随机积分导论 : 第2版

PREFACE

PREFACE TO THE FIRST EDITION

ABBREVIATIONS AND SYMBOLS

1. PRELIMINARIES

1.1 Notations and Conventions

1.2 Measurability, L' Spaces and Monotone Class Theorems

1.3 Functions of Bounded Variation and Stieltjes Integrals

1.4 Probability Space, Random Variables, Filtration

1.5 Convergence, Conditioning

1.6 Stochastic Processes

1.7 Optional Times

1.8 Two Canonical Processes

1.9 Martingales

1.10 Local Martingales

1.11 Exercises

2. DEFINITION OF THE STOCHASTIC INTEGRAL

2.1 Introduction

2.2 Predictable Sets and Processes

2.3 Stochastic Intervals

2.4 Measure on the Predictable Sets

2.5 Definition of the Stochastic Integral

2.6 Extension to Local Integrators and Integrands

2.7 Substitution Formula

2.8 A Sufficient Condition for Extendability of λz

2.9 Exercises

3. EXTENSION OF THE PREDICTABLE INTEGRANDS

3.1 Introduction

3.2 Relationship between p, O, and Adapted Processes

3.3 Extension of the Integrands

3.4 A Historical Note

3.5 Exercises

4. QUADRATIC VARIATION PROCESS

4.1 Introduction

4.2 Definition and Characterization of Quadratic Variation

4.3 Properties of Quadratic Variation for an L2-martingale

4.4 Direct Definition of/JM

4.5 Decomposition of (M)2

4.6 A Limit Theorem

4.7 Exercises

5. THE ITO FORMULA

5.1 Introduction

5.2 One-dimensional Ito Formula

5.3 Mutual Variation Process

5.4 Multi-dimensional Ito Formula

5.5 Exercises

6. APPLICATIONS OF THE ITO FORMULA

6.1 Characterization of Brownian Motion

6.2 Exponential Processes

6.3 A Family of Martingales Generated by M

6.4 Feynman-Kac Functional and the Schrodinger Equation

6.5 Exercises

7. LOCAL TIME AND TANAKA'S FORMULA

7.1 Introduction

7.2 Local Time

7.3 Tanaka's Formula

7.4 Proof of Lemma 7.2

7.5 Exercises

8. REFLECTED BROWNIAN MOTIONS

8.1 Introduction

8.2 Brownian Motion Reflected at Zero

8.3 Analytical Theory of Z via the It5 Formula

8.4 Approximations in Storage Theory

8.5 Reflected Brownian Motions in a Wedge

8.6 Alternative Derivation of Equation (8.7)

8.7 Exercises

9. GENERALIZED ITO FORMULA, CHANGE OF TIME AND MEASURE

9.1 Introduction

9.2 Generalized Ito Formula

9.3 Change of Time

9.4 Change of Measure

9.5 Exercises

10. STOCHASTIC DIFFERENTIAL EQUATIONS

10.1 Introduction

10.2 Existence and Uniqueness for Lipschitz Coefficients

10.3 Strong Markov Property of the Solution

10.4 Strong and Weak Solutions

10.5 Examples

10.6 Exercises

REFERENCES

INDEX

《随机积分导论(第2版)》由(美)钟开莱著，是一部可读性很强的讲述随机积分和随机微分方程的入门教程。将基本理论和应用巧妙结合，非常适合学习过概率论知识的研究生，学习随机积分。运用现代方法，随机积分的定义是为了可料被积函数和局部鞅，紧接着是连续鞅的变分公式ITO变化。书中包括在布朗运动的描述、鞅的Hermite多项式、Feynman-Kac泛函和Schrodinger方程。这是第二版，讨论了Cameron-Martin-Giranov变换，并且在最后一章引入随机微分方程和一些学生用的练习。读者对象：数学专业、概论论、随机统计等学科的研。数学【作者简介】作者(美)钟开莱