Table Of ContentSTOCHASTIC
CALCULUS
A Practical Introduction
PROBABILITY AND STOCHASTICS
SERIES
Edited by Richard Durrett and Mark Pinsky
Linear Stochastic Control Systems, Guanrong Chen, Goong Chen, and
Shia-Hsun Hsu
Advances in Queueing: Theory, Methods, and Open Problems,
Jewgeni H. Dshalalow
Stochastic Calculus: A Practical Introduction,
Richard Durrett
Chaos Expansion, Multiple Weiner-Ito Integrals and Applications, Christian Houdre
and Victor Perez-Abreu
White Noise Distribution Theory, Hui-Hsiung Kuo
Topics in Contemporary Probability, J. Laurie Snell
Probability
and Stochastics Series
STOCHASTIC
CALCULUS
A Practical Introduction
Richard Durrett
0
~~~F~:!~~"P
Boca Ratan London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Library of Congress Cataloging-in-Publication Data
Catalog record is available from the Library of Congress
This book contains information obtained from authentic and highly regarded sources. Reprinted material
is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable
efforts have been made to publish reliable data and information, but the authors and the publisher cannot
assume responsibility for the validity of all materials or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic
or mechanical, including photocopying, microfilming, and recording, or by any information storage or
retrieval system, without prior permission in writing from the publisher.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for
creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC
for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are
used only for identification and explanation, without intent to infringe.
Visit the CRC Press Web site at www.crcpress.com
© 1996 by CRC Press
No claim to original U.S. Government works
International Standard Book Number 0-8493-8071-5
Printed in the United States of America 6 7 8 9 0
Printed on acid-free paper
Preface
This book is the re-incarnation of my first book Brownian Motion and
Martingales in Analysis, which was published by Wadsworth in 1984. For more
than a decade I have used Chapters 1, 2, 8, and 9 of that book to give "reading
courses" to graduate students who have completed the first year graduate prob
ability course and were interested in learning more about processes that move
continuously in space and time. Taking the advice from biology that "form fol
lows function" I have taken that material on stochastic integration, stochastic
differential equations, Brownian motion and its relation to partial differential
equations to be the core of this book (Chapters 1-5). To this I have added
other practically important topics: one dimensional diffusions, semigroups and
generators, Harris chains, and weak convergence. I have struggled with this
material for almost twenty years. I now think that I understand most of it,
so to help you master it in less time, I have tried to explain it as simply and
clearly as I can.
My students' motivations for learning this material have been diverse: some
have wanted to apply ideas from probability to analysis or differential geometry,
others have gone on to do research on diffusion processes or stochastic partial
differential equations, some have been interested in applications of these ideas
to finance, or to problems in operations research. My motivation for writing
this book, like that for Probability Theory and Examples, was to simplify my life
as a teacher by bringing together in one place useful material that is scattered
in a variety of sources.
An old joke says that "if you copy from one book that is plagiarism, but
if you copy from ten books that is scholarship." From that viewpoint this is a
scholarly book. Its main contributors for the various subjects are (a) stochastic
integration and differential equations: Chung and Williams (1990), Ikeda and
Watanabe (1981), Karatzas and Shreve (1991), Protter (1990), Revuz and Yor
(1991), Rogers and Williams (1987), Stroock and Varadhan (1979); (b) partial
differential equations: Folland (1976), Friedman (1964), (1975), Port and Stone
(1978), Chung and Zhao (1995); (c) one dimensional diffusions: Karlin and
Taylor (1981); (d) semi-groups and generators: Dynkin (1965), Revuz and Yor
(1991); (e) weak convergence: Billingsley {1968), Ethier and Kurtz (1986),
Stroock and Varadhan (1979). If you bought all those books you would spend
more than $1000 but for a fraction of that cost you can have this book, the
intellectual equivalent of the ginzu knife.
Vl Preface
Shutting off the laugh-track and turning on the violins, the road from this
book's first publication in 1984 to its rebirth in 1996 has been a long and
one. In the second half of the 80's I accumulated an embarrassingly long list
of typos from the first edition. Some time at the beginning of the 90's I talked
to the editor who brought my first three books into the John
about preparing a second edition. However, after the work was done, the second
edition was personally killed Bill Roberts, the President of Brooks/Cole. At
the end of 1992 I entered into a contract with Wayne Yuhasz at CRC Press
to produce this book. In the first few months of 1993, June Meyerman typed
most of the book into TeX. In the Fall Semester of 1993 I taught a course from
this material and began to organize it into the current form. By the summer
of 1994 I thought I was almost done. At this point I had the good bad)
fortune of having Nora Guertler, a student from Lyon, visit for two months.
When she was through making an average of six corrections per page, it was
clear that the book was far from finished.
