# Stochastic Processes

Stochastic Processes

This comprehensive guide to stochastic processes gives a complete overview of the theory and addresses the most important applications. Pitched at a level accessible to beginning graduate students and researchers from applied disciplines, it is both a course book and a rich resource for individual readers. Subjects covered include Brownian motion, stochastic calculus, stochastic differential equations, Markov pro- cesses, weak convergence of processes, and semigroup theory. Applications include the Black–Scholes formula for the pricing of derivatives in financial mathematics, the Kalman–Bucy filter used in the US space program, and also theoretical applications to partial differential equations and analysis. Short, readable chapters aim for clarity rather than for full generality. More than 350 exercises are included to help readers put their new-found knowledge to the test and to prepare them for tackling the research literature.

richard f. bass is Board of Trustees Distinguished Professor in the Department of Mathematics at the University of Connecticut.

C A M B R I D G E S E R I E S I N S T A T I S T I C A L A N D P R O B A B I L I S T I C M A T H E M A T I C S

Editorial Board Z. Ghahramani (Department of Engineering, University of Cambridge)

R. Gill (Mathematical Insitute, Leiden University) F. P. Kelly (Department of Pure Mathematics and Mathematical Statistics,

University of Cambridge) B. D. Ripley (Department of Statistics, University of Oxford) S. Ross (Department of Industrial and Systems Engineering,

University of Southern California) M. Stein (Department of Statistics, University of Chicago)

This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochastic applicable mathematics. The topics range from pure and applied statistics to probability theory, operations research, optimization, and mathematical programming. The books contain clear presentations of new developments in the field and also of the state of the art in classical methods. While emphasizing rigorous treatment of theoretical methods, the books also contain applications and discussions of new techniques made possible by advances in computational practice.

A complete list of books in the series can be found at http://www.cambridge.org/statistics. Recent titles include the following:

11. Statistical Models, by A. C. Davison 12. Semiparametric Regression, by David Ruppert, M. P. Wand and R. J. Carroll 13. Exercises in Probability, by Loı̈c Chaumont and Marc Yor 14. Statistical Analysis of Stochastic Processes in Time, by J. K. Lindsey 15. Measure Theory and Filtering, by Lakhdar Aggoun and Robert Elliott 16. Essentials of Statistical Inference, by G. A. Young and R. L. Smith 17. Elements of Distribution Theory, by Thomas A. Severini 18. Statistical Mechanics of Disordered Systems, by Anton Bovier 19. The Coordinate-Free Approach to Linear Models, by Michael J. Wichura 20. Random Graph Dynamics, by Rick Durrett 21. Networks, by Peter Whittle 22. Saddlepoint Approximations with Applications, by Ronald W. Butler 23. Applied Asymptotics, by A. R. Brazzale, A. C. Davison and N. Reid 24. Random Networks for Communication, by Massimo Franceschetti and Ronald Meester 25. Design of Comparative Experiments, by R. A. Bailey 26. Symmetry Studies, by Marlos A. G. Viana 27. Model Selection and Model Averaging, by Gerda Claeskens and Nils Lid Hjort 28. Bayesian Nonparametrics, edited by Nils Lid Hjort et al. 29. From Finite Sample to Asymptotic Methods in Statistics, by Pranab K. Sen,

Julio M. Singer and Antonio C. Pedrosa de Lima 30. Brownian Motion, by Peter Mörters and Yuval Peres 31. Probability, by Rick Durrett 33. Stochastic Processes, by Richard F. Bass 34. Structured Regression for Categorical Data, by Gerhard Tutz

Stochastic Processes

Richard F. Bass University of Connecticut

CAMBRIDGE UNIVERSITY PRESS

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Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org Information on this title: www.cambridge.org/9781107008007

C© R. F. Bass 2011

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2011

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data Bass, Richard F.

