Stochastic Differential Equations and Applications

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.

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