt , X 0 = X0 > 0,

6 62

the CKLS mean-reverting gamma SDE, where the parameter γ measures the degree of nonlinearity of the relationship between Xt and its volatility. Given that σ must always be positive, Xt 0 5, 1 if α, β > 0 and γ > 0 5. In fact, when γ 0 5, 1 , Xt > 0 for all t ≥ 0 a.s.

Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect Let us examine an important modification of the logistic function (Jiang et al., 2005; Krstić and Jovanović, 2010). Specifically, it has been observed (Allee, 1931) that in high-density populations, mutually positive cooperation effects were significant (i.e., average well-being was high). Similarly, as a population decreases in volume, there could be insufficient membership to achieve these benefits. So while cooperation could enhance population performance, it was subject to the risk that, at low population densities, its loss could lead to an increased chance of extinction—a declining population may reach a density at which the per capita growth rate will be zero before population density itself reaches zero. Hence, any future declines will cause the growth rate to become negative, and this will likely be followed by extinction. In a social context, the intrinsic growth rate might be negative at low population densities due to the possibility of diminished reproductive opportunity; it is again negative at higher densities due to intraspecific competition due to overcrowding. Thus an Allee effect or underpopulation effect implies a positive association between population density and the reproduction and survival of individuals. That is, the per capita growth rate has a positive correlation with population size at low population densities. It arises when the intrinsic growth rate increases at low densities, reaches a maximum, and then declines in the direction of high population densities (Figure 6.A.1). It is important to recognize two distinct variations of the Allee effect: the strong Allee effect introduces a

Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect

Figure 6.A.1 Allee effect.

Per capita growth rates r(N), g(N ) Logistic case r(N) = r 1– N K (no Allee effect)

0

T

K

N –1 g(N) = r(N) N T (Allee effect)

population threshold (the minimal size of the population required to survive) that the population must exceed in order to grow; the weak Allee effect does not admit any such threshold. To model the Allee effect, let us form what is called the deterministic sparsity-impacted logistic per capita growth equation dNt 1 = g Nt dt Nt Nt = r 1− K

Nt −1 , T

6A1

where T is the sparsity parameter or Allee threshold (0 < T < K) and Nt T − 1 is the sparsity component of (6.A.1). A population with an initial size N0 less than T tends to zero, while a population with N0 > T tends to K. Hence, K may be viewed as a stable equilibrium point, while T depicts an unstable equilibrium. In this regard, a strong Allee effect exists if the extinction equilibrium Nt = 0 is stable given that K is a positive stable equilibrium point. (Note that Equation (6.A.1) displays a multiplicative Allee effect. An additive Allee effect can be written as dNt Nt m − = Nt r 1 − Nt + b dt K = rNt 1 −

Nt mNt − , 0 < m < 1, 0 < b < 1, K Nt + b

where m and b are termed “Allee constants.” For m > br, we have a strong Allee effect; if m < br, the weak Allee effect is obtained.)

185

186

6 Stochastic Population Growth Models

To randomize Equation (6.A.1), let us write dNt =

r Nt K −Nt Nt − T rdt + σdWt , t ≥ 0, N 0 = N0 , KT

6A2

where Wt t ≥ 0 is a standard BM process. Here, (6.A.2) is termed the sparsityimpacted logistic SDE. (See footnote 2 to this chapter for a justification of this stochastic specification.) For Nt 0, Nt K , and Nt T in (6.A.2), let us apply Itô’s formula to the expression V = ln K −Nt Nt −T = ln K − Nt −ln Nt − T = V1 −V2 : dV = dV1 − dV2 1 1 = V1 dNt + V1 dNt 2 − V2 dNt + V2 dNt 2 2 =−

dNt dNt 2 − K − Nt 2 K −Nt

2

−

dNt dNt 2 + Nt − T 2 Nt − T

2

2

A substitution of (6.A.2) into this equation (remembering that dt 2 = 0 = dt dWt and dWt 2 = dt) yields d ln

K − Nt Nt K − T vdt + σdWt =− Nt − T KT + =−

1 Nt 2 KT

K −T Nt KT

2

K − T K + T − 2Nt σ 2 dt r − σ 2 Nt

T + K −2Nt KT

6A3

dt + σdWt

Then K − Nt = Ce At , C a constant, Nt − T

6A4

where At = −

K −T KT

t

Ns 0

r − σ 2 Ns

T + K −2Ns KT

ds + σdWs

6A5

When we take into consideration the dependence upon the initial population size N0, the following two cases present themselves: 1. If T < N0 < K , then C = K − N0 N0 − T > 0 a.s., and thus T < Nt < K , t ≥ 0, so that (6.A.4) becomes Nt =

K + CT e At 1 + C e At

6A6

Appendix 6.A Deterministic and Stochastic Logistic Growth Models with an Allee Effect

2. If 0 < N0 < T , then C = K − N0 N0 −T < 0 a.s., and thus 0 < Nt < T < K , t ≥ 0, so that (6.A.4) yields Nt =

K −CT e At 1 −C e At

6A7

Clearly, (6.A.2) does not have a closed-form solution. For any initial value N0 such that 0 < N0 < K, it can be shown (Krstić and Jovanović, 2010) that there exists a unique, uniformly continuous positive solution to (6.A.2). For instance, let T < N0 < K. For T < Nt < K, let Xt t ≥ 0 be a stochastic process defined by K −Nt Nt −T

Xt = ln

6A8

Applying Itô’s formula to (6.A.8) renders dXt = f Xt dt + g Xt dWt , t ≥ 0,

6A9

where f Xt = − g Xt = −

K − T K + T e Xt TK 1 + e Xt K −T K + T e Xt TK 1 + e Xt

r −σ 2

K −T K + T e Xt e Xt − 1 2TK 1 + eXt

2

,

σ

for all t ≥ 0 and X0 = X 0 = ln K −N0 N0 − T . Under some mild restrictions (see Krstić and Jovanović (2010)), (6.A.9) has a unique continuous solution Xt , t ≥ 0, satisfying X0 = X 0 . Since (from (6.A.8)) Nt =

K + T e Xt , 1 + e Xt

we can readily demonstrate that Nt is a positive and continuous solution to (6.A.2), that is, dNt = d

K + Te Xt 1 + e Xt

= − K −T

=

K −T TK

= Nt 1 −

2

Nt K

e Xt

dXt +

1 + e Xt 2 e Xt 1 + eXt

2

Nt −1 T

1 − e Xt 1 + e Xt

K + Te Xt 1 + e Xt

dXt

2

rdt + σdWt

rdt + σdWt , t ≥ 0

187

188

6 Stochastic Population Growth Models

If 0 < N0 < T, then define the process Xt

t ≥0

as

K − Nt Nt

Nt = ln

6 A 10

Then from an application of Itô’s formula to (6.A.10), we obtain dXt = f Xt dt + g Xt dWt , t ≥ 0,

6 A 11

where f Xt = −

K − T e Xt + 1 T e Xt + 1

r − σ2

K −T e Xt −1 T e Xt −1

g Xt = −

e Xt −1 K − T e Xt − 1 2T e Xt − 1

,

σ

for all t ≥ 0 and X0 = X 0 . With Nt =

K +1

e Xt

(from (6.A.10)), it follows that Nt is a positive and continuous solution to (6.A.2) since dNt = d

=−

=

K +1

e Xt

Ke

Ke

Xt

Xt

e +1

2

dXt +

= Nt 1 −

Nt K

X

e t −1 Xt

2 e +1

Ke Xt K −T e Xt + 1 T eXt + 1

Xt

3

dXt

2

rdt + σdWt

3

Nt −1 T

rdt + σdWt , t ≥ 0

Suppose σ = 0 in (6.A.5) so that At = −

r K −T KT

t

Ns ds = −

0

r K −T Nt t KT

Then, from (6.A.4), K − Nt = Ce − Nt − T

r K −T KT

Nt t

6 A 12

Appendix 6.B Reducible SDEs

Appendix 6.B

Reducible SDEs

Our objective herein is to ascertain whether a nonlinear SDE (Gard, 1988; Gihman and Skorokhod, 1972; Kloeden and Platen, 1999) of the form dXt = f Xt , t dt + g Xt , t dWt

6B1

can be reduced, under a suitable change of variables or transformation of the form Yt = U Xt , t , to a linear SDE in Yt or dYt = a1 t + a2 t Yt dt + b1 t + b2 t Yt dWt

6B2

If ∂U ∂Xt 0, then it can be shown that a solution Xt of (6.B.1) has the form Xt = V Yt , t , where Yt is provided by (6.B.2) for appropriately defined coefficients a1(t), a2(t), b1(t), and b2(t). Applying Itô’s formula (4.18) to U(Xt, t) yields 1 dU Xt , t = Ut + fUX + g 2 UXX dt + gUX dWt , 2

6B3

with all coefficients and partial derivatives in this expression evaluated at (Xt, t). This will render (6.B.1) reducible or yield a linear SDE of the form (6.B.2) if 1 a Ut + fUX + g 2 UXX = a1 t + a2 t U Xt , t ; 2 b gUX = b1 t + b2 t U Xt , t

6B4

Looking to greater specificity, suppose (6.B.1) is an autonomous nonlinear SDE or dXt = f Xt dt + g Xt dWt

6B5

Then the transformation function takes the form Yt = U Xt and can be used to reduce (6.B.5) to the autonomous linear SDE dYt = a1 + a2 Yt dt + b1 + b2 Yt dWt

6B6

In this case, (6.B.4) becomes 1 a fUX + g 2 UXX = a1 + a2 U Xt ; 2 b gUX = b1 + b2 U Xt If g(Xt)

0 and b2

U Xt = Ce b2 B

6B7

0, then, from (6.B.7b), we obtain Xt

−

b1 , b2

6B8

where Xt

B Xt =

ds X0 g s

6B9

189

190

6 Stochastic Population Growth Models

and C and X0 are arbitrary constants of integration. Substituting (6.B.8) into (6.B.7a) yields 1 b1 , 6 B 10 C b2 A Xt + b22 −a2 e b2 B Xt = a1 − a2 2 b2 where f Xt 1 dg Xt A Xt = − 6 B 11 g Xt 2 dXt Using (6.B.10), we can derive (with a multiplicity of steps omitted) the expression b2

dA d dA =0 + g Xt dXt dXt dXt

6 B 12

Clearly, this equation holds if dA dXt = 0, or if d dXt

d dXt g Xt dA dXt dA dXt

= 0,

6 B 13

given that, from (6.B.12), b2 is chosen so that b2 = −

d dXt g Xt dA dXt dA dXt

6 B 14

Hence, a necessary and sufficient condition for the reducibility of an autonomous nonlinear SDE to an autonomous linear SDE is Equation (6.B.12), that is, (6.B.12) characterizes the conditions that f and g must satisfy for reducibility. Now a if b2

0, select U Xt = Ce b2 B

Xt

; and

b if b2 = 0, select U Xt = b1 B Xt + C,

6 B 15

where b1 must satisfy (6.B.7b). Once U(Xt) has been determined, evaluating (6.B.7a) enables us to determine (6.B.2). Then applying the inverse transformation allows us to obtain the solution to (6.B.1). Example 6.B.1

Given the SDE

1 dXt = − e − 2Xt dt + e −Xt dWt , 2 we have f Xt = − 1 2 e −2Xt and g Xt = e −Xt . Then from (6.B.11), 1 1 A Xt = − e − Xt + e −Xt ≡ 0 2 2 so that (6.B.12) holds for any b2. Set b2 = 0 and b1 = 1. Then a solution of (6.B.7b) is obtained as gUX = 1, UX = e Xt , dU Xt = e Xt dXt , and thus U Xt = e Xt via (6.B.15b). Substituting U(Xt) into (6.B.7a) yields

Appendix 6.B Reducible SDEs

0 = a1 + a2 e Xt , which holds for a1 = a2 = 0. Hence, Yt = U Xt = e Xt and (6.B.2) reduces to the linear SDE dYt = dWt , which has the solution Yt = Wt + Y0 = Wt + e X0 Since Yt = e Xt , it follows that ln Yt = Xt so that the original SDE (6.B.1) has the solution Xt = ln Wt + e X0

■

What if the transformation function Yt = U Xt is known in advance? (Remember that we are still dealing with the autonomous case.) If U(Xt) is specified up front, then the second-order Taylor expansion of U appears as 1 dYt = UX dXt + UXX dXt 2 , 6 B 16 2 where dXt = f Xt dt + g Xt dWt . A substitution of this latter equality into (6.B.16) and simplifying (with the aid of Table 4.A.1) will then yield an autonomous linear SDE. Example 6.B.2

Our objective is to solve the autonomous nonlinear SDE

dXt = f Xt dt + g Xt dWt 1 = − e −2Xt dt + e −Xt dWt 2 given that we know the transformation function to be Yt = U Xt = e Xt (see the preceding example problem). Then from (6.B.16), 1 dYt = e Xt dXt + e Xt dXt 2

2

1 = e Xt − e −2Xt dt + e −Xt dWt 2 1 1 + e Xt − e − 2Xt dt + e − Xt dWt 2 2

2

1 1 = − e − Xt dt + dWt + e Xt e −2Xt dt 2 2 = dWt , which is the same linear SDE obtained in the preceding example. The solution to this SDE then follows as usual. ■

191

193

7 Approximation and Estimation of Solutions to Stochastic Differential Equations 7.1

Introduction

It may not be possible to determine an explicit or closed-form solution to the Itô’s stochastic initial value problem depicted by the stochastic differential equation (SDE) dXt = f Xt , t dt + g Xt , t dWt , X t0 ≡ X0 , t

t0 +∞

71

Hence, we must employ numerical methods for calculating approximations to problems such as (7.1). In this regard, we must use what is called a discrete-time approximation or difference method to iteratively approximate a solution to (7.1), that is, this method utilizes a recursive algorithm that outputs the values of a discrete-time approximation at the given discretization points of a finite subinterval [t0, T] of t0 , +∞ . While the approximation is made only at the discretization points, we will always view a discrete-time approximation as a “continuous-time process” defined on [t0, T]. Looking to specifics, two essential tasks present themselves. First, the continuous-time SDE (7.1) is replaced by a discrete-time difference equation that generates the values Y0 = X0 , Y1 , Y2 , …, YN (N a positive integer) used to approximate, at given discrete-time instants t0 < t1 < t2 < < tN , the solution values X ti ; X0 , t0 , i = 1,…, N. As one might have already surmised, to obtain a “good” approximate solution, the time increments Δi = ti + 1 − ti , i = 0, 1, 2, …, should be “sufficiently small.” Second, given the stochastic nature of (7.1), the approximations Yi , i = 1, …, N should mirror the sample paths or trajectories of the solution Xt of this SDE; samples of random processes {Yi} are to be (iteratively) generated so as to approximate the solution process sample values {Xt}. So not only are the Δis required, the standard Brownian motion (BM) or Wiener increments ΔWi

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

194

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

are needed as well. The Wiener increments are viewed as sample random variables that are normally distributed with a mean of zero and variance Δi (see Section 3.5.1). How should the Yi sample values be obtained? An approximate sample path of SDE (7.1) is regarded as corresponding to a given sample path or trajectory of the underlying Wiener process. In addition, this correspondence can be realized by simulating the BM increments ΔWi via a normal pseudorandom number generator. In fact, the iterative schemes to be implemented can be thought of as “sample path simulation processes”—we simulate the entire trajectory of the solution process {Xt} of (7.1) over the entire discretization of [t0, T]. For convenience, we can choose a set of equally spaced discretization instants, for example, for N, a positive integer, ti + 1 − ti = T − t0 N ≡ Δ, i = 1,…, N, is the step size for the iteration scheme. Then we obtain the independent N 0, Δ (or, equivalently, ΔWi random increments ΔWi = W ti + 1 −W ti ΔN 0, 1 , i = 1,…, N, of the Wiener process Wt t t0 , T . (For these increments, E ΔWi = 0 and E ΔWi 2 = Δ.) Hence, the sampling of normal variates to approximate this Wiener process in (7.1) is executed by generating a set of Gaussian or normal pseudorandom numbers. Once the random increments ΔWi are generated, we can use a recursive method to calculate the sample values Yi , i = 0, 1, …, N at the points ti , i = 0, 1,…, N, of the time discretization. The types of recursive schemes that have been proven successful in approximating (7.1) are the subject of the Sections 7.2, 7.4, and 7.5.

7.2

Iterative Schemes for Approximating SDEs

Once Δ, ΔWi , i = 1,…, N, and Y0 have been specified, the approximate solutions Y1, Y2, …, YN can be calculated using an appropriate recursive formula. What sorts of recursive formulas typically work well in practice? What is our measure or yardstick for “working well?” In what follows, we shall first focus on the Euler–Maruyama (EM) and Milstein approximations to (7.1). The efficiency of these routines in rendering “good” approximations will be addressed by considering two criteria of optimality—strong and weak (orders of ) convergence. We next explore “local linearization techniques” (those of Ozaki (1992, 1993) and Shoji and Ozaki (1997, 1998)) for approximating, via simulations, solutions of SDEs. Here, criteria for “working well,” in terms of “goodness of approximation,” are provided by an assessment of the rates of convergence (in the Lp sense) of the one-step and multistep approximation errors. 7.2.1

The EM Approximation

The simplest discrete-time recursive routine used to approximate an Itô process of the form (7.1) is the EM approximation scheme. Given the time

7.2 Iterative Schemes for Approximating SDEs

discretization t0 < t1 < t2 < < tN = T of t0 , T , the EM approximation is a continuous-time stochastic process Y = Yt t t0 , T satisfying the iterative scheme Yi + 1 = Yi + f Yi , ti ti + 1 −ti + g Yi , ti W ti + 1 − W ti = Yi + f Yi , ti Δi + g Yi , ti ΔWi , i = 1, 2, …, N − 1,

72

where Y ti ≡ Yi , Y0 = X0 , Δi = ti + 1 − ti , and ΔWi = W ti + 1 − W ti . Under a regime of equidistant discretization times, we have Δi ≡ Δ = T −t0 N. What is the justification for the structure of (7.2)? Remember that the solution of (7.1) in integral form is t

Xt = X 0 +

t

f Xs , s ds + t0

g Xs , s dWs , t

t0 , T

73

t0

Clearly, (7.3) suggests that the integral equation for the true solution increment ΔXi should take the form ti + 1

Xi + 1 −Xi =

ti + 1

f Xt , t dt + ti

g Xt , t dWt ,

74

ti

and, in turn, (7.4) suggests that the difference equation scheme (7.2) should prove to be a satisfactory approximation technique since each term on the right-hand side of this expression approximates the corresponding term on the right-hand side of (7.4). So in view of (7.4), the length of the time discretization subinterval ti , ti + 1 is Δi =

ti + 1

dt; ti

and the N(0, Δi) distributed increment of the Wiener process Wt on ti , ti + 1 is ΔWi =

ti + 1

dWt = W ti + 1 − W ti

ti

If we (1) admit equidistant discretization times so that Δi ≡ Δ, i = 1,…, N − 1, (2) ΔZ, where N(0,1) is Gaussian distributed, and (3) assume that the set ΔWi coefficient functions f and g depend only on Yi and not on ti, then (7.2) can be rewritten, for approximation purposes, as Yi + 1 = Yi + f Yi Δ + g Yi

ΔZ, i = 1, 2,…, N − 1

721

(For a more formal look at the construction of Equation (7.2.1), see Appendix 7.B.) In sum, given an Itô process Xt t t0 , T solution of the SDE (7.1), the EM approximation of X is a continuous-time stochastic process {Yi} satisfying the recursive scheme (7.2) (or (7.2.1)). To simulate {Yi}, and thus the approximate sample path solution, one only needs to simulate the increments of the BM or

195

196

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

Wiener process {ΔWi}. In fact, the EM approximation uses the same BM realization as the true solution itself. What criterion should be used to assess the performance of the EM routine? 7.2.2

Strong and Weak Convergence of the EM Scheme

We noted earlier that the EM routine is a computational device for approximating an Itô process Xt by the values Yi generated by (7.2) (or (7.2.1)). Since X(T) and Y(T) are both random variables, it is reasonable to use an expression such as E X T − Y T to measure the degree of precision (error) of the approximation. More specifically, a time-discretized approximation YΔ of a continuous-time process X converges with strong order γ to the solution X at time T if there exists a constant C (not depending on Δ) such that E X T − YΔ T ≤ CΔ γ

75

for N chosen large enough so that Δ = T −t0 N 0, 1 , where X(T) is the true solution at time T and YΔ(T) the approximation. Given this error bound, it can be demonstrated that the strong order of convergence for the EM scheme is γ = 1 2 (which holds under appropriate conditions of f and g in (7.1), that is, f and g satisfy uniform growth and Lipschitz conditions with respect to X and are Hölder continuous of order 1/2 in t). The strong order of convergence criterion (7.5) indicates the rate at which the “mean endpoint error” decreases as Δ 0. A less restrictive criterion—one that considers the rate of decrease of the “error of means”— involves the notion of “weak convergence.” Specifically, a discrete-time approximation YΔ of a continuous time process X converges weakly of order β to the solution X at time T if there exists a continuously differentiable polynomial function h and a constant Ch (independent of Δ) such that E h XT − E h Y Δ T

≤ Ch Δ β , Δ

0,1

76

The EM routine converges with weak order β = 1. In general, under some smoothness conditions on f and g in (7.1), iterative approximation schemes of a particular order that converge strongly typically have a higher order of weak convergence. That is, if the step size Δ decreases k-fold, then the approximation error decreases by a factor kγ. So if, for instance, the order is 0.5 and we want to decrease the error by 100 times, we must make the step size 1002 = 10 000 times smaller. 7.2.3

The Milstein (Second-Order) Approximation

The EM approximation scheme has strong order of convergence γ = 1 2. Following Milstein (1978), it is possible to raise the strong order of convergence of the EM method to 1 by introducing a correction to the stochastic

7.2 Iterative Schemes for Approximating SDEs

increment in (7.2). That is, we can augment the EM scheme (7.2) by introducing the term 1 ∂g g 2 ∂X

ΔWi 2 − Δi

77

from the Itô–Taylor expansion (Appendix 7.A). Here, (7.7) is derived from the truncation of the stochastic Taylor expansion of the integral form of the solution Xt of (7.1).1 Hence, the Milstein scheme appears as Yi + 1 = Yi + f yi , ti Δi + g yi , ti ΔWi 1 ∂g yi , ti + g yi , ti 2 ∂X

ΔWi 2 − Δi , i = 0, 1, …, N − 1,

78

where Y ti ≡ Yi , Y0 = X0 , and t t0 , T . If we again set Δi = Δ (we have equal ΔZ, then, if the coefficient time discretizations), i = 1,…, N −1, and take ΔWi functions f and g depend only on Yi and not on ti, (7.8) can be rewritten, for approximation purposes, as Yi + 1 = Yi + f Yi Δ + g Yi

ΔZ

1 ∂g Yi Δ Z 2 − 1 , i = 1, 2, …, N − 1 + g Yi 2 ∂X

781

(Appendix 7.B offers a much more straightforward derivation of (7.8.1).) Clearly, Milstein’s approximation routine admits an additional term at each iteration relative to the EM technique. Milstein’s approximation (7.8) (or (7.8.1)), which has strong order of convergence γ = 1, converges to the Itô solution of (7.1); it thus converges to the correct stochastic solution faster than the EM scheme as the step size Δ 0. Milstein’s method is identical to the EM method if there is no “X” term in the diffusion portion of (7.1). Both the EM and Milstein schemes render successively improved approximations as the step size decreases.

