随机过程

学分：3学分

先修课程：数学分析与高等代数 (1 & 2)、离散数学与统计、线性代数与规划、金融数学、时间序列分析

课程简介：

本课程介绍了多种基于随机过程并适用于一般情况的概率模型。所含的教学内容包括随机变量、条件概率与条件期望、以及利用条件概率与条件期望计算无条件概率或无条件期望的相关方法；Markov链、指数分布与Poisson过程、以及连续时间Markov链等。特别地，本课程通过大量的实际例子（或概率模型）来阐明Markov链的构造和转移概率的计算、Markov链的状态分类、Markov的极限概率和可逆性、Poisson过程的特征、事件发生间隔与等待时间的分布、到达时间的条件分布、Poisson过程在排队论的初步应用、生灭过程的定义、构造及其极限概率的计算。

课程目标：

通过对本课程的学习，学生应掌握以下专业知识与技巧：利用条件概率与条件期望计算无条件概率或无条件期望；构造Markov链并计算其转移概率；对状态进行分类并计算极限概率；了解Markov链的可逆性并知道怎么将它应用到某些实际问题中；知道Poisson过程的定义并将其应用到排队论当中；构造生灭过程并计算其极限概率。通过对课程中经典概率模型的学习，让我们了解随机过程在运筹、网络、遗传、经济、保险、金融及可靠性中的基本应用方法与基本技巧。

考核方式：

期末闭卷考试 60%，平时课堂表现课后作业40%

课程教材：

Richard Durrett, Essentials of Stochastic Processes, 2012, ISBN: 1461436141.

Stochastic Process

Credits: 3 Credits

Pre-requisite: 

Calculus and Algebra (1 & 2); Discrete Mathematics and Statistics;Linear Algebra and Programming; Mathematical Finance; Time Series Analysis

Module Description：

This module exposes students to a variety of models available to solve common problems in real situation via using suitable stochastic process. The concept of random variables, conditional probabilities and conditional expectation will be introduced in this module, as well as the methods of calculating non-conditional probabilities and non-conditional expectation by using the conditional ones. Markov chains, the exponential distribution and Poisson process, and continuous-time Markov chains will be discussed. Their properties will be introduced one by one, and illustrated by a lot of examples, which contain some special probability models. Particularly, transition probabilities, classification of states, limiting probabilities, Time reversibility of (continuous-time) Markov chains, characteristic of Poisson process, interval and waiting time distributions, conditional distribution of the arrival times will be studied. This module will also cover the concept and properties of birth and death processes.

Learning Outcome and Objectives：

By the end of the module students should be able to:calculate non-conditional probabilities and non-conditional expectation in some situations; know how to construct a Markov chain and compute its transition probabilities in some real situations; know how to classify the states and calculate the limiting probabilities; appreciate and apply the time reversibility of Markov chains in some standard situations; understand and apply the Poisson process to solve some problems in queuing theory; formulate a birth and death process and calculate its limiting probabilities in real situation. With learning examples/models in this module, the students will be supposed to know how to apply stochastic process to operations research, network, genetics, economy, insurance, finance and reliability.

Assessment：

Assignments and Class Performance 40%

Final Exam 60%

Textbook and Reference：

Richard Durrett, Essentials of Stochastic Processes. 2012, ISBN: 1461436141.