**Graduate Studies in Mathematics**

Volume: 43;
2002;
230 pp;
Hardcover

MSC: Primary 60;

Print ISBN: 978-0-8218-2985-1

Product Code: GSM/43

List Price: $44.00

Individual Member Price: $35.20

**Electronic ISBN: 978-1-4704-2094-9
Product Code: GSM/43.E**

List Price: $44.00

Individual Member Price: $35.20

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# Introduction to the Theory of Random Processes

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*N. V. Krylov*

This book concentrates on some general facts and ideas of the theory of stochastic processes. The topics include the Wiener process, stationary processes, infinitely divisible processes, and Itô stochastic equations.

Basics of discrete time martingales are also presented and then used in one way or another throughout the book. Another common feature of the main body of the book is using stochastic integration with respect to random orthogonal measures. In particular, it is used for spectral representation of trajectories of stationary processes and for proving that Gaussian stationary processes with rational spectral densities are components of solutions to stochastic equations. In the case of infinitely divisible processes, stochastic integration allows for obtaining a representation of trajectories through jump measures. The Itô stochastic integral is also introduced as a particular case of stochastic integrals with respect to random orthogonal measures.

Although it is not possible to cover even a noticeable portion of the topics listed above in a short book, it is hoped that after having followed the material presented here, the reader will have acquired a good understanding of what kind of results are available and what kind of techniques are used to obtain them.

With more than 100 problems included, the book can serve as a text for an introductory course on stochastic processes or for independent study.

Other works by this author published by the AMS include, *Lectures on
Elliptic and Parabolic Equations in Hölder Spaces* and *Introduction to
the Theory of Diffusion Processes*.

#### Table of Contents

# Table of Contents

## Introduction to the Theory of Random Processes

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Chapter 1. Generalities 114 free
- Chapter 2. The Wiener Process 2740
- § 1. Brownian motion and the Wiener process 2740
- §2. Some properties of the Wiener process 3245
- §3. Integration against random orthogonal measures 3952
- §4. The Wiener process on [0, ∞) 5063
- §5. Markov and strong Markov properties of the Wiener process 5265
- §6. Examples of applying the strong Markov property 5770
- §7. Itô stochastic integral 6174
- §8. The structure of Itô integrable functions 6578
- §9. Hints to exercises 6982

- Chapter 3. Martingales 7184
- Chapter 4. Stationary Processes 95108
- §1. Simplest properties of second-order stationary processes 95108
- §2. Spectral decomposition of trajectories 101114
- §3. Ornstein-Uhlenbeck process 105118
- §4. Gaussian stationary processes with rational spectral densities 112125
- §5. Remarks about predicting Gaussian stationary processes with rational spectral densities 117130
- §6. Stationary processes and the Birkhoff- Khinchin theorem 119132
- §7. Hints to exercises 127140

- Chapter 5. Infinitely Divisible Processes 131144
- §1. Stochastically continuous processes with independent increments 131144
- §2. Lévy-Khinchin theorem 137150
- §3. Jump measures and their relation to Lévy measures 144157
- §4. Further comments on jump measures 154167
- §5. Representing infinitely divisible processes through jump measures 155168
- §6. Constructing infinitely divisible processes 160173
- §7. Hints to exercises 166179

- Chapter 6. Itô Stochastic Integral 169182
- §1. The classical definition 169182
- §2. Properties of the stochastic integral on H 174187
- §3. Defining the Ito integral if ∫[sup(T)][sub(0)] f[sup(2)][sub(s)] ds < ∞ 179192
- §4. Itô integral with respect to a multidimensional Wiener process 186199
- §5. Itô's formula 188201
- §6. An alternative proof of Itô's formula 195208
- §7. Examples of applying Itô's formula 200213
- §8. Girsanov's theorem 204217
- §9. Stochastic Itô equations 211224
- §10. An example of a stochastic equation 216229
- §11. The Markov property of solutions of stochastic equations 220233
- §12. Hints to exercises 225238

- Bibliography 227240
- Index 229242
- Back Cover Back Cover1245

#### Readership

Graduate students and research mathematicians, physicists, and engineers interested in the theory of random processes and its applications.

#### Reviews

The book is written in a nice and thorough style. A large number of exercises are contained.

-- Zentralblatt MATH

An attractive feature of the book, apart from the nice and meticulous style of writing, is that it contains a large number of examples and exercises (and hints for exercises—some of which are certainly quite ambitious and demanding!).

-- Mathematical Reviews