Stochastic Differential Equations and Applications
1st Edition
Volume 1
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Description
Stochastic Differential Equations and Applications, Volume 1 covers the development of the basic theory of stochastic differential equation systems. This volume is divided into nine chapters. Chapters 1 to 5 deal with the basic theory of stochastic differential equations, including discussions of the Markov processes, Brownian motion, and the stochastic integral. Chapter 6 examines the connections between solutions of partial differential equations and stochastic differential equations, while Chapter 7 describes the Girsanov’s formula that is useful in the stochastic control theory. Chapters 8 and 9 evaluate the behavior of sample paths of the solution of a stochastic differential system, as time increases to infinity. This book is intended primarily for undergraduate and graduate mathematics students.
Table of Contents
Preface
General Notation
Contents of Volume 2
1 Stochastic Processes
1. The Kolmogorov Construction of a Stochastic Process
2. Separable and Continuous Processes
3. Martingales and Stopping Times
Problems
2 Markov Processes
1. Construction of Markov Processes
2. The Feller and the Strong Markov Properties
3. Time-Homogeneous Markov Processes
Problems
3 Brownian Motion
1. Existence of Continuous Brownian Motion
2. Nondifferentiability of Brownian Motion
3. Limit Theorems
4. Brownian Motion After a Stopping Time
5. Martingales and Brownian Motion
6. Brownian Motion in n Dimensions
Problems
4 The Stochastic Integral
1. Approximation of Functions by Step Functions
2. Definition of the Stochastic Integral
3. The Indefinite Integral
4. Stochastic Integrals with Stopping Time
5. Itôs Formula
6. Applications of Itôs Formula
7. Stochastic Integrals and Differentials in n Dimensions
Problems
5 Stochastic Differential Equations
1. Existence and Uniqueness
2. Stronger Uniqueness and Existence Theorems
3. The Solution of a Stochastic Differential System as a Markov Process
4. Diffusion Processes
5. Equations Depending on a Parameter
6. The Kolmogorov Equation
Problems
6 Elliptic and Parabolic Partial Differential Equations and Their Relations to Stochastic Differential Equations
1. Square Root of a Nonnegative Definite Matrix
2. The Maximum Principle for Elliptic Equations
3. The Maximum Principle for Parabolic Equations
4. The Cauchy Problem and Fundamental Solutions for Parabolic Equations
5. Stochastic Representation of Solutions of Partial Differential Equations
Problems
7 The Cameron-Martin-Girsanov Theorem
1. A Class of Absolutely Continuous Probabilities
2. Transformation of Brownian Motion
3. Girsanov's Formula
Problems
8 Asymptotic Estimates for Solutions
1. Unboundedness of Solutions
2. Auxiliary Estimates
3. Asymptotic Estimates
4. Applications of the Asymptotic Estimates
5. The One-Dimensional Case
6. Counterexample
Problems
9 Recurrent and Transient Solutions
1. Transient Solutions
2. Recurrent Solutions
3. Rate of Wandering Out to Infinity
4. Obstacles
5. Transient Solutions for Degenerate Diffusion
6. Recurrent Solutions for Degenerate Diffusion
7. The One-Dimensional Case
Problems
Bibliographical Remarks
References
Index
Details
- No. of pages:
- 248
- Language:
- English
- Copyright:
- © Academic Press 1975
- Published:
- 1st January 1975
- Imprint:
- Academic Press
- eBook ISBN:
- 9781483217871
- Paperback ISBN:
- 9781483204444
- Hardcover ISBN:
- 9780122682018
About the Author

Avner Friedman
About the Editors

Z. W. Birnbaum

E. Lukacs
Affiliations and Expertise
Bowling Green State University
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