《随机分析基础及应用》课程介绍

本课程是在《概率论》课程的学习基础上，对《概率论》中的一些经典概念进行现代数学意义上的严格定义，并进一步对随机过程的几类经典模型和过程进行深入介绍和分析，最后介绍随机分析在金融中的应用。

本课程的主要内容包括以下几个方面：对《概率论》的一些基本概念的回顾和重新严格定义。包括概率空间与随机变量、随机变量的数字特征及分布，条件数学期望等；随机过程的定义及基本性质。包括随机过程的基本概念、数字特征、几种分类、两个基本模型的介绍：正态随机过程和泊松过程；泊松过程及其推广。包括泊松过程与指数分布的关系，到达时间间隔与等待时间的分布，复合泊松过程、剩余寿命与年龄等；马尔可夫链，主要介绍离散和连续马尔可夫链的基本概念、转移矩阵、查普曼-柯尔莫戈洛夫方程等；布朗运动。包括布朗运动轨道的性质，布朗运动的推广，布朗运动的联合分布，首中时及最大值的分布等；鞅。主要介绍条件数学期望的严格定义，并引出鞅的定义及基本性质；随机积分及伊藤公式，主要包括关于布朗运动的随机积分的定义及性质，伊藤公式的形式及应用伊藤公式求解随机微分方程，利用伊藤公式推导布莱克-舒尔斯公式。

Stochastic Analysis: Foundations and Applications

Based on the pioneer course of Probability, this course redefines some basic concepts in Probability, thus deeply introduces and analyzes some classical models and processes, finally introduces an application of stochastic analysis in finance.

The content of this course mainly includes the following:

We start with recalling and redefining some concepts in Probability. We introduce the definition and basic properties of stochastic processes, including the basic concepts, mathematical characteristics, the classification, the introduction of two basic models: normal process and Poisson process.

We give the definition of Poisson process and extensions, including the relationship between Poisson process and exponential distribution, the distributions of inter-arrival time and waiting time, compound Poisson process, the residual life and age, etc.

As an important part of stochastic process, we introduce the Markov chains, including basic concepts of discrete and continuous Markov chains, transport matrices, Chapman-Kolmogorov equations,, etc.

The central concept of this course, Brownian motion, will be discussed, including properties of Brownian motion paths, extensions of Brownian motion, joint distributions of Brownian motion, distributions of first hitting time and maximum value, etc.

The concept of martingales, including the rigorous definition of conditional expectations and martingales, the basic properties of martingales, will be introduced briefly.

Last but not the least, the most important part, we focus on the stochastic integrals and Ito’s formula, including the definitions and properties of stochastic integrals with respect to Brownian motion, the form of Ito’s formula and applying Ito’s formula to solve stochastic differential equations. We apply Ito’s formula to derive Black-Scholes formula.

《随机分析基础及应用》课程教学进度计划