1 From Equation (7.A.13), let us isolate the term t

s

L1 g XZ dWZ dWs ≈ g Xt

t0 t0

∂g Xt ∂X

t t0

Ws −Wt0 dWs

∂g ∂g 1 2 1 2 1 Ws dWs −Wt0 Wt −Wt0 = g W − W − t −t0 −Wt0 Wt + Wt20 ∂X t0 ∂X 2 t 2 t0 2 1 ∂g 1 ∂g = g Wt −Wt0 2 −Δ = g ΔWt 2 −Δ 2 ∂X 2 ∂X t

=g

197

198

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

Example 7.1

The SDE for the geometric BM model was previously given as

dXt = μXt dt + σXt dWt , X 0 ≡ X0 From (7.2.1) (or from (7.B.5) since Yt + 1 = Yt + Δ ), if we set f Xi = μXi and g Xi = σXi , then it is readily verified that the EM iteration scheme for approximating this SDE is Yi + 1 = Yi + μYi Δ + σYi ΔZ, i = 1, …, N −1

79

Also, from Example 5.2, we have 1 d ln Xt = μ − σ 2 dt + σdWt 2

7 10

Under the EM approximation of this expression, 1 ln Yi + 1 = lnYi + μ − σ 2 Δ + σ ΔZ 2

7 11

so that Yi + 1 = Yi exp

1 μ − σ 2 Δ + σ ΔZ , i = 1,…, N − 1 2

7 12

In a similar vein, using (7.8.1) (or using (7.B.12)), we can demonstrate that the Milstein approximation to the given SDE for the geometric BM process is Yi + 1 = Yi + μYi Δ + σYi ΔZ 1 + σ 2 Δ Z 2 − 1 , i = 1, …, N −1 2

7 13

Moreover, the Milstein approximation of (7.10) is 1 ln Yi + 1 = lnYi + μ − σ 2 Δ + σ ΔZ, i = 1,…, N − 1, 2

7 14

which is identical to (7.11), the EM approximation of (7.10). Hence, the Milstein scheme improves the approximation of the geometric BM process Xt, but does not improve the approximation of ln Xt for this process. ■ Example 7.2

Approximate the Cox, Ingersol, and Ross (CIR) SDE

dXt = α − β Xt dt + σ

Xt dWt , X t0 ≡ X0 > 0,

using the EM and Milstein iteration schemes. For this SDE,

7.3 The Lamperti Transformation

f Xt = α − β Xt and g Xt = σ Xt . For the EM routine we have, Yi ΔZ, i = 1, …, N − 1;

Yi + 1 = Yi + α − β Yi Δ + σ

7 15

and for the Milstein process, we have Yi ΔZ

Yi + 1 = Y i + α − β Yi Δ + σ

1 + σ 2 Δ Z 2 −1 , i = 1,…, N −1 4

7.3

■

7 16

The Lamperti Transformation

A desirable feature of the sample path trajectories that result from the approximation of SDEs is their stability. To achieve this characteristic, we now turn to an examination of a technique used to remove any state or level-dependent noise from these trajectories. In addition, this will be accomplished by transferring nonlinearities from the diffusion coefficient function g to the drift coefficient function f. To this end, suppose we have an SDE of the form dXt = f Xt , t dt + g Xt dWt , X t0 ≡ X0 ,

7 17

associated with the Wiener or BM process Xt t t0 , +∞ , where, as specified, the diffusion coefficient function g(Xt) depends only on Xt and not on t. The gist of the Lamperti transformation is that, under suitable conditions (see Appendix 7.C), an SDE such as (7.17) can always be transformed into one with a unitary diffusion coefficient. Specifically, under the Lamperti transformation Y t = F Xt = the process Yt

1 du g u

t t0 , +∞

,

7 18

u = Xt

solves the transformed SDE

dYt = fF Yt , t dt + dWt , Y t0 ≡ Y0 ,

7 19

where the transformed drift coeffunction fF(Yt, t) has the form fF =

f F − 1 Yt , t 1 − gX F −1 Yt 2 g F − 1 Yt

f Xt , t 1 − gX Xt = 2 g Xt and the transformed drift coefficient is 1.

7 20

199

200

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

Example 7.3

Let {Xt} follow a geometric BM process with SDE

dXt = μXt dt + σXt dWt , where μ and σ are constants. What is the SDE associated with the Lampertitransformed process {Yt}? From (7.19), the transformed SDE is dYt = =

μXt 1 − σ dt + dWt σXt 2 μ 1 − σ dt + dWt σ 2

(Note: from (7.18) we have, Y t = F Xt =

1 du σu

1 = ln Xt , σ u = Xt

and thus Xt = F − 1 Yt = e σYt so that Xt = e σ ln Xt fF = =

σ

= Xt as required. Clearly,

f F − 1 Yt , t 1 − gX F −1 Yt 2 g F − 1 Yt μe σYt 1 μ 1 − σ= − σ σ 2 σe σYt 2

Example 7.4

■

Let the SDE

dXt = μ1 + μ2 Xt dt + σXt dWt be associated with the Wiener process {Xt}, where μ1, μ2, and σ are constants. What is the SDE for the Lamperti process {Yt}? Using (7.19), dYt =

μ1 + μ2 Xt 1 − σ dt + dWt 2 σXt

=

μ1 μ2 1 + − σ dt + dWt σXt σ 2

or, from the preceding example problem (wherein Xt = e σYt ), dYt =

μ1 − σYt μ2 1 e + − σ dt + dWt σ σ 2

■

7.3 The Lamperti Transformation

Example 7.5 Given the SDE (7.17) for the process Xt t t0 , +∞ , determine the EM and Milstein iteration schemes for the transformed process Yt t t0 , +∞ with SDE (7.19). For the EM routine (using (7.2.1)), Yi + 1 = Yi +

f Y i , ti 1 − g X Yi 2 g Yi

Δ + ΔZ;

7 21

and for the Milstein scheme (from (7.8.1)), Yi + 1 = Yi +

f Yi , ti 1 − gX Yi 2 g Yi

Δ + ΔZ 7 22

1 + g Yi g X Yi Δ Z 2 − 1 2 Let us look at these results from a slightly different perspective. In particular, given Yt = F Xt from (7.18), let us write the inverse transformation Xt = F − 1 Yt = G Yt . From the Lamperti transformation, F Xt =

1 gX Xt and F Xt = − ; g Xt g Xt 2

7 23

and from the inverse transformation, G Yt =

1 F G Yt

G Yt = −

= g G Yt

1

2F

F G Yt

= −g Yt

2

and

G Yt G Yt 7 24

−

gX G Yt g G Yt

2

g G Yt

= g G Yt gX G Yt Next, 1. Let Yt = F Xt be determined by the Lamperti transformation. Then under dXt = f Xt , t dt + g Xt dWt , F is also an Itô process with differential 1 dF = F Xt f Xt , t + F Xt g Xt 2

2

dt + F Xt g Xt dWt

(see Itô’s formula (4.18)). Then substituting from (7.23), the preceding expression becomes

201

202

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

f Xt , t 1 − g X Xt 2 g Xt

dF =

dt + dWt

or f Xt , t 1 − gX Xt 2 g Xt

ΔYt =

Δ + ΔZ,

that is, Itô’s formula applied to transformation function F yields the EM scheme for the transformed process (7.21). 2. Let Xt = G Yt . The Taylor expansion of the inverse transformation can be written as dG = G Yt dYt + G Yt dYt

2

+ higher order ΔYt terms

Then from (7.24), the preceding expression can be written (using Table A.4.1) as 1 dG = g G Yt dYt + g G Yt gX G Yt 2

dYt

2

+ higher order ΔYt terms = g G Yt

f Xt , t 1 − g X Xt 2 g Xt

dt + dWt

1 g G Yt gX G Yt dt + higher order ΔYt terms 2 = f X t , t Δ + g Xt

ΔZ + higher order ΔYt terms

If we ignore the higher order ΔYt terms in the preceding expression, then the Taylor expansion of the inverse transformation G under (7.19) yields the EM scheme for the untransformed process. ■ Example 7.6

Given the CIR SDE

dXt = α − βXt dt + σ

Xt dWt

associated with the Wiener process {Xt}, determine the SDE for the Lamperti process {Yt}. As a first step, let us utilize (7.18) to find Y t = F Xt = 2 1 = XT σ

2

1 du g u

u = Xt

7.4 Variations on the EM and Milstein Schemes

Then Xt =

σ2 2 Yt = F −1 Yt 4

and, from (7.19), dYt = fF Yt , t dt + dWt =

α − βXt 1 1 −1 σXt − σ Xt 2 2

2

dt + dWt

A substitution of F −1 Yt into this expression yields dYt =

7.4

1 4α − βYt − 1 dt + dWt 2Yt σ 2

■

Variations on the EM and Milstein Schemes

Section 7.2 has focused on the EM and Milstein iteration schemes. However, these routines are not the “only games in town.” Other iteration techniques abound. For instance (in what follows, let Y t0 ≡ Y0 , f Yi ≡ f , and g Yi ≡ g), 1. Strong Order 1.5 Taylor Scheme (Kloeden and Platen, 1999) 1 Yi + 1 = Yi + f Δ + gΔWi + ggX ΔWi 2 −Δ 2 + fX gΔZi +

1 1 ffX + g 2 fXX 2 2

1 + fgX + g 2 gXX 2 1 + g ggXX + gX2 2

Δ

2

7 25 ΔWi Δ−ΔZi 1 ΔWi 2 − Δ ΔWi , 3

where ΔWi = ΔZ, ΔZi is N 0, 1 3 Δ3 , and COV ΔWi , ΔZi = 1 2 Δ2 . The random variable ΔZi is specified as 1 ΔVi , ΔZi = Δ ΔWi + 2 3 where ΔVi is chosen independently from ΔWi) are independent for all i.

ΔZ, that is, all of the pairs (ΔVi,

203

204

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

2. Weak Order 2.0 Taylor Scheme (Kloeden and Platen, 1999) 1 Yi + 1 = Yi + f Δ + gΔWi + ggX ΔWi 2 −Δ 2 + fX gΔZi +

1 1 ffX + fXX g 2 2 2

1 + fgX + gXX g 2 2

Δ

2

7 26

ΔWi Δ− ΔZi ,

where ΔWi = ΔZ and ΔZi is distributed as in the preceding strong order 1.5 Taylor routine. 3. Weak Order 2.0 Predictor–Corrector Scheme (Gard, 1988; Kloeden and Platen, 1999; Platen, 1980, 1981). For a predictor–corrector method, the predictor Y i + 1 is inserted into the corrector equation to give the next iteration Yi + 1 . Step 1. Consider the predictor 1 1 1 ffX + fXX g 2 Δ2 , Y i + 1 = Yi + f Δ + λi + fX g ΔW Δ + 2 2 2

7 27

where 1 λi = gΔW + ggX 2

ΔW

2

−Δ +

1 1 fgX + g 2 gXX 2 2

ΔW Δ

and ΔW N 0, Δ . Step 2. Choose the corrector Yi + t = Yi +

1 Y i + 1 + f Δ + λi 2

7 28

4. Derivative-Free Weak Order 2.0 Predictor–Corrector Scheme (Kloeden and Platen, 1999) Step 1. Determine the predictor Yi + t = Yi +

1 f u + f Δ + Ø i, 2

7 29

where the supporting value u = Yi + f Δ + gΔW , Øi=

1 g u + + g u − + 2g ΔW 4 1 + g u + −g u − 4

ΔW

2

1 − Δ Δ 2, −

7.5 Local Linearization Techniques

with supporting values u + = Yi + f Δ + g Δ, u − = Yi + f Δ− g Δ, and ΔW N 0, Δ . Step 2. Calculate the corrector Yi + 1 = Yi +

1 f Yi+1 + f Δ + Ø i 2

7 30

5. Weak Order 2.0 Milstein Scheme (Milstein, 1978) Milstein’s “second scheme” involves the approximation 1 1 Yi + 1 = Yi + f − ggX Δ + g ΔZ + ggX Δt Z 2 2 2 + Δ

3 2

+ Δ

2

1 1 1 fgX + fX g + g 2 gXX Z 2 2 2 1 1 ffX + fXX g 2 2 4

7.5

Local Linearization Techniques

7.5.1

The Ozaki Method

Given that the evolution of some dynamic phenomenon can be described by a continuous-time stochastic process, with the latter modeled as a nonlinear SDE, the Ozaki local linearization method approximates the original SDE by a linear one (Ozaki, 1992, 1993). This method enables us to obtain a sample path or trajectory of the approximated process. By discretizing the sample path, we can obtain the discretized version of the process. Looking to particulars, Ozaki’s technique is designed for time-homogeneous SDEs; it involves the approximation of the drift function of a nonlinear SDE by a linear function. Under this approximation, we should obtain a marked improvement over a simple constant approximation employed by, say, the EM scheme. Suppose we have a one-dimensional continuous-time process characterized by the one-dimensional SDE dXt = b Xt dt + g Xt dWt , t ≥ 0,

7 31

205

206

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

where b and g are twice continuously differentiable functions of X and Wt is standard BM. Since this SDE can be transformed (via Itô’s formula (4.18)) into one with a constant diffusion term,2 we need only consider the SDE dXt = b Xt dt + σdWt

7 31 1

Assume that the SDE (7.31.1) is deterministic (the stochastic nature of Xt is initially ignored). Our objective is to locally approximate b(Xt) by a linear function of Xt. So from dXt dt = b Xt , we can write d 2 Xt dXt , = bX X t 2 dt dt

7 32

where bX(Xt) is taken to be constant in the suitably restricted interval t, t + Δt . Upon integrating both sides of (7.32), first from t to u t, t + Δt , and then from t to t + Δt, we obtain the differential equation Xt + Δt = Xt +

b Xt bX X t

e bX

Xt Δt

−1

3

7 33

To translate (7.33) back to the SDE (7.31.1), we again use local linearization by setting b Xt = Kt Xt , where Kt is constant over the interval t, t + Δt . Thus, dXt = Kt Xt dt + σdWt

7 34

2 Let Yt ≡ ϕ Xt , where ϕ is twice continuously differentiable in Xt and ϕ constant. Then by Itô’s formula, the new process Yt satisfies the SDE dYt = b ϕ +

g2 ϕ 2

dt + gϕ dWt

= h Xt dt + σdWt 3 From (7.32) or, for convenience, f u t

ln

f Xt dt = bX Xt f Xt f Xu f Xt

Xt = bX Xt f Xt we have

u

dt, t

= bX Xt u −t

so that f Xu = f Xt ebX

Xt u−t

.

Set f Xu = dXu du. Then t + Δt

t + Δt

dXu = t

f Xt e bX

Xt u−t

du

t

and thus Xu

t + Δt t

=

f Xt bX Xt

e bX

Xt u−t

t + tΔt t

or (7.33).

Xt g Xt = σ, σ a

7.5 Local Linearization Techniques

Clearly, this SDE is of the autonomous linear narrow sense (ALNS) variety with a = 0 (see Table 5.1). Its solution is t + Δt

Xt + Δt = Xt e Kt Δt + σ

e Kt

t + Δt −s

dWs

7 35

t

It now remains to determine Kt. From (7.35), the conditional expectation of Xt + Δt with respect to t, or E Xt + Δt Xt = Xt e Kt Δt , coincides with the state of (7.33) at time t + Δt so that Xt e Kt Δt = Xt +

b Xt b X Xt

e bX

Xt Δt

−1

or Kt =

1 b Xt ln 1 + Δt X t bX X t

e bX

Xt Δt

−1

7 36

Given that the stochastic integral t + Δt

e Kt

t + Δt −s

dWs

t

is distributed as N(0, Vt), where Vt = σ 2

e2Kt Δt −1 4 , 2Kt

the discrete time model of (7.34), which approximates (7.31.1) at Xt 0, is bX X t

0 , with

Xt + Δt = At Xt + Bt Wt + Δt , At = e Kt Δt , e2Kt Δt −1 Bt = σ 2Kt Kt =

7.5.2

1 2

7 37

, and

1 b Xt ln 1 + Δt X t bX X t

e bX

Xt Δt

−1

The Shoji–Ozaki Method

The Shoji–Ozaki (SO) local linearization technique is an extension of the basic local linearization method of Ozaki to the case where the drift coefficient 4 Utilize footnote 3 of Chapter 5 to obtain this result.