Stochastic processes / Richard F. Bass. p. cm. – (Cambridge series in statistical and probabilistic mathematics ; 33)

Includes index. ISBN 978-1-107-00800-7 (hardback)

1. Stochastic analysis. I. Title. QA274.2.B375 2011

519.2′32 – dc23 2011023024

ISBN 978-1-107-00800-7 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Meredith, as always

Contents

Preface page xiii Frequently used notation xv

1 Basic notions 1 1.1 Processes and σ -fields 1 1.2 Laws and state spaces 3

2 Brownian motion 6 2.1 Definition and basic properties 6

3 Martingales 13 3.1 Definition and examples 13 3.2 Doob’s inequalities 14 3.3 Stopping times 15 3.4 The optional stopping theorem 17 3.5 Convergence and regularity 17 3.6 Some applications of martingales 20

4 Markov properties of Brownian motion 25 4.1 Markov properties 25 4.2 Applications 27

5 The Poisson process 32

6 Construction of Brownian motion 36 6.1 Wiener’s construction 36 6.2 Martingale methods 39

7 Path properties of Brownian motion 43

8 The continuity of paths 49

vii

viii Contents

9 Continuous semimartingales 54 9.1 Definitions 54 9.2 Square integrable martingales 55 9.3 Quadratic variation 57 9.4 The Doob–Meyer decomposition 58

10 Stochastic integrals 64 10.1 Construction 64 10.2 Extensions 69

11 Itô’s formula 71

12 Some applications of Itô’s formula 77 12.1 Lévy’s theorem 77 12.2 Time changes of martingales 78 12.3 Quadratic variation 79 12.4 Martingale representation 79 12.5 The Burkholder–Davis–Gundy inequalities 82 12.6 Stratonovich integrals 84

13 The Girsanov theorem 89 13.1 The Brownian motion case 89 13.2 An example 92

14 Local times 94 14.1 Basic properties 94 14.2 Joint continuity of local times 96 14.3 Occupation times 97

15 Skorokhod embedding 100 15.1 Preliminaries 100 15.2 Construction of the embedding 105 15.3 Embedding random walks 108

16 The general theory of processes 111 16.1 Predictable and optional processes 111 16.2 Hitting times 115 16.3 The debut and section theorems 117 16.4 Projection theorems 119 16.5 More on predictability 120 16.6 Dual projection theorems 122 16.7 The Doob–Meyer decomposition 124 16.8 Two inequalities 126

Contents ix

17 Processes with jumps 130 17.1 Decomposition of martingales 130 17.2 Stochastic integrals 133 17.3 Itô’s formula 135 17.4 The reduction theorem 139 17.5 Semimartingales 141 17.6 Exponential of a semimartingale 143 17.7 The Girsanov theorem 144

18 Poisson point processes 147

19 Framework for Markov processes 152 19.1 Introduction 152 19.2 Definition of a Markov process 153 19.3 Transition probabilities 154 19.4 An example 156 19.5 The canonical process and shift operators 158

20 Markov properties 160 20.1 Enlarging the filtration 160 20.2 The Markov property 162 20.3 Strong Markov property 164

21 Applications of the Markov properties 167 21.1 Recurrence and transience 167 21.2 Additive functionals 169 21.3 Continuity 170 21.4 Harmonic functions 171

22 Transformations of Markov processes 177 22.1 Killed processes 177 22.2 Conditioned processes 178 22.3 Time change 180 22.4 Last exit decompositions 181

23 Optimal stopping 184 23.1 Excessive functions 184 23.2 Solving the optimal stopping problem 187

24 Stochastic differential equations 192 24.1 Pathwise solutions of SDEs 192 24.2 One-dimensional SDEs 196 24.3 Examples of SDEs 198

x Contents

25 Weak solutions of SDEs 204

26 The Ray–Knight theorems 209

27 Brownian excursions 214

28 Financial mathematics 218 28.1 Finance models 218 28.2 Black–Scholes formula 220 28.3 The fundamental theorem of finance 223 28.4 Stochastic control 226

29 Filtering 229 29.1 The basic model 229 29.2 The innovation process 230 29.3 Representation of FZ-martingales 231 29.4 The filtering equation 232 29.5 Linear models 234 29.6 Kalman–Bucy filter 234

30 Convergence of probability measures 237 30.1 The portmanteau theorem 237 30.2 The Prohorov theorem 239 30.3 Metrics for weak convergence 241

31 Skorokhod representation 244

32 The space C[0, 1] 247 32.1 Tightness 247 32.2 A construction of Brownian motion 248

33 Gaussian processes 251 33.1 Reproducing kernel Hilbert spaces 251 33.2 Continuous Gaussian processes 254

34 The space D[0, 1] 259 34.1 Metrics for D[0, 1] 259 34.2 Compactness and completeness 262 34.3 The Aldous criterion 264

