**Translations of Mathematical Monographs**

1994;
271 pp;
Softcover

MSC: Primary 58; 60;
Secondary 35

Print ISBN: 978-0-8218-4600-1

Product Code: MMONO/142

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

**Electronic ISBN: 978-1-4704-4559-1
Product Code: MMONO/142.E**

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

# Introduction to the Theory of Diffusion Processes

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*N. V. Krylov*

Focusing on one of the major branches of probability theory, this book treats the large class of processes with continuous sample paths that possess the “Markov property”. The exposition is based on the theory of stochastic analysis. The diffusion processes discussed are interpreted as solutions of Itô's stochastic integral equations. The book is designed as a self-contained introduction, requiring no background in the theory of probability or even in measure theory. In particular, the theory of local continuous martingales is covered without the introduction of the idea of conditional expectation. Krylov covers such subjects as the Wiener process and its properties, the theory of stochastic integrals, stochastic differential equations and their relation to elliptic and parabolic partial differential equations, Kolmogorov's equations, and methods for proving the smoothness of probabilistic solutions of partial differential equations. With many exercises and thought-provoking problems, this book would be an excellent text for a graduate course in diffusion processes and related subjects.

#### Readership

Graduate students and researchers interested in an understanding of the important features of the theory of diffusion processes and its relationship with the theory of elliptic and parabolic second order partial differential equations.

#### Reviews & Endorsements

For those with a good background in probability theory and analysis, this book is an excellent addition to the already good collection of books. The style is very relaxed but rigorous, written in the great pedagogical tradition of the Russian masters.

-- Journal of the American Statistical Association

What makes this book different is the presentation of the material. The author starts from scratch, introducing all the necessary concepts and techniques as he needs them. This makes it easy to follow his line of thought and to get to the main topics, stochastic integrals and stochastic differential equations, without detour and without many prerequisites … invaluable help when studying from this book is a “dual” presentation of the material: All the main concepts and results are accompanied by a discussion of the intuitive idea behind them, and almost all proofs are given in a straightforward and precise manner.

-- Zentralblatt MATH

An accessible introduction to diffusion processes for working mathematicians and advanced graduate students in analysis … a provocative, instructive, and refreshing perspective from which probabilists can benefit.

-- Mathematical Reviews

The book contains ideas of the author that have not been
systematically presented in any other standard texts. Tremendous
efforts are made to explore the probabilistic solutions of partial
differential equations, reflecting the interest of the author. As an
“introduction” to the theory, the book is
*elementary* enough, even for those who have not had serious
training in probability theory … But on the other hand, the
book is *rich* enough even for specialists in the field, as it
contains many ideas which are different from the classical books on
the subject.

-- Bulletin of the AMS

This is an appealing introduction to the theory of Markov processes with continuous sample paths, based on stochastic analysis by interpreting diffusion processes as solutions of Itô's Stochastic integral equation.

-- Monatshefte für Mathematik

#### Table of Contents

# Table of Contents

## Introduction to the Theory of Diffusion Processes

- Cover Cover11
- Half title page iii4
- Title page v6
- Contents vii8
- Preface ix10
- Chapter 1. Elements of measure and integration theory 114
- Chapter 2. The Wiener process 2740
- Chapter 3. Itô’s stochastic integral 7790
- Chapter 4. Some applications of Itô’s formula 141154
- Chapter 5. Itô’s stochastic equations 165178
- Chapter 6. Further methods for investigating the smoothness of probabilistic solutions of differential equations 221234
- Appendix A. Proof of lemma II.2.4 257270
- Appendix B. Proof of theorem II.8.1 259272
- List of notations 261274
- Comments 263276
- References 267280
- Index 269282
- Recent titles in series (continued) 273286
- Back Cover Back Cover1287