207

208

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

depends upon both Xt and t. In addition, the diffusion function is to have Xt as its argument (Shoji, 1998; Shoji and Ozaki, 1997, 1998). In this regard, consider the nonhomogeneous SDE dXt = f Xt , t dt + g Xt dWt , t ≥ 0,

7 38

where f is taken to be twice continuously differentiable in X and continuously differentiable in t, and g is continuously differentiable in X. In addition, by virtue of the argument offered in footnote 2, we need only consider the nonhomogeneous SDE dXt = f Xt , t dt + σ dWt ,

7 39

where σ is constant. The development of this “new” (SO) local linearization method is motivated by that of the basic Ozaki approach. For the latter, the drift function f(Xt) was locally approximated by a linear function of Xt. In the SO local linearization process, we now focus on the local behavior of the drift function f(Xt, t), which can be expressed, via Itô’s formula, by the differential of f(Xt, t) or df =

∂f σ 2 ∂2 f ∂f + dt + dXt ∂t 2 ∂ Xt2 ∂Xt

7 40

To linearize f with respect to Xt and t, suppose that ∂2 f ∂Xt2 , ∂f ∂Xt , and ∂f ∂t, each evaluated at the point (Xt, t), are all constant on the suitably restricted time interval s, s + Δs . Hence, (7.40) can be rewritten as f Xt , t −f Xs , s =

∂f ∂2 ∂2 f + ∂t 2 ∂ Xt2

t −s +

∂f Xt − Xs ∂Xt

7 41

or f Xt , t = Ls Xt + Ms t + Ns ,

7 42

where Ls = Ms =

∂f Xs , s , ∂Xt ∂f σ 2 ∂2 f Xs , s + Xs , s , and ∂t 2 ∂ Xt2

N s = f Xs , s − = f Xs , s −

∂f ∂f σ 2 ∂2 f Xs , s + Xs , s Xs − Xs , s ∂Xt ∂t 2 ∂ Xt2

s

∂f Xs , s Xs −Ms s ∂Xt

So instead of working with (7.39), we can focus on the linear SDE

7.5 Local Linearization Techniques

dXt = Ls Xt + Ms t + Ns dt + σdWt , t ≥ s

7 43

Next, consider the transformed process Yt = e

−Ls t

Xt or, from Itô’s formula (4.18),

dYt = − Ls e −Ls t Xt + Ls Xt + Ms t + Ns e −Ls t dt + σe −Lx t dWt = Ms t + Ns e − Ls t dt + σe −Ls t dWt Then t

t

Ms u + Ns e −Ls u du +

dYt = s

s

t

e −Ls u dWu

s

or t

Ms u + Ns e − Ls u du +

Yt = Ys + s

= Ys + −

t

e − Ls u dWu

s

Ms − e −Ls t Ls t + 1 + e − Ls s Ls s + 1 L2s

Ns −Ls t − Ls s e −e + Ls

t

7 44

e −Ls u dWu

s

Substituting Yt = e −Ls t and Ys = e − Ls s into (7.44) and simplifying yields the discretized process of Xt or Xt = Xs +

f Xs , s Ls t

+σ e

Ls t −u

e Ls

t −s

−1 +

Ms Ls 2

e Ls

t −s

− 1 − Ls t − s 7 45

dWu

s

Since the preceding approximation holds locally or on s, s + Δs , (7.45) can be rewritten as Xs + Δs = Xs + +σ

f Xs , s Ls Δs Ms e −1 + 2 Ls Ls s + Δs

e

Ls s + Δs −u

e Ls Δs − 1 − Ls Δs 7 45 1

dWu

s

We know that the last term on the right-hand side of (7.45.1) is distributed as N(0, Vs), where Vs = σ 2

e2Ls Δs − 1 2Ls

Hence, we may express the discrete time model of (7.43), which approximates (7.39) at Xs 0 , with Ls 0, as

209

210

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

Xs + Δs = A Xs Xs + B Xs Ws + Δs ,

7 46

where A Xs = 1 +

f Xs , s Ls Δs Ms e −1 + Xs L s Xs Ls 2

e Ls Δs − 1 − Ls Δs

and B Xs = σ

Example 7.7

e2Lx Δs − 1 2Ls

1 2

Given the homogeneous SDE

dXt = θXt2 dt + σdWt , t ≥ 0, with θ and σ constant, determine the discretized process associated with (1) the Ozaki local linearization method and (2) the SO local linearization method. 1. Given that b Xt = 2θXt and bX Xt = 2θ, we may utilize (7.37) to obtain Kt =

1 ln 1 + Xt e2θΔt − 1 , Δt

and thus we can readily determine that At = e Kt Δt and e2Kt Δt Bt = σ 2Kt

1 2

Hence, the Ozaki local linearization method renders the process discretization Xt + Δt = At Xt + Bt Wt + Δt 2. From (7.46), f Xs , s = θXs2 ; Ls = 2θXs ; and Ms = θσ 2 . Then A Xs = 1 +

1 2θXs Δs σ2 e −1 + 2 4θXs 3

e2θXs Δs −1 − 2θXs Δs ;

and B Xs = σ

e4θXs Δs − 1 4θXs

1 2

Hence, the SO local linearization method yields the process discretization Xs + Δs = A Xs Xs + B Xs Ws + Δs

■

7.5 Local Linearization Techniques

7.5.3

The Rate of Convergence of the Local Linearization Method

We noted in Section 7.5.2 that Shoji and Ozaki (1997, 1998) have offered an upgrade of the Ozaki (1992, 1993) local linearization method used to facilitate the discrete approximation of continuous-time processes (and, as we shall see later on, estimate the parameters of the same) (Shoji, 1998). The following question arises: How close does the process approximated by the SO method come to the true process? To answer this question, we obviously need to assess the goodness of the approximation of the local linearization technique by considering the rate of convergence of the approximation error. In this regard, two types of approximation errors are defined, the errors resulting from one-step and multistep approximations, respectively. (While the one-step approximation error must be considered when estimating parameters, the rate of convergence of the multistep approximation error is of paramount importance when constructing a sample path of a stochastic process. This is because the multistep approximation error reflects the cumulative error of the local linearization.) Looking to specifics, let us define the one-step and multistep approximation errors as the difference between the state of the true process and that of the approximating process given that the current state of the true stochastic process, Xs, is known and bounded. Moreover, the evaluation of the rate of convergence will be cast in Lp terms. To this end, Shoji (1998) offers two theorems. The first theorem involves the rate of convergence of the one-step approximation in the Lp sense. Theorem 7.1 Shoji’s Theorem 1 Suppose Xt and X t are the true stochastic process and the approximate process derived from the SO local linearization method, respectively. Let a pth order p error of one-step-ahead prediction be defined as Es Xt − X t , with s ≤ t ≤ T . Then the rate of convergence of Es Xt − X t p

Es Xt −X t = O t − s

2p

p 1 p

is 2, or

,5

5 The large O notation is utilized to represent the limiting behavior of a function when its argument tends to some value (or to +∞). That is, let f(x) and g(x) be real-valued functions defined on the same real domain, and let x tend to a limit. Then f(x) = O(g(x)) means that |f(x)/g(x)| remains bounded, or f x ≤ M g x for all x ≥ x0 and all M > 0. Thus f x = O 1 implies that f(x) is itself a bounded function; un = O 1 implies that un is the nth term of a bounded sequence or un ≤ M for all M > 0. Similarly, f h = O h p means that h −p f h is bounded as h 0. In sum, O(g(x)) is the quantity whose ratio to g(x) remains bounded as x tends to a limit.

211

212

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

where Es represents conditional expectation at time s. We next state a theorem that pertains to the convergence of the multistep approximation error. Theorem 7.2 Shoji’s Theorem 2 Let t be fixed for s ≤ t ≤ T and tk 0 ≤ k ≤ n be a n-partition of the time interval [s, t] with Δt = tk −tk −1 . Consider a step-by-step approximation of the integration Xt −Xs =

t

t

f Xu , u du + σ dWu

s

s

by using the SO local linearization method. Then the convergence of the stepby-step approximation is O(Δt) in the Lp sense. Generally speaking, the results of numerical experiments lead to the conclusion that the errors induced by the local linearization method are much smaller than the errors generated by, say, the EM method. In fact, differences between these two methods for multistep approximations are very pronounced. The efficiency of the local linearization method in multistep approximations follows from the structure of the stochastic integration process, that is, in the local linearization method, the stochastic integration component uses the information on increments of a BM process with a discrete time interval; but in the EM method, the sum of the increments of a BM process is reduced to Wt + 1 −Wt (with all intermediate information lost). Hence, the local linearization method is preferred over the EM method. A process discretized by the local linearization technique is normally distributed, and thus we can readily simulate a random variable following this distribution. In short, samples generated by the local linearization method can be treated as a very good approximation to the realization of the true process.

Appendix 7.A

Stochastic Taylor Expansions

Our objective here is to determine the stochastic analogue of the classical or deterministic Taylor expansion of a real-valued function h Xt R R, where h is taken to be continuously differentiable with a linear growth bound (there exists a constant K > 0 such that h Xt 2 ≤ K 2 1 + Xt 2 for all t t0 , T ). We begin by deriving the deterministic Taylor formula. To this end, our starting point is the solution X t ≡ Xt of the initial value problem dXt = f X t , X t0 = X0 , dt

7A1

Appendix 7.A Stochastic Taylor Expansions

t

t0 , T , with t0 ≤ t ≤ T . In integral form, the solution appears as t

X t = X0 +

7A2

f Xs ds t0

Given h(Xt), we have, via the chain rule, dh Xt ∂ h Xt = ∂X dt

∂ h Xt ∂X

dXt = f Xt dt

If we employ the operator L=f

∂ , ∂X

7A3

then the integral equation (7.A.2) can be written as t

h Xt = h X0 +

t0 , T

Lh Xs ds, t

7A4

t0

(Note that when h X ≡ X, Lh = f so that (7.A.4) becomes (7.A.2).) Next, if we now apply (7.A.4) to h(Xs) in the preceding integrand term, we obtain s

h Xs = h X0 +

Lh Xz dz t0

Then the integral in (7.A.4) becomes t

t

s

Lh Xs ds = t0

Lh X0 + t0

L Lh Xz dz ds t0

In addition, thus (7.A.4) can be rewritten as t

h Xt = h X0 +

s

Lh X0 + t0

L Lh Xz dz ds t0

t

= h X0 +

t

s

L2 h Xz dzds

Lh X0 ds + t0

7A5

t0 t0

= h X0 + t − t0 Lh X0 +

t

s

L2 h Xz dzds t0 t0

A second application of (7.A.4), but this time to the term h(Xz) in the integrand of the double integral in (7.A.5), yields z

h Xz = h X0 +

Lh Xu du t0

Then the double integral term in (7.A.5) becomes

213

214

7 Approximation and Estimation of Solutions to Stochastic Differential Equations t

s

t

s

z

L2 h Xz dzds =

L 2 h X0 +

t0 t0

t0 t0

L2 Lh Xu

dzds,

t0

and thus (7.A.5) appears as h Xt = h X0 + t − t0 Lh X0 t

s

z

L 2 h X0 +

+ t0 t0

L3 h Xu du dzds t0 t

= h X0 + t − t0 Lh X0 +

s

t

t0 t0

= h X0 + t − t0 Lh X0 + L2 h X0

z

L3 h X0 dudzds t0 t0 t0

t

s − t0 ds +

t0

= h X0 + t − t0 Lh X0 +

s

L2 h X0 dzds + t

s

z

L3 h X0 dudzds t0 t0 t0

1 t − t0 2 L 2 h X0 + R 3 , 2 7A6

where t

s

z

L3 X0 dudzds

R3 = t0 t0 t0

is called the third-order remainder term. If we continue iterating in this fashion, we obtain the deterministic Taylor formula with integral form of the remainder r

h Xt = h X0 + l=1

t

t − t0 l l L h X0 + Rr + 1 , r = 1, 2, …, l

7A7

t0 , T , where the (r + 1)st-order remainder term is t

sr + 1

s2

Rr + 1 = t0 t0

L r + 1 h Xs1 ds1 …dsr + 1

t0

(Note that we are implicitly assuming that h is r + 1 times continuously differentiable.) If L = ∂ ∂X, the more familiar version of (7.A.7) is r

h Xt = h X0 + l=1

t − t0 l ∂l h X0 + Rr + 1 , r = 1, 2, 3, …, t l ∂X l

t0 , T 7A71

We next look to the development of the stochastic Taylor formula with properties analogous to (7.A.7.1). As we shall now see, the said development is based

Appendix 7.A Stochastic Taylor Expansions

on the iterated application of Itô’s formula.6 To this end, suppose we have the SDE d Xt = f Xt dt + g Xt dWt , t t0 , T , or, in integral form, t

t

X t = X0 +

f Xs ds +

g Xs dWs ,

t0

7A9

t0

where it is assumed that f and g are sufficiently smooth real-valued functions satisfying a linear growth bound (i.e., f Xt 2 ≤ K 2 1 + Xt 2 and g Xt 2 ≤ K 2 1 + Xt 2 , respectively, for constant K > 0 and t t0 , T ). For h Xt R R twice continuously differentiable, ∂ 1 ∂2 h X s + g Xs 2 2 h X s ∂X 2 ∂X

t

h Xt = h X 0 +

f Xs t0

∂ + g Xs h Xs dWs , t ∂X t0

ds

t

7 A 10

t0 , T

If we introduce the operators L0 = f

∂ 1 2 ∂2 ∂ + g , L1 = g , ∂X 2 ∂X 2 ∂X

7 A 11

then (7.A.10) can be expressed as t

t

L0 h Xs ds +

h Xt = h X0 + t0

L1 h Xs dWs , t

t0 , T

7 A 12

t0

(Note that for h X = X, L0 h = f and L1 h = g and thus (7.A.12) reduces to (7.A.9).) Now, if we apply (7.A.12) to the functions h = f and h = g in (7.A.9), we obtain the stochastic Taylor formula or the Itô–Taylor expansion of the solution Xt of (7.A.9) as follows: t

s

Xt = X 0 + t0

t0

t t0

s

L0 g Xz dz + t0

Yt −Ys =

t s

7 A 13

t

ds + g X0

dWs + R,

t0

6 For Yt = V Xt , t , t

L1 g Xz dWz ds t0

t

= X 0 + f X0

L1 f Xz dWz ds t0

s

g X0 +

+

s

L0 f Xz dz +

f X0 +

t0

t0 , T , we can express the Itô’s formula for any 0 ≤ s ≤ t ≤ T as

∂V ∂V 1 2 ∂2 V du + +f + g ∂t ∂X 2 ∂X 2

t

g s

∂V dWu . ∂X

(7.A.8)

215

216

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

where the remainder term R appears as t

s

t

s

L0 f Xz dzds +

R= t0 t0 t

L1 f Xz dWz ds t0 t0

s

t

s

L0 g Xz dzdWs +

+

L1 g Xz dWz dWs

t0 t0

t0 t0

In the preceding specification of the Itô–Taylor expansion, the Itô formula was applied only once. By continuously expanding the integrand functions of the multiple integrals in R, higher order stochastic Taylor expansions are obtained. Example 7.A.1 For r = 2, find the deterministic (second-order) Taylor expansion of the real-valued function h Xt = ae bXt , t t0 , T , at the point X t0 = X0 = 1. From (7.A.7.1), ∂h X0 1 ∂2 h X0 + t − t0 2 + R3 2 ∂X ∂X 2 1 = ae b + t −t0 abe b + t − t0 2 ab2 e b + R3 2

h Xt = h X 0 + t − t 0

= ae b 1 + t −t0 b +

1 t − t 0 2 b2 + R 3 2

■

Example 7.A.2 For f Xt = μXt and g Xt = σXt , determine the Itô–Taylor expansion (7.A.13) at X0 = 1. It is readily verified that Xt = X0 + μ t −t0 + σ Wt − Wt0 + R t

= X0 + μ t −t0 + σ Wt − Wt0 +

s

f t0 t0

t

s

+

g t0 t0 t

s

+

g t0 t0

∂f Xz dWz ds + ∂X

t

s

f t0 t0

∂f Xz dzds ∂X

∂g Xz dzdWs ∂X

∂g Xz dWz dWs ∂X

= X0 + μ t −t0 + σ Wt − Wt0 +

t

s

μ2 Xt dzds

t0 t0 t

s

+

μσXt dWz ds +

t0 t0 t

s

+ t0 t0

t

s

t0 t0

σ 2 Xt dWz dWs

μσXt dzdWs

Appendix 7.B The EM and Milstein Discretizations t

= X0 + μ t −t0 + σ Wt − Wt0 +

μ2 Xt s − t0 ds

t0 t

+

t

μσXt Ws − Wt0 ds +

t0 t

+

μσXt s − t0 dWs

t0

σ 2 Xt Ws −Wt0 dWs

t0

1 = X0 + μ t −t0 + σ Wt − Wt0 + μ2 Xt t − t0 2

2

+ μσXt Ws − Wt0 t −t0 + μσXt s − t0 Wt − Wt0 1 + σ 2 Xt Wt −Wt0 2

Appendix 7.B 7.B.1

2 ■

The EM and Milstein Discretizations

The EM Scheme

Suppose a stochastic process (Maruyama, 1955)

Xt

t t0 , +∞

follows an SDE of the form

dXt = f Xt dt + g Xt dWt , X t0 ≡ X0 ,

7B1

where the coefficient functions f and g depend only on the argument Xt and not on t itself. For uniform time steps Δt ≡ Δ, integrating (7.B.1) from t to t + Δ yields t+Δ

Xt + Δ = Xt +

t +Δ

f Xu du + t

g Xu dWu

7B2

t

Looking to the second term on the right-hand side of (7.B.2), we have t+Δ

t+Δ

f Xu du≈ f Xt t

du = f Xt Δ;

7B3

t

and from the third term on the right-hand side of the same, t+Δ t

g Xu dWu ≈ g Xt

t +Δ

dWu t

= g Xt Wt + Δ − Wt = g Xt ΔWt

7B4

217

218

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

Since we know that ΔWt tion as

ΔN 0, 1 = ΔZ, we can write the EM approxima-

Xt + Δ = Xt + f X t Δ + g X t 7.B.2

ΔZ

7B5

The Milstein Scheme

The Milstein approximation augments the EM scheme by a second-order correction term. That is, the accuracy of the approximation is increased by introducing Itô expansions of the coefficient functions f Xt = ft and g Xt = gt . By Itô’s formula (4.18), the SDEs for these functions are, respectively, 1 a dft = ft ft + gt2 ft 2

dt + gt ft dWt and 7B6

1 b dgt = ft gt + gt2 gt dt + gt gt dWt 2

The integral form of the ft and gt coefficient functions at times t < s < t + Δ are thus s

1 fu fu + gu2 fu 2

a fs = f t + t s

s

du + t

1 fu gu + gu2 gu du + 2

b gs = gt + t

gu fu dWu and 7B7

s

gu gu dWu

t

A substitution of these expressions into (7.B.2) yields t+Δ

Xt + Δ = Xt +

s

t

t

t+Δ

s

gt +

+ t

t

s

1 fu fu + gu2 fu 2

ft +

du + t

1 fu gu + gu2 gu du + 2

gu fu dWu ds

s t

7B8

gu gu dWu ds

If we ignore terms of order higher than one and retain the term involving dWudWs, then (7.B.8) becomes t+Δ

Xt + Δ = X t + f t

t+Δ

ds + gt t

t +Δ s

dWs + t

t

t

gu gu dWu dWs

7B9

(Note that the first three terms on the right-hand side of this expression constitute the EM approximation (7.B.5).) Looking to the last term on the righthand side of (7.B.9), we have

Appendix 7.C The Lamperti Transformation t+Δ s t

t

gu gu dWu dWs ≈ gt gt

t+Δ s

dWu dWs t

t

t+Δ

= gt gt

Ws − Wt dWs

t t+Δ

= gt gt

7 B 10 Ws dWs − Wt Wt + Δ − Wt

t t +Δ

= gt gt

t

Ws dWs −Wt Wt + Δ + Wt2

If we harken back to Example 4.1, we see that t+X t

1 1 1 Ws dWs = Wt2+ Δ − Wt2 − Δ 2 2 2

7 B 11

Then a substitution of this result into (7.B.10) renders t+Δ s t

1 gu gu dWu dWs ≈ gt gt Wt + Δ − Wt 2 − Δ 2 t

7 B 10 1

1 = gt gt ΔWt 2 − Δ 2 In addition with ΔWt

ΔN 0, 1 , we finally obtain, from (7.B.9) and (7.B.11),

1 Xt + Δ = Xt + ft Δ + gt ΔZ + gt gt Δ Z 2 −1 2

Appendix 7.C

7 B 12

The Lamperti Transformation

Let Xt t 0, +∞ be a Weiner process governed by the SDE dXt = f Xt , t dt + g Xt dWt , X 0 ≡ X0 , where t is not an explicit argument of the diffusion coefficient function g (Lamperti, 1964). For this type of SDE, we have Theorem C.1. Theorem C.1 Suppose

(Lamperti Transformation)

Y = F Xt , t =

1 du g u

,

7C1

u = Xt

where g Xt > 0 for all (Xt, t) and F S diffusion and is governed by the SDE

R is one-to-one onto. Then Yt has unit

219

220

7 Approximation and Estimation of Solutions to Stochastic Differential Equations

dYt =

f F −1 Y , t 1 − gX F −1 Y 2 g F −1 Y

f Xt , t 1 − gX Xt = 2 g Xt

dt + dWt 7C2

dt + dWt , Y t0 ≡ Y0

To verify this result, we need only call upon the Itô formula (4.A.5) or 1 V Xt , t = Vt + fVX + g 2 VXX dt + gVX dWt 2 An application of this expression to (7.C.1) yields, via the fundamental theorem, dYt = 0 + f Xt , t + g Xt =

f Xt , t g Xt

1 g Xt

1 g Xt

1 + g Xt 2

2

dWt

1 − gX Xt , t dt + dWt 2

−

g X Xt g Xt 2

dt

221

8 Estimation of Parameters of Stochastic Differential Equations 8.1

Introduction

Suppose we have a time-homogeneous stochastic differential equation (SDE) of the form dXt = f Xt ; θ dt + g Xt ; θ dWt , t ≥ 0,

81

and we seek to obtain an estimate of the unknown parameter vector θ θ, where θ is an open-bounded parameter set in Rk. While this task might, at first blush, seem rather straightforward, it is no trivial matter to carry out the actual estimation process. This is because a continuous-time diffusion can be observed only at discrete time points. Hence, the transition probability density function, and thus the associated likelihood function,1 is not explicitly computable, that is, it does not have a closed-form representation. To address this issue, a variety of statistical procedures have been developed that, to varying degrees, have been successful in providing “good” estimates of θ from a single sample of observations at discrete times. In fact, a representative assortment of surveys on the estimation of θ have been offered by Cysne (2004), Shoji and Ozaki (1998), Hurn et al. (2006), Jensen and Poulson (2002), Durham and Gallant (2002), Nicolau (2004), Iacus (2008), Shoji and Ozaki (1997), and Sѳrensen (2004), to name but a few. While the presentation herein pertaining to the estimation of diffusion process parameters will not be, by any stretch of the imagination, exhaustive in nature, the primary focus will be on the application of the pseudo-maximum likelihood (PML) method. This technique is fairly straightforward, is efficient, and has yielded some very encouraging results.