35 Applications of weak convergence 269 35.1 Donsker invariance principle 269 35.2 Brownian bridge 273 35.3 Empirical processes 275

Contents xi

36 Semigroups 279 36.1 Constructing the process 279 36.2 Examples 283

37 Infinitesimal generators 286 37.1 Semigroup properties 286 37.2 The Hille–Yosida theorem 292 37.3 Nondivergence form elliptic operators 296 37.4 Generators of Lévy processes 297

38 Dirichlet forms 302 38.1 Framework 303 38.2 Construction of the semigroup 304 38.3 Divergence form elliptic operators 307

39 Markov processes and SDEs 312 39.1 Markov properties 312 39.2 SDEs and PDEs 314 39.3 Martingale problems 315

40 Solving partial differential equations 319 40.1 Poisson’s equation 319 40.2 Dirichlet problem 320 40.3 Cauchy problem 321 40.4 Schrödinger operators 323

41 One-dimensional diffusions 326 41.1 Regularity 326 41.2 Scale functions 327 41.3 Speed measures 329 41.4 The uniqueness theorem 333 41.5 Time change 334 41.6 Examples 336

42 Lévy processes 339 42.1 Examples 339 42.2 Construction of Lévy processes 340 42.3 Representation of Lévy processes 344

Appendices A Basic probability 348 A.1 First notions 348 A.2 Independence 353 A.3 Convergence 355 A.4 Uniform integrability 356

xii Contents

A.5 Conditional expectation 357 A.6 Stopping times 359 A.7 Martingales 359 A.8 Optional stopping 360 A.9 Doob’s inequalities 361 A.10 Martingale convergence theorem 362 A.11 Strong law of large numbers 364 A.12 Weak convergence 367 A.13 Characteristic functions 370 A.14 Uniqueness and characteristic functions 372 A.15 The central limit theorem 372 A.16 Gaussian random variables 374

B Some results from analysis 378 B.1 The monotone class theorem 378 B.2 The Schwartz class 379

C Regular conditional probabilities 380

D Kolmogorov extension theorem 382

References 385 Index 387

Preface

Why study stochastic processes? This branch of probability theory offers sophisticated theo- rems and proofs, such as the existence of Brownian motion, the Doob–Meyer decomposition, and the Kolmogorov continuity criterion. At the same time stochastic processes also have far-reaching applications: the explosive growth in options and derivatives in financial mar- kets throughout the world derives from the Black–Scholes formula, while NASA relies on the Kalman–Bucy method to filter signals from satellites and probes sent into outer space.

A graduate student taking a year-long course in probability theory first learns about sequences of random variables and topics such as laws of large numbers, central limit theorems, and discrete time martingales. In the second half of the course, the student will then turn to stochastic processes, which is the subject of this text. Topics covered here are Brownian motion, stochastic integrals, stochastic differential equations, Markov processes, the Black–Scholes formula of financial mathematics, the Kalman–Bucy filter, as well as many more.

The 42 chapters of this book can be grouped into seven parts. The first part consists of Chapters 1–8, where some of the basic processes and ideas are introduced, including Brownian motion. The next group of chapters, Chapters 9–15, introduce the theory of stochastic calculus, including stochastic integrals and Itô’s formula. Chapters 16–18 explore jump processes. This requires a study of the foundations of stochastic processes, which is also known as the general theory of processes. Next we take up Markov processes in Chapters 19–23. A formidable obstacle to the study of Markov processes is the notation, and I have attempted to make this as accessible as possible. Chapters 24–29 involve stochastic differential equations. Two very important applications, to financial mathematics and to filtering, appear in Chapters 28 and 29, respectively. Probability measures on metric spaces and the weak convergence of random variables taking values in a metric space prove to be relevant to the study of stochastic processes. These and related topics are treated in Chapters 30–35. We then return to Markov processes, namely, their construction and some important examples, in Chapters 36–42. Tools used in the construction include infinitesimal generators, Dirichlet forms, and solutions to stochastic differential equations, while two important examples that we consider are diffusions on the real line and Lévy processes.

The prerequisites to this book are a sound knowledge of basic measure theory and a course in the classical aspects of probability. The probability topics needed are provided (with proofs) in an appendix.

There is far too much material in this book to cover in a single semester, and even too much for a full year. I recommend that as a minimum the following chapters be studied: Chapters 1–5, Chapters 9–13, Chapters 19–21, and Chapter 24. If possible, include either

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