随机积分和微分方程：第2版

Introduction

1 Preliminaries

1 Basic Definitions and Notation

2 Martingales

3 The Poisson Process and Brownian Motion

4 Levv Processes

5 Why the Usual Hypotheses?

6 Local Martingales

7 Stieltjes Integration and Change of Variables

8 Naive Stochastic Integration is Impossible

Bibliographic Notes

Exercises for Chapter 1

2 Semimartingales and Stochastic Integrals

1 Introduction to Semimartingales

2 Stability Properties of Semimartingales

3 Elementary Examples of Semimartingales

4 Stochastic Integrals

5 Properties of Stochastic Integrals

6 The Quadratic Variation of a Semimartingale

7 Itos Formula (Change of Variables)

8 Applications of Itos Formula

Bibliographic Notes

Exercises for Chapter 2

3 Semimartingales and Decomposable Processes

1 Introduction

2 The Classification of Stopping Times

3 The Doob-Meyer Decompositions

4 Quasimartingales

5 Compensators

6 The Fundamental Theorem of Local Martingales

7 Classical Semimartingales

8 Girsanovs Theorem

9 The Bichteler-Dellacherie Theorem

Bibliographic Notes

Exercises for Chapter 3

4 General Stochastic Integration and Local Times

1 Introduction

2 Stochastic Integration for Predictable Integrands

3 Martingale Representation

4 Martingale Duality and the Jacod-Yor Theorem on

Martingale Representation

5 Examples of Martingale Representation

6 Stochastic Integration Depending on a Parameter

7 Local Times

8 Az6mas Martingale

9 Sigma Martingales

Bibliographic Notes

Exercises for Chapter 4

5 Stochastic Differential Equations

1 Introduction

2 The H___p Norms for Semimartingales

3 Existence and Uniqueness of Solutions

4 Stability of Stochastic Differential Equations

5 Fisk-Stratonovich Integrals and Differential Equations

6 The Markov Nature of Solutions

7 Flows of Stochastic Differential Equations: Continuity and

Differentiability

8 Flows as Diffeomorphisms: The Continuous Case

9 General Stochastic Exponentials and Linear Equations

10 Flows as Diffeomorphisms: The General Case

11 Eclectic Useful Results on Stochastic Differential Equations

Bibliographic Notes

Exercises for Chapter 5

6 Expansion of Filtrations

1 Introduction

2 Initial Expansions

3 Progressive Expansions

4 Time Reversal

Bibliographic Notes

Exercises for Chapter 6

References

Subject Index

本书是第2版（全英文版）。第1版本的《随机积分和微分方程》问世13年以来，有关这方面的书不断涌现，特别是在数学金融方面具有很强应用性的书更是发展迅速。然而没有一本书是真正用函数解析法来表达半鞅和随机积分，这使得新的方法并没有得到很好的应用。尽管这本书不再适合称其为一种新的方法。然而新版本的及时出现，在很大程度上完善了原版本。
　　这版较第1版做了一些调整，并且增加了不少新的内容。第3章增加了停时的分类和Bichteler-Dellacherie定理；第4张增加了鞅表示的Jacod-Yor定理、鞅表示的例子以及Sigma鞅；增加了新的一章第6章。并且每章的后面增加了不少练习，这些可以作为学习本教材的很好的补充。

《随机积分和微分方程(第2版)》是真正用函数解析法来表达半鞅和随机积分，这使得新的方法并没有得到很好的应用。尽管这《随机积分和微分方程(第2版)》不再适合称其为一种新的方法。然而新版本的及时出现，在很大程度上完善了原版本。