1 For a review of the concept of the likelihood function and the particulars of the maximum likelihood (ML) procedure, see Appendix 8.A.

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

222

8 Estimation of Parameters of Stochastic Differential Equations

8.2 The Transition Probability Density Function Is Known To put the PML method in perspective, let us first consider the case in which Equation (8.1) is readily solvable and the transition probability density function is known and thus of closed form. In this circumstance, exact ML estimation is feasible, and the construction of the pseudo-likelihood function is not necessary. Let us rewrite (8.1) as

Example 8.1

dXt = μXt dt + σXt dWt , t ≥ 0

82

Clearly, this is the case of geometric Brownian motion (BM) first introduced in Example 5.2. If we set Yt = ln Xt , then, by Itô’s formula (4.18), dYt = μ − 12 σ 2 dt + σdWt . Hence, the transition densities of Yt follow a normal distribution. Upon integrating this SDE, we get 1 ln Xt = ln X0 + μ − σ 2 t + σWt 2

83

or Xt = X0 e

μ− 12 σ 2 t + σWt

,

84

with E Xt = X0 e μt , V Xt = X02 e2μt eσ t − 1 2

(see Example 5.9). For i = 1, 2,…, 1 ln X ti − ln X ti −1 = μ − σ 2 2

ti − ti−1 + σ W ti −W ti − 1 , ti ≥ 0, 85

where W ti −W ti−1 X ti = X t i − 1 e where Zi x ti

N 0, ti −ti −1 (see Equation (3.6iii)), and thus μ − 12 σ 2

ti − ti− 1 + σ ti −ti −1 Zi

,

86

N 0, 1 , i = 1, 2,….. If we set x ti = ln X ti −ln X ti −1 , then N

1 μ − σ2 2

ti − ti−1 , σ 2 ti −ti −1

,

8.2 The Transition Probability Density Function Is Known

that is, x(ti) has a log-normal2 transition density of the form p x ti ,Δti ; μ, σ =

1 σX ti

x ti − μ− 1 2 σ 2 Δti exp − 2σ 2 Δti 2πΔti

2

, 87

where Δti = ti − ti − 1 . So if the sample random variables X ti , i = 1, …,n, are drawn from a log-normal population with probability density function (8.7), then the likelihood function of the sample is n

L μ, σ;x t1 , …, x tn , Δti , n =

p x ti , Δti ;μ, σ i=1

n

1 X ti i=1

=

2πσ 2 Δti

−

n − 1 2 e 2σ 2 Δti

n i=1

x ti − μ− 12σ 2 Δti

2

,

and thus the log-likelihood function appears as n

ln L = −

n n ln X ti − ln 2π − ln σ 2 Δti 2 2 i=1

n 1 1 x ti − μ − σ 2 Δti − 2 2σ Δti i = 1 2

2

88

As required, we seek to obtain the values of μ and σ (denote them as μ and σ) such that L = L μ,σ;x t1 , …, x tn , Δti , n = arg max L μ, σ;x t1 , …, x tn , Δti , n To this end, let us look to the first-order conditions for a maximum. Setting ∂ln L ∂μ = 0 and ∂ln L ∂σ 2 = 0 yields, respectively, estimates of the log-normal mean m and variance v or n

a m=

i=1

x ti

n

1 = μ − σ 2 Δti 2 89

and n

b v=

i=1

x ti −m n

2

= σ 2 Δti

Then estimates of the geometric BM parameters μ and σ are, respectively, 2 For a discussion of the log-normal distribution, see Appendix 8.B.

223

224

8 Estimation of Parameters of Stochastic Differential Equations

a μ= b σ=

m+ 1 2 v and Δti

8 10

v Δti

(It is convenient to set Δti = Δ = constant = 1 n .) ■ Example 8.2 We noted in Equation (5.35) that the mean-reverting Ornstein– Uhlenbeck process is given by the SDE dXt = α μ −Xt dt + σdWt , X 0 = X0 , and that a particular solution to the same is Xt = X0 e −αt + μ 1 −e − αt + σ

t

e −α t −s dWs

8 11

0

(see Equation (5.36)). Since Wt is a Wiener process, σ

t

e −α t −s dWs

N 0,

0

σ2 1 −e −2αt 2α

(see Example 5.10). Hence, Xt is normally distributed with E Xt = μ + X0 −μ e −αt , V Xt =

σ2 1 −e −2αt 2α

8 12

For Δti = ti − ti− 1 , i = 1 2,…, X ti has a normal transition probability density function of the form p X ti ,Δti ;α, μ, σ = 2π

−

1 2

σ2 1 −e − 2αΔti 2α

× exp −

−

1 2

X ti − μ− X ti −1 − μ e −αΔti σ 2 α 1 − e −2αΔti

2

8 13 If the sample random variables X ti , i = 1,…, n, are drawn from a normal population with probability density function (8.13), then the likelihood function of the sample is

8.2 The Transition Probability Density Function Is Known n

L α,μ, σ;x t1 ,…,x tn , Δti , n =

p x ti , Δti ;α, μ, σ i=1

= 2π

−

n 2

σ2 2α

× exp −

−

n 2

n i=1

1 −e − 2αΔti

−

1 2

α n X ti −μ − X ti − 1 − μ e −αΔti σ2 i = 1 1 − e −αΔti

,

8 14 and thus the log-likelihood function has the form n n σ2 1 n − ln 1− e −2αΔti ln L = − ln 2π − ln 2 2 2 i=1 2α α n X ti −μ − X ti− 1 − μ e − αΔti − 2 σ i=1 1 −e − 2αΔti

2

8 15

As expected, the first-order conditions for a maximum of (8.14) ∂ln L ∂α = ∂ln L ∂μ = ∂ln L ∂σ = 0 generate a highly nonlinear system of simultaneous equations that typically can be solved by numerical (iterative) methods. ■ Example 8.3 We previously specified the Cox, Ingersol, and Ross (CIR) square-root SDE (see Example 6.9) as dXt = α −βXt dt + σ

Xt dWt , X 0 = X0 > 0,

where the CIR process is well defined if α, β, and σ are all positive and 2α > σ 2 . Since this diffusion process has a known transition probability density function that exists in a closed or explicit form, we may use the exact ML technique to determine the parameter vector θ = α, β, σ . It has been demonstrated by Feller (1951a) that the transition density of the CIR process has the following form: given Xt at time t, the density of Xt + Δt at time t + Δt is q

p Xt + Δt Xt , Δt;α, β, σ = ce − u− v

v 2 Iq 2 uv , u

where c= Iq

σ2

2β 2α , u = cXt e −βΔt , v = cXt + Δt , q = 2 − 1, − βΔt σ 1 −e

is the modified Bessel function of the first kind of order q,

8 16

225

226

8 Estimation of Parameters of Stochastic Differential Equations ∞

Iq x = k =0

and Γ

x 2

1 k Γ kq + 1

2k + q

, x R,

is the gamma function Γ z =

∞

x z − 1 e − x dx, z

R+

0

(For the density (8.16), the distribution function is noncentral chi-squared with 2q +2 degrees of freedom and noncentrality parameter 2u.) Given that Xt t ≥ 0 is a Markov process, the likelihood function of the sample, for Δ = Δt = ti − ti−1 = constant, i = 1,…, n, is L α, β, σ;X t0 , X t1 , …, X tn , Δ, n = n

p X t0 ;α, β, σ

p X ti X ti−1 , Δ;α, β, σ , i=1

and the log-likelihood function appears as ln L α,β,σ;X t0 , X t1 , …, X tn , Δ, n = n

ln p X ti X ti− 1 , Δ;α, β, σ

8 17

i=1

under the assumption that the distribution of the initial condition X(t0) is independent of θ = α, β, σ . Upon combining (8.16) and (8.17), we obtain the loglikelihood function for the CIR process ln L

n

= n lnc +

i=1

q vt i + ln Iq 2 uti −1 vti − uti −1 − vti + ln 2 uti− 1

, 8 18

where uti− 1 = cX ti−1 e −βΔt and vti = cX ti Then the exact ML estimates of the parameters of the CIR process are obtained as θ = α,β,σ = arg max ln L θ

■

8.3 The Transition Probability Density Function Is Unknown

8.3 The Transition Probability Density Function Is Unknown We now turn to the particulars of the PML method. What exactly is a pseudolikelihood function? Think of it as an approximation to the joint probability distribution of a set of random variables; it replaces the true or exact likelihood when conducting an ML operation. That is, if some conditional moments of a diffusion process are known but not the true transition density (p) itself, then, under certain restrictions, it might be feasible to estimate θ from a pseudotransition density function h. So while h does not properly belong to the family of the exact transition density function, it is consonant in terms of the moments of p. The h density is thus a surrogate for p and is termed the “pseudo-transition density.” To obtain the pseudo-likelihood function, let us first note that a diffusion process Xt t ≥ 0 is a Markov process. In this regard, suppose X t0 = X0 and X ti ≡ Xi ,i = 1,…, n, is a sequence of n + 1 historical observations on the random variable Xt that are sampled at nonstochastic dates t0 < t1 < < tn . To generate PML estimates of θ, we need to first form the joint density of the sample or the pseudo-likelihood function. n

L θ, X1 ,…, Xn , Δi , n = h X0 ; θ

h Δi , Xi−1 ,Xi ; θ , 3

8 19

i=1

where Δi = ti −ti− 1 , i = 1, …, n, h Δi , Xi−1 , Xi ; θ is the pseudo-transition density function, and h(X0; θ) is the density of the initial state. As usual, we shall work with the logarithm of the pseudo-likelihood function or n

ln L = ln h X0 ; θ +

ln h Δi , Xi− 1 , Xi ; θ

8 20

i=1

If the initial condition X0 does not depend on θ, then (8.19) becomes n

L θ, X1 ,…, Xn , Δi , n =

h Δi , Xi− 1 , Xi ; θ ,

8 19 1

i=1

and thus (8.20) is rewritten as n

lnh Δi , Xi−1 , Xi ; θ

ln L = i=1

3 See Appendix 8.C for the derivation of this expression via the Markov property.

8 20 1

227

228

8 Estimation of Parameters of Stochastic Differential Equations

The PML estimate of θ is then n

ln h Δi , Xi−1 , Xi ; θ

θ = arg max θ θ

8.3.1

i=1

Parameter Estimation via Approximation Methods

In what follows, we shall focus on process-based solutions to the SDE. dXt = f Xt ; θ dt + g Xt ; θ dWt , X 0 = X0

8 21

That is, we shall consider three discrete-time iteration schemes that are often employed to obtain a practicable pseudo-likelihood function. These are the Euler–Maruyama (EM) scheme, the Ozaki (1985) local linearization method, and the Shoji–Ozaki or SO (1994) new local linearization method. (In what follows, we shall assume, for convenience, that Δti = Δ = constant for all i.) 8.3.1.1 The EM Routine

Let the preceding SDE (Equation (8.21)) appear as dXt = f Xt ; θ dt + σdWt ,

8 22

where g Xt ; θ = σ = constant, σ is not one of the components of θ, and f(Xt; θ) is piecewise constant (e.g., f(Xt; θ) is constant on a small time interval t,t + Δt ). Given the EM scheme (7.2) or Xt + Δt = Xt + f Xt ; θ Δ + σ Wt + Δ −Wt ,

8 23

we know that the increments Xt + Δ − Xt are independent Gaussian random variables with mean f(Xt; θ)Δ and variance σ 2Δ. Then the pseudo-transition density function of the discretized process appears as h Δ,Xi− 1 , Xi ; θ, σ =

1 2πσ 2 Δ

e−

;

Xi −Xi −1 −f Xi −1 θ Δ 2σ 2 Δ

2

,

8 24

and thus the logarithm of the pseudo-likelihood function is n ln L θ, σ; X1 ,…, Xn , Δ, n = ln h X0 ; θ, σ − ln 2πσ 2 Δ 2 1 n Xi − Xi − 1 −f Xi − 1 ; θ Δ − σ2Δ 2 i=1

2

8 25

8.3 The Transition Probability Density Function Is Unknown

Then the PML estimates of θ and σ, denoted, respectively, as θ and σ, are those that maximize (8.25).4 Example 8.4 We noted earlier (Example 8.2) that the Ornstein–Uhlenbeck (OU) process is given by the SED dXt = α μ −Xt dt + σdWt

8 26

Then approximating the solution to this SDE via the EM routine gives Xt + Δ = Xt + α μ −Xt Δ + σ Wt + Δ −Wt ,

8 27

where the increments Xt + Δ − Xt are independent Gaussian random variables with mean α μ −Xt Δ and variance σ 2Δ. Then the pseudo-transition density of the discretized process has the form 1

h Δ,Xi− 1 ,Xi ; α, μ, σ =

2πσ 2 Δ

e−

Xi −Xi− 1 −α μ −Xi −1 Δ 2σ 2 Δ

2

,

8 28

and the associated pseudo log-likelihood function appears as n ln L α, μ, σ; X1 ,…, Xn , Δ, n = ln h X0 ; α, μ, σ − ln 2πσ 2 Δ 2 n 1 Xi −Xi − 1 − α μ − Xi −1 Δ − 2 i=1 2σ 2 Δ

2

■

8 29 It is important to note that the approximation afforded by the EM routine is “good” (1) if Δt is very small (otherwise the parameter estimates are biased), (2) if the following holds Polynomial Growth Condition: there exist positive L and m values (both independent of θ) such that f X; θ ≤ L 1 + X

m

,θ

θ,

and (3) if nΔ3n 0. If the third condition holds, the PML estimator obtained from (8.25) is consistent and asymptotically efficient. (For an elaboration of 4 If (8.23) is replaced by Xt + Δ + Xt + f Xt ; θ Δ + g Xt ; θ Wt + Δ −Wt ,

8 23 1

then the independent Gaussian increments Xt + Δ − Xt are Gaussian with mean f(Xt; θ)Δ and variance g(Xt; θ)2Δ. In this circumstance, the pseudo-transition density is 2 X −X −f Xi −1 ; θ Δ − i i −1 2 1 2g Xi −1 ; θ Δ . h Δ, Xi −1 , Xi ; θ = 2πΔg X ; θ e i −1

229

230

8 Estimation of Parameters of Stochastic Differential Equations

these three requirements, see Iacus (2008, pp. 122–125) and the references listed therein.) 8.3.1.2 The Ozaki Routine

Suppose we have a time-homogeneous SDE of the form dXt = f Xt ; θ dt + σdWt , t ≥ 0, X 0 = X0 , where σ is constant. Given the Ozaki (1992, 1993) local linearization iteration scheme, Xt + Δ = A t Xt + B t W t + Δ , where At = e

Kt Δ

e2Kt Δ − 1 ,Bt = σ 2Kt

1 2

,

and 1 f Xt Kt = ln 1 + Δ X t fX X t

e fX

Xt Δ

−1

(see Equation (7.37)), the conditional distribution Xt + Δ Xt is Gaussian with mean E Xt + Δ = e Kt Δ Xt and variance Vt = σ 2 e2Kt Δ −1 2Kt . Then the pseudo-transition density function of the discretized process has the form h Δ,Xi− 1 , Xi ; θ, σ =

1 1 e− 2 2πVi− 1

Xi −eKi −1Δ Xi −1

2

Vi− 1

,

8 30

and thus the logarithm of the pseudo-likelihood function is ln L Δ, Xi−1 , Xi ; θ, σ = ln h X0 ; θ, σ − −

1 n ln 2πVi −1 2 i=1

1 n Xi − eKi− 1 Δ Xi− 1 2 i=1 Vi−1

2

1 n = ln h X0 ; θ, σ − ln 2πVi −1 2 i=1 −

1 n Xi − Ei−1 2 , 2 i=1 Vi− 1

8 31

8.3 The Transition Probability Density Function Is Unknown

where E i = Xi +

f Xi fX X t

e fX

Xi Δ

− 1 , Vi = σ 2

e2Ki Δ − 1 2Ki

Then maximizing (8.31) with respect to θ and σ yields the PML estimates θ and σ, respectively. Example 8.5 Given the SDE dXt = θXt3 dt + σdWt ,t ≥ 0, determine the logarithmic expression for the pseudo-likelihood function via the Ozaki local linearization method. Proceeding as before, we need to first calculate 1 1 2 Ki = ln 1 + e3θXi Δ −1 Δ 3 Then e Ki Δ X i = X t +

Xi 3θXi2 Δ e −1 = Ei , 3

1 + 1 3 e3θ Xi Δ − 1 2

Vi = σ

2

2 Δ ln 1 + 1 3 e3θ

2

Xi2 Δ

−1

−1

Next, a substitution of Ei− 1 and Vi− 1 into (8.31) gives the desired result.

■

Example 8.6 One of the key assumptions regarding the implementation of the Ozaki local linearization technique is that the diffusion term has a constant coefficient, that is, we have an SDE of the form (7.31.1). If this is not the case, then we can usually employ the Lamperti transformation Y t = F Xt =

1 du g u

u = Xt

to achieve this form. This renders the transformed SDE dYt = fF Yt , t dt + dWt , Y 0 = Y0 , where the transformed drift coefficient is fF Y t , t =

f F −1 Yt , t 1 − gX F −1 Yt 2 g F −1 Yt

=

f Xt , t 1 − g X Xt 2 g Xt

In this regard, suppose our objective is to apply the Ozaki local linearization method to the SDE dXt = μ1 + μ2 Xt dt + σXt dWt , X 0 = X0

231

232

8 Estimation of Parameters of Stochastic Differential Equations

For our purposes, let σ remain as the coefficient on Wt. Then set Y t = F Xt =

1 du u

= ln Xt u = Xt

so that the transformed SDE appears as dYt = fF Yt dt + σdWt = μ1 Xt−1 + μ2 −

1 dt + σdWt 2

= μ1 e −Yt + μ2 −

1 dt + σdWt 2

To obtain the pseudo-likelihood function, we need to first find (in terms of the transformed variable Yt) 1 μ e − Yi + μ2 − 1 2 Ki = ln 1 + 1 Δ Yi −μ1 e −Y1

e − μ1 e

−Yi

Δ

−1 8 32

1 1 1 = ln 1 − Yi−1 1 + μ − e Yi Δ μ1 2 2

e

−μ1 e − Yi Δ

−1

Then e Ki Δ Y i = Y i − 1 + Vi = σ 2

1 1 μ − e Yi μ1 2 2

e − μ1 e

−Yi

Δ

− 1 = Ei ,

e2Ki Δ − 1 , 2Ki

and thus the logarithm of the pseudo-likelihood function can be written as ln L Δ,Xi− 1 , Xi ; μ1 , μ2 , σ = ln h Y0 ; μ1 , μ2 , σ −

1 n 1 n Yi −Ei −1 ln 2πVi−1 − 2 i=1 2 i=1 Vi −1 n

+

ln i=0

2

8 33

1 , Xi

as determined earlier, where Yi = ln Xi . Why does the last term on the right-hand side of (8.33) need to be included in this expression? The answer to this question has to do with executing the Lamperti transformation and its effect on the (pseudo) log-likelihood function. Appendix 8.D covers the rationale for, and details of, this change-of-variable adjustment. ■

8.3 The Transition Probability Density Function Is Unknown

8.3.1.3 The SO Routine

Given an SDE of the form dXt = f Xt ; θ dt + σdWt (see the discussion underlying Equations (7.38) and (7.39)); the SO new or revised local linearization method that approximates this SDE at Xt 0 , with Lt 0, is Xt + Δ = A X t X t + B X t W t + Δ , where A Xt = 1 +

f Xt Lt Δ Mt e −1 + Xt L2t Xt Lt

e2Lt Δ − 1 B Xt = σ 2Lt

e Lt Δ −1 − Lt Δ ,

1 2

, Lt = fX Xt , and Mt =

σ2 fXX Xt 2

(see Equation (7.46)). Here, the conditional distribution of Xt + Δ Xt is Gaussian with mean E Xt + Δ = A Xt Xt and variance Vt = σ 2

e2Lt Δ −1 2Lt

Let us set E i = A Xi Xi = Xi +

f Xi Li Δ Mi e −1 + 2 Li Li

e Li Δ − 1 − Li Δ

8 34

and Vi = σ 2

e2Li Δ − 1 2Li

8 35

Then the logarithm of the pseudo-likelihood function is ln L Δ,Xi −1 , Xi ; θ, σ = ln h X0 ; θ, σ −

1 n ln 2πVi −1 2 i=1

1 n Xi − Ei− 1 − 2 i=1 Vi− 1

2

8 36

and may be maximized accordingly. Example 8.7 Suppose our objective is to specify the logarithmic expression for the pseudo-likelihood function using the SO local linearization routine given the SDE dXt = θXt3 dt + σdWt , t ≥ 0

233

234

8 Estimation of Parameters of Stochastic Differential Equations

Given f Xt ; θ = θXt3 , it follows that Li = 3θXi2 ,Mi = σ 2 3θXi , and thus 1 2 σ2 2 Ei = Xi + Xi e3θXi Δ − 1 + e3θXi Δ −1 − 3θXi2 Δ 3 3θ Xi3 with e6θ Xi Δ − 1 6θ Xi2 2

Vi = σ 2

Then substituting Ei−1 and Vi−1 into (8.36) yields the desired result.