During the 1994-95 academic year most of my time was devoted to prepar
the second edition of my first year graduate textbook Probability: Theory
and Examples. After that experience my brain cells could not bear to work
on another book for another several months, but toward the end of 1995 they
decided "it is now or never." The delightful Cornell tradition of a long winter
break, which for me stretched from early December to late January, provided
just enough time to finally finish the book.
I am grateful to my students who have read various versions of this book
and also made numerous comments: Don Allers, Hassan Allouba, Robert Bat
tig, Marty Hill, Min-jeong Kang, Susan Lee, Gang Ma, and Nikhil Shah. Earlier
in the process, before I started writing, Heike Dengler, David Lando and I spent
a semester reading Protter (1990) and Jacod and Shiryaev (1987), an enterprise
which contributed greatly to my education.
The ancient history of the revision process has unfortunately been lost. At
the time of the proposed second edition, I transferred a number oflists of typos
to my copy of the book, but I have no record of the people who supplied the
lists. I remember getting a number of corrections from Mike Brennan and Ruth
Williams, and it is impossible to forget the story of Robin Pemantle who took
Brownian Motion, Martingales and Analysis as his only math book for a year
long trek through the South Seas and later showed me his fully annotated copy.
However, I must apologize to others whose contributions were recorded but
whose names were lost. Flame me at [email protected] and I'll have something
to say about you in the next edition.
Rick Durrett
About the Author
Rick Durrett received his Ph.D. in Operations research from Stanford
University in 1976. He taught in the Mathematics Department at UCLA for
nine years before becoming a Professor of Mathematics at Cornell University.
He was a Sloan Fellow 1981-83, Guggenheim Fellow 1988-89, and spoke at the
International Congress of Math in Kyoto 1990.
Durrett is the author of a graduate textbook, Probability: Theory and
Examples, and an undergraduate one, The Essentials of Probability. He has
written almost 100 papers with a total of 38 co-authors and seen 19 students
complete their Ph.D. 's under his direction. His recent research focuses on the
applications of stochastic spatial models to various problems in biology.
Stochastic Calculus: A Practical Introduction
1. Brownian Motion
1.1 Definition and Construction 1
1.2 Markov Property, Blumenthal's 0-1 Law 7
1.3 Stopping Times, Strong Markov Property 18
1.4 First Formulas 26
2. Stochastic Integration
2.1 Integrands: Predictable Processes 33
2.2 Integrators: Continuous Local Martingales 37
2.3 Variance and Covariance Processes 42
2.4 Integration w.r.t. Bounded Martingales 52
2.5 The Kunita-Watanabe Inequality 59
2.6 Integration w.r.t. Local Martingales 63
2.7 Change of Variables, Ita's Formula 68
2.8 Integration w.r.t. Semimartingales 70
2.9 Associative Law 74
2.10 Functions of Several Semimartingales 76
Chapter Summary 79
2.11 Meyer-Tanaka Formula, Local Time 82
2.12 Girsanov's Formula 90
3. Brownian Motion, II
3.1 Recurrence and Transience 95
3.2 Occupation Times 100
3.3 Exit Times 105
3.4 Change of Time, Levy's Theorem 111
3.5 Burkholder Davis Gundy Inequalities 116
3.6 Martingales Adapted to Brownian Filtrations 119
4. Partial Differential Equations
A. Parabolic Equations
4.1 The Heat Equation 126
4.2 The Inhomogeneous Equation 130
4.3 The Feynman-Kac Formula 137
B. Elliptic Equations
4.4 The Dirichlet Problem 143
4.5 Poisson's Equation 151
4.6 The Schrodinger Equation 156
C. Applications to Brownian Motion
4. 7 Exit Distributions for the Ball 164
4.8 Occupation Times for the Ball 167
4.9 Laplace Transforms, Arcsine Law 170
5. Stochastic Differential Equations
5.1 Examples 177
5.2 Ita's Approach 183
5.3 Extension 190
5.4 Weak Solutions 196
5.5 Change of Measure 202
5.6 Change of Time 207
6. One Dimensional Diffusions
6.1 Construction 211
6.2 Feller's Test 214
6.3 Recurrence and Transience 219
6.4 Green's Functions 222
6.5 Boundary Behavior 229
6.6 Applications to Higher Dimensions 234
7. Diffusions as Markov Processes
7.1 Semigroups and Generators 245
7.2 Examples 250
7.3 Transition Probabilities 255
7.4 Harris Chains 258
7.5 Convergence Theorems 268
8. Weak Convergence
8.1 In Metric Spaces 271
8.2 Prokhorov's Theorems 276
8.3 The Space C 282
8.4 Skorohod's Existence Theorem for SDE 285
8.5 Donsker's Theorem 287
8.6 The Space D 293
8. 7 Convergence to Diffusions 296
8.8 Examples 305
Solutions to Exercises 311
References 335
Index 339
