随机过程基础

1. Review of Probability

1.1 Events and Probability

1.2 Random Variables

1.3 Conditional Probability and Independence

1.4 Solutions

2. Conditional Expectation

2.1 Conditioning on an Event

2.2 Conditioning on a Discrete Random Variable

2.3 Conditioning on an Arbitrary Random Variable

2.4 Conditioning on a a-Field

2.5 General Properties

2.5 Various Exercises on Conditional Expectation

2.7 Solutions

3 Martingales in Discrete Time

3.1 SequencesofRandomVariables

3.2 Filtrations

3.3 Martingales

3.4 Games or Uhance

3.5 StoppingTimes

3.5 Optional Stopping Theorem

3.7 Solutions

4 Martingale Inequalities and Convergence

4.1 Doobs Martingale Inequalities

4.2 Doobs Martingale Convergence Theorem

4.3 Uniform Integrability and L1 Convergence of Martingales

4.4 Solutions

5. Markov Chains

5.1 First Examples and Definitions

5.2 Classification of States

5.3 Long-Time Behaviour of Markov Chains： General Case

5.4 Long-Time Behaviour of Markov Chains with Finite State

Space

5.5 Solutions

6. Stochastic Processes in Continuous Time

6.1 General Notions

6.2 Poisson Process

6.2.1 Exponential Distribution and Lack of Memory

6.2.2 Construction of the Poisson Process

6.2.3 Poisson Process Starts from Scratch at Time t

6.2.4 Various Exercises on the Poisson Process

6.3 Brownian Motion

6.3.1 Definition and Basic Properties

6.3.2 Increments of Brownian Motion

6.3.3 Sample Paths

6.3.4 Doobs Maximal L2 Inequality for Brownian Motion

6.3.5 Various Exercises on Brownian Motion

6.4 Solutions

7. Ito Stochastic Calculus

7.1 Ito Stochastic Integral： Definition

7.2 Examples

7.3 Properties of the Stochastic Integral

7.4 Stochastic Differential and It5 Formula

7.5 Stochastic Differential Equations

7.6 Solutions

Index

随机过程在数学、科学和工程中有着越来越广泛的应用。《随机过程基础》包括随机过程一些基本而又重要的内容：条件期望，Markov链，Poisson过程和Brown运动；同时也包括Ito积分和随机微分方程等应用范围越来越广的内容。《随机过程基础》的习题是其基本内容的延伸，而且有十分完整的解答，非常适合高年级本科生和研究生自学使用或用作教学参考书。

随机过程理论在数学、科学和工程中有着越来越广泛的应用。《随机过程基础》包括随机过程一些基本而又重要的内容：条件期望，Markov链，Poisson过程和Brown运动：同时也包括Ito积分和随机微分方程等应用范围越来越广的内容。
　　这是一本难得的好书，它1999年出版，到2000年已经是第3次印刷，到2003年则重印到第6次。