随机微分方程在数学以外的许多领域有着广泛的应用，它对数学领域中的许多分支起着有效的联结作用。本书是《Universitext》丛书之一，是一部理想的研究生教材，内容做了较大的修改和补充，包括鞅表示论、变分不等式和随机控制等内容，书后附有部分习题解答和提示。

本书为全英文版，适合数学专业研究生阅读参考。

随机微分方程在数学以外的许多领域有着广泛的应用，它对数学领域中的许多分支起着有效的联结作用。本书是《Universitext》丛书之一，是一部理想的研究生教材。我们曾影印出版了第2版和第4版，第6版与第4版相比，内容做了较大的修改和补充，增加了90页的篇幅(近1／3内容)，包括鞅表示论、变分不等式和随机控制等内容，书后附有部分习题解答和提示。

Introduction

1.1 Stochastic Analogs of Classical Differential Equations

1.2 Filtering Problems

1.3 Stochastic Approach to Deterministic Boundary Value Problems

1.4 Optimal Stopping

1.5 Stochastic Control

1.6 Mathematical Finance

Some Mathematical Preliminaries

2.1 Probability Spaces, Random Variables and Stochastic Processes

2.2 An Important Example: Brownian Motion

Exercises

Ito Integrals

3.1 Construction of the It5 Integral

3.2 Some properties of the It5 integral

3.3 Extensions of the Ito integral

Exercises

The Ito Formula and the Martingale Representation

Theorem

4.1 The 1-dimensional It5 formula

4.2 The Multi-dimensional It5 Formula

4.3 The Martingale Representation Theorem

Exercises

Stochastic Differential Equations

5.1 Examples and Some Solution Methods 

5.2 An Existence and Uniqueness Result

5.3 Weak and Strong Solutions 

Exercises

6  The Filtering Problem

6.1 Introduction

6.2 The 1-Dimensional Linear Filtering Problem

6.3 The Multidimensional Linear Filtering Problem

Exercises

7  Diffusions: Basic Properties

7.1 The Markov Property 

7.2 The Strong Markov Property

7.3 The Generator of an It5 Diffusion

7.4 The Dynkin Formula

7.5 The Characteristic Operator

Exercises

8  Other Topics in Diffusion Theory

8.1 Kolmogorov's Backward Equation. The Resolvent

8.2 The Feynman-Kac Formula. Killing

8.3 The Martingale Problem

8.4 When is an It5 Process a Diffusion?

8.5 Random Time Change

8.6 The Girsanov Theorem

Exercises

9  Applications to Boundary Value Problems

 9.1 The Combined Dirichlet-Poisson Problem. Uniqueness

 9.2 The Dirichlet Problem. Regular Points

 9.3 The Poisson Problem

 Exercises

10 Application to Optimal Stopping 

 10.1 The Time-Homogeneous Case

 10.2 The Time-Inhomogeneous Case

 10.3 Optimal Stopping Problems Involving an Integral

 10.4 Connection with Variational Inequalities

 Exercises

11 Application to Stochastic Control 

 11.1 Statement of the Problem

 11.2 The Ha.milton-Jacobi-Bellman Equation

 11.3 Stochastic control problems with terminal conditions

 Exercises

12 Application to Mathematical Finance

 12.1 Market, portfolio and arbitrage

 12.2 Attainability and Completeness

 12.3 Option Pricing

Exercises

Appendix A: Normal Random Variables

Appendix B: Conditional Expectation

Appendix C: Uniform Integrability and Martingale

Convergence

Appendix D: An Approximation Result

Solutions and Additional Hints to Some of the Exercises..

References

List of Frequently Used Notation and Symbols

Index