■

Example 8.8 We determined in Example 8.6 that, under the Lamperti transformation, the SDE dXt = μ1 + μ2 Xt dt + σXt dWt , X 0 = X0 , was converted to dYt = fY Yt dt + σdWt = μ1 e −Yt + μ2 −

1 dt + σdWt 2

To obtain the logarithm of the pseudo-likelihood function for the SO routine, we need to calculate Li =

∂ σ 2 ∂2 fY Y i , M i = fY Yi , ∂Y 2 ∂Y 2

E i = Yi +

fY Yi Li Δ Mi e −1 + 2 Li Li

e Li Δ − 1 −Li Δ ,

8 37

and Vi (determined from (8.35)). So for fY Yi = μ1 e −Yi + μ2 − 1 2 , we have Li = − μi e −Yi , Mi =

σ2 μ e −Yi , 2 1

and thus Ei = Yi − 1 +

1 1 μ − e Yi μ1 2 2

e −μ1 e

−Yi

Δ

σ 2 e Yi − μ1 e −Yi Δ + e −1 + μ1 e −Yi Δ , 2μ1

−1 8 38

Appendix 8.A The ML Technique −Y

Vi = σ 2

e − 2μ1 e i Δ −1 , − 2μ1 e −Yi

8 39

where Yi = ln Xi . Then, from (8.38) and (8.39), substituting Ei − 1 and Vi −1 into (8.33) gives us the desired result. ■

Appendix 8.A

The ML Technique

Generally speaking, the method of maximum likelihood is a data reduction technique that yields statistics that are used to summarize sample information. As we shall soon see, this method requires knowledge of the form of the population probability density function. To obtain an estimate of some unknown population parameter θ, assume that the sample random variables X1, …, Xn have been drawn from a population with probability density function p(x; θ). Given that the Xi ,i = 1,…,n are independent and identically distributed, their joint probability density function has the form n

L x1 ,…, xn ; θ, n =

p xi ; θ

8A1

i=1

Here, θ is fixed and the arguments are the variables xi ,i = 1, …,n. But if the xi’s are treated as realizations of the sample random variables and θ is taken to be a variable and no longer held constant, then (8.A.1) is termed the likelihood function of the sample and can be written as n

L θ;x1 ,…, xn , n =

p xi ; θ

8A2

i=1

to highlight its dependence on θ. For computational expedience, L will be transformed to the logarithmic-likelihood function n

ln p xi ; θ

ln L θ;x1 ,…, xn , n =

8A21

i=1

Here, (A.8.2) depicts, in terms of θ, the a priori probability of obtaining the observed random sample. In addition, as θ varies over some admissible range for fixed realization xi , i = 1,…, n, the said probability does likewise. In short, the logarithmic likelihood function expresses the probability of the observed random sample as a function of θ. So with θ treated as a variable in ln L, the method of ML is based upon the principle of maximum likelihood: select as an estimate of θ that value of the parameter, θ, that maximizes the probability of observing the given random sample. So to find θ, we need only maximize ln L

235

236

8 Estimation of Parameters of Stochastic Differential Equations

with respect to θ. In this regard, if L is a twice-differentiable function of θ, then a necessary condition for ln L to attain a maximum at θ = θ is d ln L dθ

=0 5

8A3

θ=θ

Hence, all we need to do is set ∂ln L ∂θ = 0 and solve for the value of θ,θ, which makes this derivative vanish. If θ = g x1 ,…, xn , n is the value of θ that maximizes ln L, then θ is termed the maximum likelihood estimate of θ; it is the realization of the maximum likelihood estimator T = g x1 ,…,xn , n and represents the parameter value most likely to have generated the sample realizations x1 ,i = 1,…,n. If θ is held fixed, then the population density p(x; θ) is completely specified. But if the xi’s are held fixed and θ is variable, then we seek to determine from which density (indexed by θ) the given set of sample values was most likely to have been drawn, that is, we want to determine from which density the likelihood is largest that the sample was obtained. This determination can be made by finding the value of θ, θ, for which L = L θ; x1 ,…, xn , n = arg maxθ L θ;x1 ,…,xn , n − − θ thus makes the probability of getting the observed sample greatest in the sense that it is the value of θ that would generate the observed sample most often.6 Example 8.A.1

Let X1, …, Xn be a set of sample random variables drawn from −1

a normal population with probability density function p x; μ, σ = 2πσ exp 2 2 2 − x− μ 2σ , −∞ < x < ∞. Let us find the ML estimates of μ and σ . To do so, it requires that we generalize (A.8.3). That is, if the likelihood function 5 In what follows, we shall usually deal with lnL rather than L itself since lnL is a strictly monotonic function of θ, and thus the maximum of L occurs at the same θ as the maximum of lnL. Thus, maximizing lnL is equivalent to maximizing L since d lnL dθ = 1 L dL dθ = 0 implies dL dθ = 0. In addition, it is assumed that θ is an interior point of the set of admissible θs and that the sufficient condition for a maximum of lnL at θ is satisfied, that is, d 2 lnL dθ2 θ = θ

< 0.

6 The notation arg max is an abbreviation of the phrase “the argument of the maximum.” It is the set of points of a given argument for which the given function attains its greatest value. Hence, arg max L θ; x1 ,…, xn , n is the set of points θ for which L attains its largest value. This set may be empty, a singleton, and have multiple elements. For instance, if g x = 1 − x , then arg maxx 1− x = 0 . The arg max operator is complementary to the max operator max g(x) in that the latter returns the maximum value of g(x) instead of the point(s) that produces that value, that is, maxx g x = maxx 1− x = 1.

Appendix 8.A The ML Technique

depends on h parameters θ1, …, θh, then the first-order conditions for ln L θ1 ,…,θh ;x1 ,…, xn , n to attain a maximum at θj = θj ,j = 1,…,h, are ∂ln L ∂θj

= 0, j = 1, …, h

8A31

θj = θj

Hence, we need only set ∂ln L ∂θj = 0,j = 1,…,h, and solve the resulting simultaneous equation system for the maximum likelihood estimates θj = g j x1 ,…, xn , n . Thus, the θj s are the realizations of the maximum likelihood estimators T j = g j x1 ,…, xn , n , j = 1, …, h. The ML estimates of μ and σ 2 are, respectively, the values μ and σ 2 for which n

p xi ; μ, σ 2

L μ, σ 2 ;x1 ,…, xn , n = i=1

= 2πσ 2

−

n 2 e−

n i=1

xi −μ

2

2σ 2

or 2 n n 1 n xi −μ ln L = − ln 2π − ln σ 2 − 2 2 2 2σ i = 1

8A4

attains a maximum. Then from (A.8.3.1) and (8.A.4), a ∂

ln L 1 n = xi − μ = 0, ∂μ σ 2 i = 1

b ∂

2 ln L n 1 n =− 2+ 4 xi − μ = 0 2 ∂σ 2σ 2σ i = 1

8A5

n

x = X (the ML estimate of the mean μ of a From (8.A.5a), μ = 1 n i=1 i normal population is the sample realization of the mean estimator n x ). In addition, from (8.A.5b) and μ = X, we obtain X= 1 n i=1 i n

σ2 = 1 n

i=1

xi − X

2

(the ML estimate of the variance of a normal popula-

tion is the realization of the variance estimator s20 = 1 n well-known result that X is an unbiased estimator of μ,

i=1 and s20 n

mator of σ . An unbiased estimator of σ is s = 1 n− 1 2

2

2

2

n

xi − X ). (It is a is a biased esti-

i=1

2

xi −X .)

■

The ML technique will generally yield an efficient or minimum variance unbiased estimate of a parameter θ, if one exists. That is, if a ML estimator θ can be found and θ is unbiased for θ, then θ will typically be an efficient or best unbiased estimator of θ. In addition, ML estimators are consistent estimators of population parameters.

237

238

8 Estimation of Parameters of Stochastic Differential Equations

Appendix 8.B

The Log-Normal Probability Distribution

Suppose a continuous random variable X is normally distributed with mean μ and standard deviation σ (i.e., X N μ, σ ). Then X’s probability density function has the form p x; μ, σ =

1 1 e−2 2πσ

x−μ 2 σ

, −∞ < x < +∞

8B1

Let us now consider the notion of “functions of random variables.” That is, given a continuous random variable X with probability density function f(x), suppose another random variable Y can be written as a function of X or Y = u(X). How can we determine the probability density function of Y? If we can find Y’s cumulative distribution function G y =P Y ≤y =P u x ≤y y

=

−∞

8B2

ux

f x dx =

f x dx,

−∞

then its probability density is given by g y =G y =f u x u x

8B3

(provided u (x) exists). In this regard, let X N μ, σ and let y = e X . Then the cumulative distribution of Y is G y = P Y ≤ y = P e X ≤ y = P X ≤ ln y , y > 0 Hence, from (8.B.2), ln x

G y =

−∞

1 − 12 e 2πσ

x−μ 2 σ

dx, y > 0,

and thus the probability density function of Y is g y = G y = f u x u x = f ln y =

1 1 e−2 2πσy

ln y− μ 2 σ

1 y

8B4

, y > 0,

the log-normal distribution. Thus, the random variable Y has a log-normal distribution if X = ln Y has a normal distribution. Stated alternatively, if X is a random variable with a normal distribution, then Y = e X has a log-normal distribution. The parameters of the log-normal distribution are μ and σ—the mean

Appendix 8.C The Markov Property, Transitional Densities

of ln Y is μ, and the standard deviation of ln Y is σ. In addition, for X = ln Y N μ, σ , the mean and variance of the log-normal variable Y = e X are, respectively, E Y = E e X = e μ + 2 σ and 1 2

V Y = V e X = e2μ + σ e σ − 1 2

2

Appendix 8.C The Markov Property, Transitional Densities, and the Likelihood Function of the Sample Let X = Xt t t0 , T be the path of a Markov process defined on the probability space (Ω, A, P) and which assumes continuous values in R. (Here, A is the σ-algebra of Borel sets B in R.) We know that a process such as this is called a diffusion process. Also, for X ti ≡ Xi , i = 1, …, n, let P(X1, …, Xn; θ) be the joint probability of observing the path when θ is the true parameter, where it is assumed that Xi B for all i = 1,…, n and all Borel sets B R. From the definition of conditional probability, P X1 ,…,Xn ; θ = P Xn Xn− 1 , …, Xn ;θ P X1 ,…,Xn −1 ; θ

8C1

With X as Markov process, let us write the Markov property in terms of transition probabilities as P Xn Xn −1 , …, X1 ;θ = P Xn Xn −1 ;θ (once the present is known, the past and future are independent). Hence, (8.C.1) simplifies to P X1 ,…,Xn ; θ = P Xn Xn− 1 ;θ P X1 ,…, Xn −1 ; θ

8C2

Applying the Markov property to the term P X1 ,…, Xn−1 ; θ in (8.C.2) yields P X1 ,…,Xn −1 ; θ = P Xn− 1 Xn −2 ;θ P X1 ,…,Xn −2 ; θ

8C3

Then inserting (8.C.3) into (8.C.2) gives P X1 ,…,Xn ; θ = P Xn Xn− 1 ;θ P Xn− 1 Xn− 2 ;θ P X1 ,…,Xn− 3 ; θ 8C4 If we repeat this process of applying the Markov property n times in succession, we ultimately obtain P X1 ,…,Xn ; θ = P Xn Xn− 1 ;θ P Xn− 1 Xn− 2 ;θ

P X2 X1 ;θ P X0 ; θ 8C5

We know that for i = 1, …, n, every joint probability P(X1, …, Xn; θ) of the stochastic process X = Xt t t0 , T has a probability density p(X1, …, Xn; θ). Hence, we can express the Markov property in terms of the transition density function

239

240

8 Estimation of Parameters of Stochastic Differential Equations

as p Xn X1 , …, Xn−1 ;θ = p Xn Xn− 1 ;θ . From the definition of the conditional density function, we can write the joint density as p X1 ,…, Xn ; θ = p Xn Xn −1 , …, X1 ;θ p X1 ,…, Xn−1 ; θ = p Xn Xn −1 ;θ p X1 ,…, Xn ; θ via the Markov property applied to transition densities. (The reader can now see where all this is going.) By continuing this process we ultimately obtain p X1 ,…, Xn ; θ = p Xn Xn− 1 ;θ p Xn− 1 Xn −2 ;θ

p X2 X1 ;θ p X0 ; θ , 8C6

where p(X0; θ) is the density of the initial state. Thus, the conditional densities on the right-hand side of (8.C.6) are the transition probability densities of the Markov process. With the Xi , i = 1, …, n, independent and identically distributed, their joint probability density function has the form n

L X1 ,…,Xn ; θ, n = p X0 ; θ

p Xi Xi− 1 ;θ i=1

If we now treat the Xis as realizations of the sample random variables and θ is deemed variable, then the likelihood function of the sample is n

L θ;x1 ,…,xn , n = p x0 ; θ

p xi xi −1 ;θ

8C7

i=1

We noted in Sections 3.5.2 and 5.3.2 that a continuous-time Markov process is homogeneous (with respect to time) if all of its transition densities depend only on the time difference t-s rather than on specific values of s and t, that is, p x, s; t, y = p t − s, x, y ,0 ≤ s ≤ t. For instance, a standard Brownian motion (SBM) or Wiener process Wt t ≥ 0 is a homogeneous Markov process with transition density p x, s; t, y =

1 2π t −s

e − 2 y −x 1

2

t −s

,0 ≤ s ≤ t,

given that Wt t ≥ 0 is Gaussian with independent increments for which W0 = 0 a s , E Wt = 0, and V Wt = t − s,0 ≤ s ≤ t. In terms of our immediately preceding notation, we can write the transition density as p x, ti−1 , ti , y = p Δi , x, y . If Δi = ti −ti − 1 = Δ = constant, then this density is written as p(Δ, x, y) or, for our purposes, as p Δ,Xi − 1 , Xi . Under this convention, (8.C.7) becomes

Appendix 8.D Change of Variable n

L θ, x1 ,…, xn , n = p x0 ; θ

p Δ, xi− 1 , xi ; θ

8C8

i=1

or, more generally, Equation (8.16).

Appendix 8.D

Change of Variable

Suppose we are interested in obtaining the probability density function of a continuous random variable Y given that we have the probability density function of a random variable X and a function y = h(x) connecting the variables X and Y. Suppose further that h(x) is differentiable and either increasing or decreasing for all values within the range of X so that the inverse function x = h −1 y exists for all of the corresponding values of Y. Furthermore, h − 1 is taken to be differentiable except when h x = 0. Under these conditions, we state

Theorem 8.D.1 If the probability density function of the random variable X is given by p(x) and y = h x is differentiable and either increasing or decreasing for all values within the range of X for which p x 0, then the probability density function of Y is given by dx dy

p y =p x =p h

−1

y

dx dx , dy dy

8D1 0,

where dx/dy is the derivative of the inverse function x = h −1 y . Example 8.D.1 be specified as px =

Let the probability density function of the random variable X

e −x , x > 0; ,x ≤ 0

0

and let the random variable Y assume values that are given by y = h x = + x1 2 . Clearly, x = h − 1 y = y2 . Since dx dy = 2y, we obtain, from (8.D.1), py =

e− y 0

2

2y = 2ye − y , y > 0; 2

,y ≤ 0

■

241

242

8 Estimation of Parameters of Stochastic Differential Equations

Next, let us consider the transformation from two continuous random variables to two new random variables. More specifically, suppose we are given two continuous random variables Y1 and Y2 with joint probability density function p(y1, y2). Moreover, these random variables are taken to be related (on a one-toone basis) to the two random variables X1 and X2 by the known functions x1 = u1 y1 , y2 and x2 = u2 y1 , y2 that have unique (single-valued) inverses y1 = v1 x1 ,x2 and y2 = v2 x1 , x2 . If we denote the joint probability density function of the random variables X1 and X2 by p(x1, x2), then p x1 ,x2 = p y1 , y2 J = p v1 x1 , x2 , v2 x1 , x2

8D2

J,

where |J| is the absolute value of the Jacobian determinant

J=

∂y1 ∂y1 ∂x1 ∂x2 = ∂y2 ∂y2 ∂x1 ∂x2

∂ y1 ,y2 ∂ x1 ,x2

Example 8.D.2 p y1 ,y2 = e

8D3

Let − y1 + y 2

,0 ≤ y1 , y2 < +∞,

with x1 = u1 y1 , y2 = y1 + y2 and x2 = u2 y1 , y2 =

y1 y2

or, after a bit of algebra, y1 = v1 x1 ,x2 = x1 x2 1 + x2

−1

and y2 = v2 x1 ,x2 = x1 1 + x2

−1

Then, from (8.D.3), x2 x1 1 + x2 −1 1 − x2 1 + x2 1 + x2 J= 1 −x 1 + x2 − 2 1 + x2

−1

=−

x1 , 1 + x2 2

and thus, from (8.D.2), p x1 ,x2 = e −x1

x1 1 + x2

2

, 0 ≤ x1 , x2 < +∞

■

In general, we can now turn to the transformation from n continuous random variables to n new random variables. That is, suppose we have n continuous random variables y1, y2, …, yn with joint probability density function p(y1, y2, …, yn).

Appendix 8.D Change of Variable

As required, these random variables are related (in a one-to-one fashion) to the n random variables x1, x2, …, xn by the known functions xi = ui y1 ,y2 ,…, yn , i = 1,…, n, which have unique (single-valued) inverses yi = vi x1 ,x2 ,…, xn , i = 1,…, n Let p(x1, x2, …, xn) represent the joint probability density function of the random variables Xi , i = 1,…, n. Then p x1 ,x2 ,…, xn = p y1 , y2 ,…, yn J = p v1 x1 , x2 ,…, xn , …, vn x1 ,x2 ,…,xn

J,

8D4

where |J| is the absolute value of the nth-order Jacobian determinant

J=

∂ y1 ,y2 ,…, yn ∂ x1 ,x2 ,…, xn

∂y1 ∂y1 ∂y1 … ∂x1 ∂x2 ∂xn ∂y2 ∂y2 ∂y2 … ∂x ∂xn 1 ∂x2 =

8D5

∂yn ∂yn ∂yn … ∂x1 ∂x2 ∂xn Now, how does all this relate to the structure of Equation (8.30)? Going back to Equation (8.D.4), we can write ln p x1 , x2 ,…, xn = ln p y1 , y2 ,…, yn + ln J Since yi = lnxi ,i = 1, …, n, in Example 8.5 1 0 … 0 x1 1 … 0 0 x 2 J= =

n i=1

0 0 …

1 xi

1 xn

since the determinant of a diagonal matrix equals the product of the elements on the main diagonal. Then, n

ln J = ln i−1

as required.

n 1 1 = ln xi xi i=1

243

245

Appendix A A Review of Some Fundamental Calculus Concepts A.1

Limit of a Function

Let y be a real-valued function of x and written y = f x , with f defined at all points x within some interval about point x0. f is said to approach a limit L as x x0 if lim f x = L

x

A1

x0

Under what conditions does L exist? Let us denote the left-hand limit of x at x0 (the limit of f as x x0 from below) as x

lim f x ; x0 −

and the right-hand limit of f at x0 (the limit of f as x x

x0 from above) as

lim f x x0 +

Now, if lim f x = lim f x = L, then the limit L exists.1 x

A.2

x0 −

x

x0 +

Continuity

If L = f x0 in (A.2), then f is said to be continuous at a point x = x0 , that is, lim f x = f x0

x

x0

1 A function y = f x is said to be bounded on a set A if there is a number M such that f x ≤ M for all x A. f is monotone increasing (resp. monotone decreasing) on A if f x1 ≤ f x2 (resp. f x1 ≥ f x2 ) for x1 < x2 and x1 , x2 A. If f is bounded and monotone increasing on a, b = x a < x < b , then the limit limx x0 − f x exists. If f is bounded and monotone decreasing on (a, b), then limx x0 + f x exists.

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

246

Appendix A: A Review of Some Fundamental Calculus Concepts

Furthermore, a function f is termed left-continuous at x0 if x

lim f x = f x0 x0 −

(f x

f x0 as x x

x0 from below); and right-continuous at x0 if

lim f x = f x0 x0 +

(f x f x0 as x x0 from above). So if f is continuous at x0, then it is both left-continuous and right-continuous at that point. Looked at in another fashion, a real-valued function y = f x is continuous at a point x = x0 if the increment in f over a “small” interval is “small,” that is, Δf x = f x − f x0

0 as Δx = x− x0

0

or, for h = x −x0 , Δf x = f x0 + h − f x0

0 as h

0

f is said to be continuous if it is continuous at each point of its domain. A function is termed regular if it has both left- and right-hand limits at any point of its domain, and has one-sided limits at its boundary. A regular rightcontinuous function (RRC) is one that is right-continuous with left-handed limits. A function may fail to be continuous at x = x0 if either f (x) does not approach any limit as x x0 , or because it approaches a limit that is different from f (x0). In this regard, a discontinuity refers to a point at which f is defined but not continuous, or to a point at which f is not defined (e.g., f x = x −1 is discontinuous at x = 0). So if f (x0) is defined and L = limx x0 f x exists but L f x0 , then f has a discontinuity at x0. When limx x0 f x does not exist, x0 is termed an essential discontinuity for f.

A.3

Differentiable Functions

A function y = f x , x a, b , is said to be differentiable at a point x = x0 a, b if f is defined on a neighborhood of x0 and has a derivative at x0 defined by lim

Δx

Δy

0 Δx

= lim x

x0

f x − f x0 f x0 + h − f x0 = lim h 0 x− x0 h

= f x0

or =

dy dx

A2

x = x0

Under what conditions does f (x0) exist? Let us denote the left-hand derivative of f at x0 as

Appendix A: A Review of Some Fundamental Calculus Concepts

x

lim

x0 −

f x − f x0 = f − x0 ; x− x0

the right-hand derivative of f at x0 is represented as

x

lim

x0 +

f x0 − f x0 = f + x0 x−x0

Then if f is differentiable at x0, we must have f − x0 = f + x0 . Note: if f is differentiable at x0, then it is also continuous there; but if f is continuous at x0, this does not necessarily imply that f is differentiable at that point. f is said to be differentiable if it is differentiable at each point of its domain.

A.4

Differentials

Let the real-valued function y = f x be differentiable at x = x0 . If dx is taken to be an independent variable whose value is arbitrary, then the expression dy = f x0 dx

A3

is called the differential of f at x0, with dy serving as the dependent variable. Thus, the differential of f is a homogeneous linear function of the independent variable dx. So for x held fixed, dy is the value of the differential for any chosen dx. (We can also regard the differential as a function of x in that it is defined at each point where the derivative of f exists.) Next, suppose that the real-valued function of x = g t is defined for α < t < β, and that a < g t < b. Also, let y = f x be defined for a < x < b. Then if we replace x by g(t) in f, we obtain the composite function y = F t = f g t . How do we differentiate a composite function? The answer is provided by Theorem A.1. Theorem A.1 Composite Function (Chain) Rule Suppose g is differentiable at the point t0 t α < t < β , with x0 = g t0 . Suppose also that f is differentiable at x0. Then the composite function F t = f g t is differentiable at t0, and F t0 = f x 0 g t0 Given this result, we can readily specify the differential of a composite function as follows. If y = f x and x = g t so that y = f g t = F t , then dy = F t dt = f x g t dt = f x dx

A4

(Note: in (A.4), t is the independent variable; x and dx are not independent—as they were in (A.3).)

247

248

Appendix A: A Review of Some Fundamental Calculus Concepts

Looking to the differential of a multivariate function, suppose v = f x, y , where x, y R2 . Think of the total differential of f, df, as a linear combination of dx and dy, where dx and dy are independent variables. In practice, we write df = fx dx + fy dy,

A5

where fx and fy are assumed to exist at the point (x, y). (We can also regard df as a function of (x, y) if (x, y) is a point at which f is differentiable.) However, the existence of the partial derivatives at (x, y) is not sufficient to guarantee the existence of the total differential df. Indeed, there are functions for which fx and fy both exist at a point (x, y), and yet df does not exist. However, if fx and fy are defined throughout a neighborhood of the point (x0, y0), and are continuous at that point, then df exists (or f is differentiable) at (x0, y0). In addition, at each point where df does exist, it is defined for all values of dx and dy. In general, if v = f x, y has continuous first partial derivatives in a set A (the domain of f ), then f has a total differential df = fx dx + fy dy at every point (x, y) of A. An important application of the derivative concept is Theorem A.2, the law of the mean. Theorem A.2 Mean Value Theorem for Derivatives Suppose a real-valued function y = f x is continuous over a, b = x a ≤ x ≤ b and differentiable over (a, b). Then there is a point x = x0 a, b such that f b − f a = f x0 b − a

A.5

Integration

A.5.1

Definite Integral

A6

Suppose a function y = f x is defined and continuous over [a, b]. The definite integral of f over [a, b] is the number b

f x dx, a

with f(x) termed the integrand. How is the value of this number determined? To answer this question, we shall perform the following stepwise construction: 1. Select an integer n ≥ 1 and subdivide [a, b] into n subintervals by picking the points x0, x1, …, xn, with a = x0 < x1 < < xn = b, and setting Δxi = xi − xi−1 ,i = 1, …, n. (Here, we have formed what is commonly called a partition P = a = x0 , x1 , …, xn = b of [a, b].) 2. Within the ith subinterval Δxi, select an arbitrary point xi such that xi− 1 ≤ xi ≤ xi ,i = 1,…, n (Figure A.1a).

Appendix A: A Review of Some Fundamental Calculus Concepts

(a) y

a

xi–1 x′i xi

x

b

(b) y

Mi mi

a

b

Δxi

x

Figure A.1 (a) An approximating rectangle. (b) Largest and smallest values of f in Δxi.

3. Determine the value of f(x) at each of the xi s and form the approximating sum f x1 Δx1 + f x2 Δx2 +

+ f xn Δxn

4. Take the limit of the approximating sum as n increases without bound and the largest of the Δxis approaches zero, that is, find n

b

f xi Δxi ,

f x dx = lim

L= a

P

0

A7

i=1

where the fineness or mesh of P is P = max Δxi , i = 1, …, n . When the indicated limit exists, the area under f(x) from a to b is L. The process of

249

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Appendix A: A Review of Some Fundamental Calculus Concepts

determining this limit is termed Riemann integration, and the actual limit is called the definite integral of f from a to b. Under what conditions is a function integrable over (a, b)? The answer depends upon the behavior of the limit in (A.7). Let mi be the smallest value (a greatest lower bound) of f(x) in the subinterval Δxi. If xi is chosen as the point at which f(x) equals mi for each Δxi, then we can define the lower approximating sum or lower Darboux sum as n

mi Δxi

s= i=1

In similar fashion, for Mi the largest value (or least upper bound) of f(x) in Δxi, and xi is selected as the point at which f(x) equals Mi for each Δxi, then the upper approximating sum or upper Darboux sum is n

Mi Δxi

S= i=1

n

f xi Δxi ≤ S (Figure A.1b). Clearly, mi ≤ f xi ≤ Mi , i = 1,…, n, and thus s ≤ i=1 Now, if we can demonstrate that both s and S approach the same limit L as n f xi Δxi L. P 0, then i=1 To this end, let m and M denote, respectively, the greatest lower bound and least upper bound of f on (a, b). Then m ≤ mi and Mi ≤ M. With s and S contained between fixed bounds, we can conclude that s has a least upper bound I (called the lower integral of f over (a, b)), and S has a greatest lower bound J (termed upper integral of f over (a, b)), or s ≤ I ≤ J ≤ S. In fact, I and J always exist if f is bounded on (a, b). Moreover, if we consider s and S for all possible choices of subintervals Δxi over (a, b), then lim s = I and lim S = J

P

0

P

0

Hence, our answer to the preceding question pertaining to the integrability of f over (a, b) is provided by Theorem A.3. Theorem A.3 A bounded function f(x) is integrable over (a, b) if and only if its lower and upper integrals are equal or b

f x dx = I = J a

We also note briefly the following i. A function f(x) is integrable over [a, b] if it is continuous over this interval. ii. A bounded function is integrable in any interval in which it is monotonic (e.g., f x1 ≤ f x2 if x1 < x2 ).

Appendix A: A Review of Some Fundamental Calculus Concepts

iii. A bounded function is integrable in any interval in which it has only a finite number of discontinuities. iv. A bounded function with an infinite number of discontinuities is integrable if and only if its points of discontinuity form a set of measure zero. (A set of points has a zero measure if it is possible to enclose them in a countable set of intervals whose total length is arbitrarily small.) v. If f(x) and g(x) are integrable on [a, b], a < b, and f x ≤ g x , then b

b

f x dx ≤

g x dx

a

a

vi. If f(x) is integrable in (a, b), a < b, then so is |f(x)| and b

f x dx ≤

a

A.5.2

b

f x dx a

Variable Limits of Integration

Let the real-valued function f(t) be integrable over [a, b] and let x an integral can be expressed as a function of its upper limit as

a, b . Then

x

F x =

A8

f t dt a

Here, F(x) is differentiable and continuous over any interval (a, b) in which f(x) is integrable. Looking to the derivative of F(x) we have Theorem A.4, an existence theorem. Theorem A.4 dF d = dx dx

If f (x) is integrable in (a, b), then b

f t dt = f x , x

a, b ,

A9

a

for all x where f(x) is continuous. (Similarly, d dx

b

f t dt = − f x

x

Note: this theorem legitimizes the existence of a function F(x) whose derivative is a given function f (x) at each of its points of continuity. The importance of Theorem A.4 is the following. We found that if f(x) is continuous in (a, b), then dF dx = f x for all x a, b . Hence, the differential equation dF = f x dx has the general solution y = F x + C, where C is an x

arbitrary constant and F x =

f t dt for fixed a and x a member of [a, b], that a

251

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Appendix A: A Review of Some Fundamental Calculus Concepts

is, (A.9) involves all functions for which F x = f x . Here, F(x) is termed the indefinite integral of f (x). We next have Theorem A.5, which provides us with a systematic process or standard for evaluating definite integrals. Theorem A.5 Fundamental Theorem of Integral Calculus If f (x) is integrable on (a, b) and F(x) is any function having f (x) as a derivative, then b

f x dx = F x a

b a

= F b −F a

A 10

Finally, Theorem A.6 offers a procedure for transforming definite integrals. Given that

Theorem A.6

i. f(x) is continuous on [a, b]; ii. Ø (t) and Ø t are continuous and Ø t iii. Ø α = a and Ø β = b,

a, b whenever t

α, β ; and

it follows that, for g t = f Ø t Ø t , β

b

f x dx = a

A.5.3

α

β

g t =

α

f Ø t Ø t dt

A 11

Parametric Integrals

Suppose we have a situation in which a parameter θ is present: (1) under the integral sign (θ is an argument of the integrand f ); or (2) as a limit of integration; or (3) θ possibly appears in both places. A.5.3.1 Parameter in the Integrand

Suppose the real-valued function f is defined at a point (x, θ) in the rectangular region K = x, θ a ≤ x ≤ b, c ≤ θ ≤ d , and let the integral appear as F θ =

b

f x, θ dx

a

What properties of F can be gleaned from those of f ? Specifically, if f is continuous at each point of K, then F is continuous for each θ c, d . In addition, if f (x, θ) is integrable with respect to x for each value of θ, and ∂f ∂θ exists and is continuous in x and θ throughout K, then F is differentiable with respect to θ and F θ =

b a

∂f dx ∂θ

A 12

Appendix A: A Review of Some Fundamental Calculus Concepts

A.5.3.2 Parameter in the Limits of Integration

Suppose v

G u, v =

f x dx u

If u = u θ and v = v θ , then G u θ ,v θ =

vθ uθ

f x dx = F θ

Then from the chain rule and Theorem (A.4), F θ =

∂G ∂u ∂G ∂v + ∂u ∂θ ∂v ∂θ

= −f u

A 13

∂u ∂v +f v ∂θ ∂θ

A.5.3.3 Parameter in the Integrand and in the Limits of Integration

Let F θ =

u

f x, θ dx

a

For u = u θ , let uθ

G u, v =

f x, v dx, a

where u = u θ and v = θ. Then, via the chain rule and Theorem (A.4), ∂G ∂G = f u, v , = ∂u ∂v

∂f x, v ∂v

u a

dx,

and F θ =

∂G ∂u ∂G ∂v + ∂u ∂θ ∂v ∂θ

∂u + = f u, v ∂θ

u a

∂f x, v ∂v

A 14 dx,

where ∂v ∂θ = 1. A.5.4

Riemann–Stieltjes Integral

As we shall now see, the Riemann–Stieltjes (RS) integral is a generalization of the ordinary Riemann integral. Suppose we have two real-valued functions f (x)

253

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Appendix A: A Review of Some Fundamental Calculus Concepts

and g (x), both defined on [a, b]. Then the RS integral of f with respect to g can be expressed as b

A 15

f x dg x a

Here, f is the integrand and g is termed the integrator. In fact, if g x = x, then (A.15) becomes the Riemann integral (A.7). How is the RS integral defined? To answer this question, let us form a partition P of [a, b] as a = x0 ,x1 ,…, xn = b . This partition of [a, b] renders n subintervals Δxi = xi −xi − 1 ,i = 1,…,n, where the fineness or mesh of P is P = max Δxi , i = 1,…, n . In addition, select a set of n points xi ,x2 ,…, xn and place one of them in each of the aforementioned subintervals. Next, define n

b

f xi g xi − g xi−1

f x dx = lim P

a

0

A 16

i=1

provided that a unique limit exists. (Note: given f, if g is constant on [a, b], then g xi −g xi− 1 = 0 for all i and thus (A.16) is zero.) To address the issue of the existence of the limit in (A.16), let us employ the following generalizations of the lower and upper approximating sums (Darboux sums). Given a partition P and, say, a nondecreasing function g on [a, b], define the upper Darboux sum of f with respect to g as n

sup f xi g xi −g xi− 1

U=

i = 1 xi Δxi

In addition, define the lower Darboux sum of f with respect to g as n

L=

inf f xi g xi − g xi− 1

i = 1 x Δxi

Then f is said to be RS integrable with respect to g if lim U − L = 0

P

0

Under what conditions on f and g will the immediately preceding limit (or the limit in (A.16)) exist? Before answering this question, we need to consider the following definition pertaining to the function g. For a given partition P of [a, b], the variation of g on [a, b] is specified as n

g xi − g xi− 1

Vg P = i=1

Then the total variation of g on [a, b] is defined as Vg a, b = sup Vg P P

Appendix A: A Review of Some Fundamental Calculus Concepts

(i.e., we are taking the least upper bound of the set of all sums Vg(P)) or Vg a, b = lim Vg P P

0

If Vg a, b < +∞, then g is said to be of finite (bounded) variation on [a, b]. Given this concept of bounded variation, we now look to Theorem A.7. Theorem A.7 Existence Theorem for RS Integrals For real-valued functions f (x) and g (x) defined on [a, b], if f is continuous and g is of bounded variation on [a, b], then the RS integral exists. Note: a real-valued function that is monotone on a given interval satisfies the condition of being of bounded variation on that interval, for example, for g monotone on [a, b], Vg a, b = g a −g b . So for the existence of the RS integral, it is sufficient to define the integral with respect to monotone functions. To summarize, given that f is continuous on [a, b], conditions on g that are sufficient to guarantee the existence of the RS integral are the following: i. g is nondecreasing on [a, b]. ii. g is nonincreasing on [a, b]. iii. g is the sum of a nondecreasing function and a nonincreasing function on [a, b]. A few of the important features of the RS integral are the following: 1. If f is Riemann integrable and g has a continuous derivative, then b

b

f x dg x = s

f x g x dx a Riemann integral

RS integral

2. The RS integral allows integration by parts, or b

f x dg x = f b g b − f a g a −

a

b

g x df x a

3. Suppose f is continuous and g is discontinuous with a finite number of discontinuities at which g’s function value “jumps” and then remains constant in the open intervals between the points of discontinuity. To see this, suppose [a, b] is divided into n parts by the partition a = x0 ,x1 ,…, xn = b and let g(x) assume the constant value ci in the interior of the ith subinterval (Figure A.2). For f continuous on [a, b], it can be demonstrated that b

f x dg x = f x0 c1 − g x0 + f x1 c2 − c1

a

+

+ f xn −1 cn −cn− 1 + f xn g xn −cn

255

256

Appendix A: A Review of Some Fundamental Calculus Concepts

Figure A.2 The function g exhibits a finite number of discontinuities.

g g(b) Cn C2 C1 g(a) x1

x0=a

x2

xn–1

A.6

Taylor’s Formula

A.6.1

A Single Independent Variable

xn=b

x

Theorem A.8 Taylor’s Theorem Let the real-valued function y = f x and its first n + 1 (n ≥ 0) derivatives be continuous throughout a closed interval [a, b] containing the point x0. Then the value of f at any point x near x0 is f x = f x0 + f x0 x− x0 +

1 f 2

x0 x− x0

2

1 n f x0 x− x0 n + Rn + 1 , n where the remainder term Rn + 1 has the form +

Rn + 1 =

1 n

A 17

+

n

x

x− t f

n+1

A 18

t dt

x0

Equation (A.17) is known as Taylor’s formula with integral remainder. While the remainder term can assume various forms, we shall employ Lagrange’s form of the remainder, or Rn + 1 =

1 f n+1

n+1

ξ x− x0

n+1

, x0 < ξ < x 2

A 19

2 This result is a consequence of the mean value theorem for integrals: let g(t), h(t) be continuous on [α, β] with h t ≥ 0 for all t α, β . Then for some t = ξ such that α < ξ < β, β α

g t h t dt = g ξ

β α

h t dt

Appendix A: A Review of Some Fundamental Calculus Concepts

Then combining (A.17) and (A.19) renders Taylor’s formula with Lagrange’s form of the remainder, or f x = f x0 + f x0 x− x0 + +

+

1 f n

n

1 f 2

x0 x− x0

n

x0 x −x0 1 f n+1

+

2

n+1

ξ x− x0

n+1

, x0 < ξ < x A 20

(Note: Taylor’s formula with Lagrange’s form of the remainder is an extension of the mean value theorem for derivatives, that is, (A.20) coincides with (A.6) if n = 0.) Suppose we set x = x0 + h. Then x0 < ξ < x0 + h can be expressed as ξ = x0 + θh, 0 < θ < 1, so that (A.20) becomes f x = f x0 + f x0 x− x0 + +

+

1 f n

n

1 f 2

x0 x− x0

n

x0 x −x0 1 f n+1

+

2

n+1

x0 + θh

n+1

,0 < θ < 1 A 21

A.6.2

Generalized Taylor’s Formula with Remainder

Suppose h = x −x0 is an (n × 1) vector with components hi = xi −x0i ,i = 1, …,n. Then the jth order differential of the real-valued function y = f x at point x0 interior to a closed region K R n may be written as d j f x 0 , h = h1

∂ + ∂x1

+ hn

∂ ∂xn

j

f , j = 0 1,…, n x0

For instance, d 0 f x0 , h = f X 0 , n

d 1 f x0 , h =

hr r=1 n

∂f x0 , ∂xr

n

d 2 f x0 , h =

hr hs r=1 s=1

∂2 f x 0 ,… ∂xr ∂xs

In this regard, we have Theorem A.9 Generalized Taylor’s Formula with Remainder Let the real-valued function y = f x , x R n , and its first n + 1 (n ≥ 0) derivatives be continuous throughout a closed region K of Rn containing the point x0 Then the value of f at any point x near x0 is

257

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Appendix A: A Review of Some Fundamental Calculus Concepts

f x = f x0 + df x0 , h + +

+

1 2 d f x0 , h 2

A 22

1 n d f x0 , h + Rn + 1 , n

where Rn + 1 is a remainder term of the form Rn + 1 =

1 d n + 1 f x0 + θh, h ,0 < θ < 1 n+1

A 23

An important special case of (A.22) occurs when x R2 , that is, n

f x1 ,x2 =

1 ∂ ∂ h1 + h2 j ∂x ∂x 1 2 j=0 +

1 n+1

h1

j

f x01 , x02

∂ ∂ + h2 ∂x1 ∂x2

n+1 x01 + θh1 , x02 + θh2

f, 0 0 such that S f , P − L < ε whenever P < δ Here, L is termed the Lebesgue integral of f on [a, b] and denoted

a, b

fdμ

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

260

Appendix B : The Lebesgue Integral

y yi y′i yi–1

a

E1

E2

E3

b

x

Figure B.1 Lebesgue integration.

Looking to Figure B.1, yi is the height of the ith rectangle and μ(Ei) is the length of its base. For the particular subinterval Δyi considered, B = E1 E2 E3 = x a, b yi− 1 < f x < yi . Then μ B = μ E1 + μ E2 + μ E3 and the area associated with yi , yi−1 < yi < yi , is yi μ B —the amount contributed to the total integral. An alternative procedure for defining the Lebesgue integral (one that mirrors somewhat the development of the Riemann integral) is the following. Given the preceding partition P along with the definitions of l, u, and Ei , i = 1,…, n, set E0 = x a, b f x = l and y −1 = l. Then we can define the upper and lower Lebesgue sums as n

n

yi μ Ei and sp =

Sp = i=0

yi−1 μ Ei , i=0

respectively. Then the upper Lebesgue integral is defined as U = inf P Sp and the lower Lebesgue integral is specified as L = supP sp . Hence, a bounded Lebesgue measurable function f on [a, b] is Lebesgue integrable if lim L −U = 0,

P

0

In which case L = U and the common value is termed the Lebesgue integral of f on [a, b] and written f

a, b

fdμ

It is important to note (and the reader might already have guessed) that if a, b R is continuous on a finite closed interval, then the Lebesgue integral b a, b

fdμ exists and has the same value as the Riemann integral

f x dx. a

261

Appendix C Lebesgue–Stieltjes Integral As we shall now see, the Lebesgue–Stieltjes (LS) integral is based on the general definition of the Lebesgue integral; it is the ordinary Lebesgue integral with respect to the LS measure. Let F x :R R be a monotone, nondecreasing real-valued function (or, equivalently, F is of bounded variation on R) which is everywhere right-continuous. In addition, let μF a, b = F b −F a for each semi-open interval (a, b], that is, μF is a non-negative and finitely additive measure function for the class of intervals of the form (a, b]. Since μF can be extended to the σ-ring B of Borel sets in R, this extension defines μF on the completion of B with respect to μF . Hence, the finite Borel measure corresponding to F, μF, is called the LS measure corresponding to F, and the resulting class of sets is termed LS measurable for F. We know that a general function F x R R is a distribution function (see Equation (2.2)) if i. F is monotone increasing and continuous from the right; and ii. F x 0 as x −∞ and F x 1 as x +∞ . Clearly, a distribution function F can be used to define an LS measure μF on a σ-algebra FF within the measure space (R, FF , μF), where FF is the class of sets that are LS measurable for F. Given the preceding discussion, suppose F is a distribution function with corresponding LS measure μF and let f be a bounded Borel-measurable function. (Remember that f is Borel measurable if x f x B is a Borel set for each Borel set B R.) Then the LS integral of f with respect to F is f x dF x = f x dμF x

C1

(Note: we can also view the LS integral as an ordinary Lebesgue integral with respect to the LS measure that is associated with any function F of bounded

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

262

Appendix C : Lebesgue–Stieltjes Integral

variation on R. Stated alternatively, in defining the LS integral we have shifted our focus from a nondecreasing, right-continuous function to a measure function on R, that is, we have adopted the view that if F is nondecreasing and right-continuous, then f dF x represents the Lebesgue integral of f using the measure generated by F.) When f(x) is a continuous, bounded real-valued function and F(x) is a nondecreasing real-valued function, the LS integral is equivalent to the Riemann–Stieltjes (RS) integral. The reason why the LS integral is important for our purposes is that it can be used to evaluate expectations via the distribution function. Suppose we have a probability space of the form (R, B, P), where B is the Borel σ-algebra on R and P is a probability measure on B. As a practical matter, the characterization of P on B can be made by a distribution function F x = P −∞ ,x , with dF(x) representing the LS probability measure associated with F. This is because there exists a one-to-one correspondence between the distribution function F on R and the probability measure P on B. In fact, for a, b B, P a, b = F b − F a . Since a random variable on (R, B, P) can be viewed as a measurable function f(x), its expectation can be written as E f x =

f x dF x , which is the LS R

integral of f with respect to F. More formally, if X is a random variable with distribution function F(x) and f (x) is a measurable function on R, with f (x) integrable, then f x dP x =

E f x =

f x dF x

R

C2

R

Note that for f X = X, E X =

xdF x , provided this integral is finite or exists. R

263

Appendix D A Brief Review of Ordinary Differential Equations D.1

Introduction

A basic problem encountered in the integral calculus is the determination of a function y = f x when its derivative is a given function of x, g(x). In this regard, given that dy = g x or dy = g x dx, dx the function y can be expressed as

D1

y = g x dx = c, c an arbitrary constant Equation (D.1) is an elementary example of an ordinary differential equation (ODE).

D.2

The First-Order Differential Equation

Consider the ODE of the form F x, y,

dy = 0, dx

D2

where F is a given function of the arguments x, y, and dy/dx. A solution to (D.2) will typically be a relation of the form G x, y, c = 0, c an arbitrary constant

D3

Given (D.3), we may obtain an ODE that is solved by (D.3) by differentiating this equation with respect to x, ∂G ∂G dy + = 0, ∂x ∂y dx

D4

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

264

Appendix D: A Brief Review of Ordinary Differential Equations

and eliminating c from (D.3) and (D.4). For instance, suppose G x, y, c = y − cx = 0

D5

so that dy −c = 0 dx

D6

If we eliminate c from the preceding two equations, we obtain the ODE y−x

dy =0 dx

D7

Does (D.5) represent all solutions to (D.7)? A moment’s reflection on this issue will elicit a no answer. A general theorem pertaining to the question of all solutions of the ODE dy dx = f x, y is provided by Theorem D.1. Theorem D.1 Existence and Uniqueness Let the functions f and ∂f ∂y be continuous in some rectangular region R = x, y a < x < b, c < y < d of the xy-plane containing the point (x0, y0). Then within some subinterval x0 − h < x < x0 + h of a < x < b, there is a unique solution y = y x of the ODE dy dx = f x, y with the initial condition y0 = y x0 . The conditions in this theorem are sufficient to guarantee the existence of a unique solution of dy dx = f x, y , y0 = y x0 . (Note: if f does not satisfy the hypotheses of the theorem, a unique solution may still exist.) The expression G x, y, c = 0 that contains all solutions of F x, y, dy dx = 0 in R will be termed the general solution of this ODE in R. A solution obtained from the general solution of an ODE via the assignment of a particular value of c will be called a particular solution.

D.3

Separation of Variables

The simplest class of ODEs dy dx = f x, y that can be solved in a straightforward fashion are the equations in which the variables are separable in a way such that N y

dy +M x =0 dx

or M x dx + N y dy = 0

D8

Appendix D: A Brief Review of Ordinary Differential Equations

Then the general solution of (D.8) is M x dx + N y dy = c, c an arbitrary constant For instance, suppose dy + xe − y = 0 dx Then e y dy + xdx = 0 (see (D.8)), e y dy + xdx = c or 1 e y + x2 = c 2 Then solving for y yields 1 y = ln c − x2 2 (Any restriction on c?) A class of first-order ODEs that can be rendered separable by a change of variable is dy y =g dx x

D9

Here, g is said to be homogeneous in that it depends only on the ratio y/x (or x/y) and not on x and y separately. If we set v = y x so that dy dx = g v , where x is the independent variable and v is taken to be the new dependent variable, then, if we can find v as a function of x, y = vx will ultimately be a function of x and a solution to (D.9) follows. To find v, use y = v x to obtain dy dv =v+x dx dx Then (D.9) becomes v+x

dv =g v , dx

and thus the variables v and x are separable as dv dx =0 + v−g v x

See D 8

For example, let us solve

265

266

Appendix D: A Brief Review of Ordinary Differential Equations

x

y dy − xe − x − y = 0 dx

(Clearly this expression is homogeneous once both sides are divided by x.) Setting y = vx, we obtain x v+x

dv − xe −v −vx = 0 dx

or dv −v − e = 0, dx and thus x

e v dv − dx x = 0 (separability holds). Then integrating yields e v −ln x = ln c, e v = ln cx, v = ln ln cx or y = vx = xln ln cx , cx > 1

D.4

Exact ODEs

Given the function g x, y = c,

D 10

the associated ODE appears as dg =

∂g ∂g dx + dy = 0 ∂x ∂y

This said, suppose we have an ODE of the form M x, y dx + N x, y dy = 0

D 11

If M and N are the partial derivatives of some function g(x, y), with M x, y =

∂g ∂g , N x, y = , ∂x ∂y

D 12

then we can argue that if (D.10) and (D.11) hold, it must be the case that (D.12) also holds. That is, if there exists a function g(x, y) for which (D.12) is true, then

Appendix D: A Brief Review of Ordinary Differential Equations

the solution of the ODE (D.11) is provided by (D.10). In this instance, the expression M x, y dx + N x, y dy is an exact differential, dg, and the ODE (D.11) is called an exact ODE. A necessary and sufficient condition on the functions M and N which ensures that (D.11) be an exact ODE is that ∂M x, y ∂N x, y = ∂y ∂x

D 13

An obvious question that presents itself at this point in our discussion is “When (D.13) holds, can we always find a function g for which (D.12) also holds?” Let’s start with determining a g such that ∂g x, y =M ∂x

D 14

Given g such that (D.14) holds, can we also find N = ∂g ∂y? From (D.14), x

g x, y =

M t, y dt + h y

D 15

(we integrate partially with respect to x), where h(y) is an arbitrary function of integration. Then from (D.15), ∂g ∂ = ∂y ∂y x

=

x

M t, y dt + h y

∂M t, y dt + h y ∂y

D 16

Substituting ∂g ∂y = N x, y into (D.16) yields h y = N x, y −

x

∂M t, y dt ∂y

D 16 1

Our next step is to determine h(y). While not obvious, the right-hand side of (D.16.1) is a function of y only. To verify this, let us differentiate the right-hand side of this expression with respect to x so as to obtain Nx x, y −My x, y = 0 via (D.13). Since the right-hand side of (D.16.1) is independent of x, a single integration with respect to y gives h(y) or h(y) can be obtained from h y =N−

∂ M∂x ∂y

267

268

Appendix D: A Brief Review of Ordinary Differential Equations

Finally, substituting form h(y) in (D.15) gives x

g x, y =

y

M t, y dt +

N x, s −

x

Ms t, s dt ds

D 17

For example, we can readily determine that the ODE 2xy2 + 2y dx + 2x2 y + 2x dy = 0 M x, y

N x, y

is exact since ∂M ∂N = 4xy + 2 = ∂y ∂x To obtain a solution to this exact ODE, let us start with ∂g = M = 2xy2 + 2y, ∂x g=

2xy2 + 2y ∂x + h y

= x2 y2 + 2xy + h y , and thus ∂g = 2x2 y + 2x + h y ∂y Since also ∂g = N = 2x2 y + 2x, ∂y it follows that h y = 2x2 y + 2x− = 2x2 y + 2x−

∂ M∂x ∂y ∂ 2 2 x y + 2xy ∂y

= 2x2 y + 2x− 2x2 y + 2x = 0 or h y = c0 = constant. The solution to the original ODE is thus g = c or g = x2 y2 + 2xy = c, where c0 has been absorbed into the constant c.

Appendix D: A Brief Review of Ordinary Differential Equations

How about the ODE 2x + 4y dx + 2x− 2y dy = 0 M x, y

N x, y

Is it exact? Since ∂M =4 ∂y

∂N = 2, ∂x

the answer is no.

D.5

Integrating Factor

Suppose an ODE is not exact. In this circumstance, our objective is to transform a non-exact ODE to an exact one by multiplying the non-exact one by a suitable function of x and y. That is, given M x, y dx + N x, y dy = 0, with ∂M ∂y

D 18

∂N ∂x, can (D.18) be made exact by forming

μMdx + μNdy = 0,

D 19

where μ = μ x, y . If any such μ exists, then it is called an integrating factor of (D.18). The requirement for exactness of (D.19) is ∂ μM ∂ = ∂y ∂μ M+ ∂y

μN , ∂x ∂M ∂μ ∂N μ= N+ μ, ∂y ∂x ∂y

or ∂μ ∂μ ∂N ∂M − M− N= μ ∂y ∂x ∂x ∂y

D 20

So if a function μ satisfying (D.20) can be found, then (D.19) will be exact. The solution of (D.19) can then be obtained by the method of Section D.4 and will appear implicitly as g x, y = c. Note that this expression also defines the solution to (D.18) since μ can be eliminated from all terms of (D.19). Rather than trying to solve the partial differential equation (D.20) for μ, let us, as a compromise, take an alternative tack. Specifically, we shall consider the conditions on M and N that will enable us to obtain an integrating factor μ of a particular form. That is, we shall restrict the equation for μ to one that depends on x alone. In this instance, we set ∂μ ∂y = 0 and rewrite (D.20) as 1 dμ ∂M ∂y − ∂N ∂x = N μ dx

D 20 1

269

270

Appendix D: A Brief Review of Ordinary Differential Equations

Since the left-hand side of this equation depends only on x, we should be able to find an integrating factor μ = μ x such that the right-hand side of (D.20.1) also depends on x alone. If we set the right-hand side of (D.20.1) equal to r(x), then our integrating factor has the form 1 dμ =r x , μ dx and thus

μ=e

r x dx D 21

For instance, let us solve the ODE 3xy + y2 dx + x2 + xy dy = 0 M x, y

N x, y

Since ∂M = 3x + 2y ∂y

∂N = 2x + y, ∂x

this ODE is not exact. From (D.20.1), 1 dμ 3x + 2y − 2x + y 1 = = 2 μ dx x x + xy or dμ μ = dx x, and thus lnμ = ln x or μ x = x. Then from (D.19), 3x2 y + xy2 dx + x3 + x2 y dy = 0 M x, y

N x, y

Clearly, this ODE is exact since ∂M ∂N = 3x2 + 2xy = = 3x2 + 2xy ∂y ∂x Then it can be shown that the solution to this exact ODE is 1 g x, y = x3 y + x2 y2 = c 2 An important example of an ODE in which an integrating factor μ can be found which depends only on x is that of a linear ODE of the form dy +R x y=Q x dx

D 22

Appendix D: A Brief Review of Ordinary Differential Equations

or R x y −Q x dx + dy = 0

D 22 1

N x, y = 1

M x, y

(Note: this ODE is linear since y and dy/dx appear linearly in (D.22).) Then we can determine ∂M ∂y − ∂N ∂x =R x N so that the integrating factor has the form

μ=e

R x dx D 23

Now, multiplying both sides of (D.22) by this integrating factor gives

e

R x dx R x dx dy +R x y =e Qx dx

or R x dx d e y dx

R x dx =e

Qx

Then from this latter expression, we get the solution to (D.22) or R x dx

R x dx y= e

e

Q x dx + c

For example, consider the linear ODE dy +y=x dx (Here, R = 1 and Q = x.) From (D.23), an integrating factor is μ=e

dx

= ex

Thus, via (D.24), the solution is e x y = xe x dx + c

D 24

271

272

Appendix D: A Brief Review of Ordinary Differential Equations

Employing integration by parts

udv = uv − vdu , we ultimately obtain

y = x −1 + ce −x

D.6

Variation of Parameters

Let us return to the linear ODE (D.22) or dy +R x y=Q x dx

D 22

and express its homogeneous version as dyh + R x yh = 0 dx

D 25

or dyh + R x dx = 0 yh The solution to this homogeneous ODE is −

R x dx

yh = c0 e

D 26

c0 u x , c0 a parameter, where −

R x dx

u x =e

D 27

Suppose we now guess that the full solution to (D.22) has the form y=v x u x

D 28

(Clearly, we have replaced the arbitrary parameter c0 in (D.26) by a term dependent on x, that is, we are varying the parameter c0.) If we substitute (D.28) into (D.22), we obtain an ODE for v(x) as follows:

Appendix D: A Brief Review of Ordinary Differential Equations

dv x du x u x +v x +R x v x u x =Q x , dx dx dyh + R x yh = 0 dx dv x u x =Q x , dx Qx dx, dv x = ux vx =

D 29

Qx dx + c ux

Then inserting (D.27) and (D.29) into (D.28) ultimately yields y=u x v x −

R x dx

R x dx

dx + c

Qxe

e

For instance, suppose we have the ODE dy −y = e x dx Here, R x = − 1,Q x = e x , and thus, from (D.27), −

R x dx

u x =e

dx =e

Then (D.30) becomes y = ex

= ex

exe

−

dx dx + c

e x e −x dx + c

= e x x + c = xe x + ce x

= ex

D 30

273

275

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Maruyama, G. Continuous Markov Processes and Stochastic Equations. Rend. Circ. Mat. Palermo 4: 48–90 (1955). McKean, H. P. Stochastic Integrals. New York: Academic Press (1969). Milstein, G. N. A Method of Second-Order Accuracy Integration of Stochastic Differential Equations. Theory Probab. Appl. 23: 396–401 (1978). Nicolau, J. Introduction to the Estimation of Stochastic Differential Equations Based on Discrete Observations. Autumn School and International Conference, Stochastic Finance (2004). Øksendal, B. Stochastic Differential Equations, 6th ed. New York: Springer (2013). Ozaki, T. Non-Linear Time Series Models and Dynamic Systems. Handbook of Statistics, Vol. 5, Hannan, E. ed. North-Holland/Amsterdam: Elsevier: 25–83 (1985). Ozaki, T. A Bridge Between Nonlinear Time Series Models and Nonlinear Stochastic Dynamical Systems: A Local Linearization Approach. Stat. Sin. 2: 25–28 (1992). Ozaki, T. A Local Linearization Approach to Nonlinear Filtering. Int. J. Control. 57: 75–96 (1993). Platen, E. Weak Convergence of Approximations to Itô Integral Equations. Z. Angew. Math. Mech. 60: 609–614 (1980). Platen, E. An Approximation Method for a Class of Itô Processes. Liet. Mat. Rink. 21 (1): 121–133 (1981). Richards, F. J. A Flexible Growth Function for Empirical Use. J. Exp. Bot. 10: 290–300 (1959). Shoji, L. A Comparative Study of Maximum Likelihood Estimators for Nonlinear Dynamical Systems Models. Int. J. Control. 71 (3): 391–404 (1998). Shoji, I., Ozaki, T. Estimation for a Continuous Time Stochastic Process: A New Local Linearization Approach. Res. Memo., No. 524. Tokyo: The Institute of Statistical Mathematics (1994). Shoji, L., Ozaki, T. Comparative Study of Estimation Methods for Continuous Time Stochastic Processes. J. Time Ser. Anal. 18: 485–506 (1997). Shoji, L., Ozaki, T. Estimation for Nonlinear Stochastic Differential Equations by Local Linearization Method. Stoch. Anal. Appl. 16: 733–752 (1998). Simmons, G. F. Introduction to Topology and Modern Analysis. New York: McGraw-Hill Book Co., Inc. (1963). Sѳrensen, H. Parametric Inference for Diffusion Processes Observed at Discrete Points in Time: A Survey. Int. Stat. Rev. 72 (3): 337–354 (2004). Steele, J. M. Stochastic Calculus and Financial Applications. New York: Springer (2010). Taylor, S. J. Introduction to Measure and Integration. New York: Cambridge University Press (1973). Tvedt, J. Market Structure, Freight Rates and Assets in Bulk Shipping. Bergen: Norwegian School of Economics and Business Administration (1995). Verhulst, P. F. Notice Sur la loi que la Population Suit Dans Son Accroissement. Correspondence Mathematique et Physique. Ghent 10: 113–121 (1838).

277

279

Index a

b

absolute convergence 35 absolute growth rate, instantaneous 168 adapted process 89, 90, 101 addition process 27 additive Allee effect 185 additive identity 27 additive inverse 27 additive noise 172, 182 additive set 17 σ-additive set function 17 σ-algebra of sets 15, 16 Allee effect 173, 184–188 Allee threshold 185 almost sure convergence 73 approximation of SDE Euler–Maruyama scheme 194–196, 203–205, 217–218, 228–230 Milstein scheme 196–199, 203–205, 218, 219 Ozaki routine 230–232 parameter estimation methods 228–235 SO routine 233–235 arg max 236 asymptotically stable in the large 155 autonomous linear narrow sense (ALNS) 182, 207 autonomous SDE 136 linear 137 nonlinear 188–191

Banach space 28 Bernoulli’s equation 174 Bessel function 225 bivariate cumulative distribution function 55, 60 bivariate normal distribution 67 bivariate probability density function 59 bivariate probability mass function 55 Borel–Cantelli (BC) lemmas 78, 79 Borel-measurable functions 26, 133 Borel σ-algebra 15–16, 24 Borel set 16, 21, 23, 40, 105, 106, 239, 261 bounded set, real numbers 6 bounded variation 105, 121, 255, 261 Brownian bridge processes 145, 147, 149, 150 Brownian motion (BM) construction of 106–109 geometric process 138, 139, 150, 170, 200, 222 as Markov process 104–106 nondifferentiability 110, 111 normalized process 104 one-dimensional 114, 120, 133 standard (see standard Brownian motion (SBM)) stochastic integral 116 white noise process 109

Stochastic Differential Equations: An Introduction with Applications in Population Dynamics Modeling, First Edition. Michael J. Panik. © 2017 John Wiley & Sons, Inc. Published 2017 by John Wiley & Sons, Inc.

280

Index

c carrying capacity 171 Cauchy convergence criterion 9, 74 Cauchy sequence 9 central limit theorem 80 central moment, rth 52 change of variables 241–243 Chebyshev’s inequality 80, 81 CIR growth equation 183 CKLS mean-reverting gamma SDE 184 class 1 classical Banach spaces 29–31 closure of a set 11 coefficient of correlation 64 complement (set theory) 2, 3 complete measure space 21 complete metric space 9 completeness property 6 complete probability space 40, 90, 102, 114, 120, 127, 133 composite function 247 conditional Cauchy–Schwarz inequality 84 conditional Chebyshev’s inequality 84 conditional expectation 66, 67, 81–84 conditional mean 68 conditional probability 56 density function 60, 61, 68, 69, 71 mass function 57 conditional probability distribution 57 conditional variance 66–69 continuous bivariate probability distribution 59 continuous functions 12–13 continuous multivariate cumulative distribution function 70 continuous multivariate marginal probability mass function 70 continuous random variables 52, 54, 59, 60, 63, 64, 66, 67, 238, 241, 242 continuous stochastic process 88, 246 continuous-time martingales 96, 97, 103 continuous-time population growth model 168 continuous-time stochastic process 85, 88, 90, 193 contracting sequence, of sets 13

convergence continuous-time martingale 97 discrete-time martingale 93–96 in distribution 75–77 of expectations 76, 78 in Lp, p > 0 73–75 in mean 74 in mean square 74 in probability 75 in quadratic mean 74 of random variables 79, 80 rate of 211–212 of sequences of events 78, 79 of sequences of functions 35, 36 step-by-step approximation 212 strong and weak 196 countably additive set function 17 countably infinite set 1 covariance 64 covariance function 109 Cox, Ingersol, and Ross (CIR) SDE 182, 183, 198, 199, 202, 203, 225, 226

d Darboux sum 250, 254 d-dimensional Itô process with stochastic differential equation 127 decrescent function 155 definite integrals 248–251 demographic stochasticity 167 De Morgan’s laws 4 density-dependent population growth equation 170 density function, transition probability 222–235 derived set 11 deterministic generalized logistic growth models 174–177 deterministic Gompertz growth models 177–179 deterministic linear growth models 181, 182 deterministic logistic growth models 170–174, 184–188 deterministic negative exponential growth models 179, 180 deterministic population growth model 168, 169

Index

deterministic sparsity-impacted logistic per capita growth equation 185 deterministic Taylor formula 214 differentiable functions 246–247 differential form 122–124, 153 differentials 247, 248 diffusion coefficient functions 133, 154, 199, 219 diffusion process 153, 154, 221, 225, 227, 239 Dirac delta function 104 Dirac point mass function 109 discontinuity 42, 44, 246, 251, 255 discrete bivariate probability distribution 55 discrete bivariate random variable 54 discrete multivariate cumulative distribution function 69 discrete multivariate marginal probability mass function 70 discrete random variables 42, 44, 50–53, 55–58, 63, 64, 66, 67 discrete-time approximation 193, 196 discrete-time martingales 91–96 discrete time model 207, 209 discrete time stochastic process 85 disjoint set 2 distribution function 42–49, 261 cumulative 42, 44, 62, 75, 76, 80, 238 limiting 76 distributive laws (set theory) 3 domain of function 17 dominated convergence theorem (DCT) 36, 78, 95 Donsker’s invariance principle 108 Doob’s continuous-time stopping theorem 96 Doob’s convergence theorem 97 Doob’s regularization theorem 99 Dow Jones industrial average 85 drift coefficient 104, 133, 154, 199, 207

e element 1 elementary process 115, 116 element inclusion 1

EM approximation scheme see Euler–Maruyama (EM) approximation scheme empty set 1 environmental stochasticity 167 equilibrium solution 155–159 equivalent process 115 essential discontinuity 246 Euclidean norm 29 Euclidean space 29 Euler–Maruyama (EM) approximation scheme 194–196, 217, 218, 228–230 strong and weak convergence 196 variations on 203–205 exact ML estimation 222 exact ordinary differential equation 266–269 existence of moments 148–149 existence of solution 134–136 existence theorem 134, 148, 153, 154, 157, 251 for RS integrals 255–256 expanding sequence, of sets 13 exponential growth models, negative 179, 180 extended real number system 6

f family 1 Fatou’s lemma 36 feedback term 174, 180 field 6 filtered probability space 89–91, 99 finite-dimensional distribution 86–88, 110 σ-finite function 18 finite intersection property 12 finite set 1 first-order differential equation 263–264 function space 28

g gamma function 226 Gaussian process 88, 109, 118 geometric mean-reversion process γ-Hölder 98

145

281

282

Index

globally asymptotically stable equilibrium 155 globally stochastically asymptotically stable equilibrium 158 global solution 136, 137, 155, 157 Gompertz growth models 177–179 greatest lower bound 6

h Haar functions 107 Hilbert space orthonormal basis 30 of random variables 65 Hölder function 98 homogeneous variety, linear

159–161

i indefinite integrals 107, 252 indefinite Itô stochastic integral 119 independent random variables 61, 66–68 indistinguishable process 86 inequalities Chebyshev’s 80, 81 conditional Cauchy-Schwarz 84 continuous-time martingale 97, 98 discrete-time martingale 97 Markov’s 80 moment 148, 149 infimum 6 infinite set 1 informational stochasticity 167, 168 inner product 30 inner product norm 30 inner product space 30 instantaneous absolute growth rate 168 instantaneous relative growth rate 168, 171 integrable stochastic process 88 integral of measurable function 33–35 of measurable function on measurable set 34, 35 of non-negative measurable function 33 of non-negative simple function 32, 33 integrand 248, 254 parameter in 252, 253

integrating factors 162–164, 269–272 integration 248–256 definite integral 248–251 limits of 251–252 parametric integrals 252–253 RS integral 253–256 integration by parts formula 121, 130 integrator 254 intersection (set theory) 2, 3 inverse transformation 190, 201, 202 isometry 115, 116 Itô chain rule 120, 121 Itô coefficient matching 138–142, 145 Itô diffusion 153 Itô integral 114–121, 126 Itô isometry 115, 116, 126, 184 Itô multiplication table 129–131, 137, 160 Itô process 120, 121, 127, 128, 194–196 Itô representation theorem 126 Itô’s formula 120, 129, 130, 137, 139, 142, 151, 186–189, 206, 215, 218, 222 integral form 122, 123 multidimensional 127, 128, 130, 131 one-dimensional 120–126 Itô’s product formula 130 Itô’s stochastic initial value problem 193 Itô stochastic integral 114–120 Itô–Taylor expansion 197, 215, 216, 217

j Jacobian determinant 242, 243 joint probability 54 density function 69 mass function 69

k Kolmogorov continuity theorem 98, 99 Kolmogorov existence theorem 87, 88, 109–110

l Lagrange’s form of the remainder 256 Lamperti transformation 199–203, 219, 220, 231, 232, 234 large O notation 211

Index

L1-bounded martingale 93, 94 L2-bounded martingale convergence theorem 95, 96 least upper bound 6 Lebesgue integration 259, 260 Lebesgue measurable functions 26, 27 Lebesgue measurable sets 23 Lebesgue–Stieltjes (LS) integral 261, 262 Lebesgue sum 259 left-continuous function 88, 246 likelihood function 221, 223, 226, 235, 239–241 limit inferior of sequence 9, 10, 14 limiting function 35 limit of a function 245 limit point 11 limit set 15 limits of integration 69, 251–253 limits of sequences 8–10 limit superior of sequence 9, 10, 14 limit to growth parameter 179 linear growth condition, SDE 135, 136 linear growth models 181, 182 linearity, of conditional expectation 83 linear SDEs 136–154 autonomous 137 homogeneous 137 homogeneous variety 159–161 Markov processes 152–154 product form 159–162 properties of solutions 147–152 strong solutions to 137–147 linear space 27 linear subspace 27 linear variety 161–162 Lipschitz continuity 134, 135 local linearization techniques 194 Ozaki method 205–207 rate of convergence 211, 212 Shoji–Ozaki method 207–210 locally compact space 12 logarithmic-likelihood function 235 logistic growth models 170–174, 184–188 logistic population growth function 171 log-normal probability distribution 238–239

Lyapunov function 156–159 Lyapunov stability 155, 157

m marginal cumulative distribution functions 60 marginal probability density function 60, 68, 71 distribution 56 mass function 55 Markov process 88, 105 Brownian motion 104–106 homogeneous 105, 153, 240 likelihood function 226 linear SDEs 152–154 strong 106 transition probability 105 Markov property 88, 105, 239–241 Markov’s inequality 80 martingales continuous-time 96, 97, 103 discrete-time 91–96 inequalities 97, 98 left limit 97 property 91 representation theorem 126, 127 right-continuous 97 square-integrable 119, 126, 127 stopping time 93 transform theorem 92, 93 uniformly integrable 94 usual conditions in 96 master set 1 maximum likelihood estimator 236, 237 maximum likelihood (ML) technique 235–237 max operator 236 mean reversion 144, 145 stochastic square-root growth model with 182–184 Э-measurable 24 measurable functions complete 21 outer 19–21 Э-measurable sets 16 measurable space 16

283

284

Index

measurable stochastic process 88 measure functions 18, 19 measure space 19 mensuration stochasticity 167 mesh size 116 method of maximum likelihood 235 metric outer measure 20, 21 metric spaces 7 Milstein (second-order) approximation scheme 196–199, 203–205, 218, 219 ML technique see maximum likelihood (ML) technique moment about zero, rth 52 moment inequalities 148, 149 monotone convergence theorem (MCT) 35, 36, 77 monotone function 18, 135, 245 monotone sequence, of sets 13 multidimensional Itô formula 127, 128, 130, 131 multiple random variables continuous case 59–63 discrete case 54–59 expectations and moments 63–69 multivariate discrete and continuous cases 69–72 multiplicative Allee effect 185 multiplicative identity 27 multiplicative noise term 171, 175, 178, 180, 181 multivariate conditional probability mass function 70 multivariate normal distribution 71 multivariate standard normal distribution 72 mutually exclusive set 2

n natural filtration 90, 102 negative definite function 155, 156 net birth rate 168, 169 noise 113, 114, 167 additive 172, 182 multiplicative 171, 175, 178, 180–182 white 109, 114 nonanticipating process 89 noncentral chi-squared distribution 226

nondecreasing sequence, of sets 13 nondifferentiability, Brownian motion 110, 111 normalized Brownian motion (BM) process 104 normed linear spaces bounded continuous real-valued functions 28, 29 bounded real-valued functions 27, 28 classical Banach spaces 29–31 null vector 30

o one-dimensional Brownian motion (BM) 114, 120, 133 one-dimensional Itô formula 120–126 ordered field 6 ordinary differential equation (ODE) 113, 168, 177, 263, 269–270 exact ODE 139, 266–269 first-order differential equation 263–264 general solution 264 integrating factor 269–272 linear 271, 272 Lyapunov stability for 155–158 particular solution 264 separation of variables 264–266 variation of parameters 272, 273 Ornstein–Uhlenbeck (OU) process 144, 145, 151, 224, 229 orthogonal increments 119 orthonormal basis 31 orthonormal set 30 outer measure functions 19–21 Ozaki local linearization method 205–207, 230–232

p parameter estimation methods 228–235 parameter space 85 parameters, variation of 272, 273 parametric integrals 252, 253 partial-averaging property, of conditional expectation 82 partial sum process 100 particular solution, of ODE 264

Index

partition 248 path regularity, stochastic process 98, 99 path uniqueness of solution 135 per capita growth rate 168, 171, 184, 185 PML method see pseudo-maximum likelihood (PML) method p-norm 29 P-null set 90, 102 point of closure 10, 11 point-set theory 10–12 polynomial growth condition 229, 230 population growth model continuous-time 168 deterministic 168, 169 stochastic 169, 170 positive definite function 155 predictor–corrector method 204 principle of maximum likelihood 235 probability density function 70 probability distributions 42–49 probability mass 54 probability space 37–42 complete 40, 90, 102, 114, 120, 127, 133 effective 56 filtered 89, 90, 133, 135 reduced/effective 56, 81 stochastic process 86 submartingale 95 supermartingale on 96 product form 159 linear SDEs 159–162 product rule 160, 161 progressively measurable process 90 proper subset 1 pseudo-likelihood function 222, 227, 228, 230–234 pseudo-maximum likelihood (PML) method 221, 222, 227–229 pseudo-transition density function 227, 228, 230 pth-order moment 147, 148

q quadratic variation process

88, 102

r radially unbounded function 155, 159 Radon–Nikodỳm derivative 83 Radon–Nikodỳm theorem 82–84 randomerror 113 random process see stochastic process random variables continuous 52, 54, 59, 60, 63, 64, 66, 67, 238, 241, 242 convergence of 72–80 discrete 42, 44, 50–53, 55–58, 63, 64, 66, 67 dominated convergence theorem 78 expectation of 49–52 limiting 72 moments of 52–54 monotone convergence theorem 77 multiple (see multiple random variables) rate of convergence 211, 212 real numbers lower bound 6 system 6 upper bound 6 reduction method 175, 178 regular function 246 regular outer measures 20 regular right-continuous function (RRC) 246 relative growth rate 113 instantaneous 168, 171 representation theorem, martingales 126–127 retraction 13 retraction mapping 13 Richards growth function 174 Riemann integration 250 Riemann–Stieltjes (RS) integral 253–256 right-continuous function 88, 246 σ-ring 15

s sample path 86 continuous 107, 108, 119, 153, 157 SBM process 103 simulation processes 194 stochastic process 86, 211 sample space, effective 56

285

286

Index

saturation level 174 SBM see standard Brownian motion (SBM) scalar multiplication 27 scaling property 103 Schauder functions 107 second-order Taylor expansion 122, 129, 131, 137, 191 sequence of sets 4 operations on 13–15 set of measure zero 21, 25, 35, 40, 251 sets algebra 2–4 difference between 2, 3 indicator/characteristic function 24 and set inclusion 1–2 Shoji–Ozaki (SO) local linearization techniques 207–210, 233–235 Shoji’s theorem 211–212 simple function 25 single-valued functions 4–5 solutions 134 equilibrium 155, 157 existence and uniqueness of 134–136 global 136 properties of 147–152 strong 135, 137–147 trivial 155, 157 weak 135 space 1 sparsity component 185 sparsity-impacted logistic SDE 186 sparsity parameter 185 speed of adjustment 182, 183 square-integrable 31, 88 martingale 119, 126–127 stability of stochastic differential equation (SDE) 154–159 stable equilibrium 155, 158–159, 185 standard Brownian motion (SBM) 100–104, 193, 240 states 85 state space 85 stationary process 88, 102, 105, 144 in wide sense 109 stochastically asymptotically stable equilibrium 158, 159 stochastically asymptotic stable in the large 158

stochastically bounded solution 147 stochastically equivalent process 86 stochastically stable equilibrium 158 stochastically unstable equilibrium 158 stochastic differential equation (SDE) 133 artificial solution 164 autonomous 136 Brownian bridge processes 145, 147, 149, 150 geometric Brownian motion 138, 139, 150, 170, 200, 222 homogeneous 210 integrating factors 162–164 Itô coefficient matching 138–142, 145 linear (see linear SDEs) OU process 144, 145, 151 reducible 189–191 stability 154–159 taxonomy and solutions 146 test for exactness 139, 140 variation of parameters 162, 164–165 stochastic generalized logistic growth models 174–177 stochastic Gompertz growth models 177–179 stochastic integral 114–120, 125, 126 additivity 118 Gaussian 118 linearity 118 measurability 118 stochastic linear growth models 181, 182 stochastic logistic growth models 170–174, 184–188 stochastic Lyapunov function 158 stochastic mean-reverting square-root growth equation 183 stochastic negative exponential growth models 179, 180 stochastic population growth model 169, 170 stochastic process 98, 99 adapted 89, 90 characteristics of 88, 89 continuous time 85, 88, 90 discrete time 85 filtration 89, 90 finite-dimensional distributions 86–88 Gaussian 88

Index

integrable 88 left-continuous 88 measurable 88 path regularity 98, 99 right-continuous 88 sample paths 86 square-integrable 88 usual conditions 90 version/modification 86 stochastic square-root growth model 182–184 stochastic Taylor expansions 212–217 stochastic Taylor formula 215 stopped process 93 stopping time, martingale 93, 96 strong convergence 196 strong law of large numbers 79 strongly unique solution 135 strong Markov process 106 strong Markov property 106 strong solutions 135 to linear SDEs 137–147 submartingales 93, 95–98 subsets 1 classes of 15, 16 subspace 7 supermartingales 93, 95–98 supremum 6 sure convergence 72 symmetric difference (set theory) 2, 3 symmetric random walk 99, 100 Brownian motion as limit of 108, 109 symmetry property 103

t tail event, of sequence 78 tail σ-algebra 78 Taylor expansion inverse transformation 202 second-order 122, 129, 131, 137, 191 stochastic 212–217 Taylor’s formula with integral remainder 256 Taylor’s theorem 256–258 remainder 257, 258 single independent variable 256, 257 t-continuous stochastic process 119

third-order remainder term 214 time-homogeneous SDE 205, 221, 230 time-inversion property 103 time space 85 topological space 15 Tower law 84 tower property 89–90 transition density 154, 239–241 transition probability 153 density function 222–235 Markov process 105 trivial solution 155–157, 159

u unbounded variation 103 underpopulation effect 184 uniform Lipschitz condition 134, 135 uniformly integrable 94, 96, 97 uniformly stochastically bounded process 147 union (set theory) 2, 3 uniqueness of solution 134–136 uniqueness theorem 134, 148, 153, 154, 157 unit vector 30 universal set 1 unstable equilibrium 155, 158, 185 upper and lower bounds 6

v variables, separation of 264–266 variance 64 variance coefficient 104 variation of parameters 144, 162, 164–165, 272–273 vector space 27

w weak Allee effect 185 weak convergence 196 weak law of large numbers 79, 80 weakly unique solution 135 weak solution 135 white noise 109, 114 Wiener process 100–104, 154, 167, 194–196, 200, 202, 224, 240

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