课程大纲
COURSE SYLLABUS
1.
课程代码/名称
Course Code/Title
MAT7093 随机分析 Stochastic Analysis
2.
课程性质
Compulsory/Elective
选修
Elective
3.
课程学分/学时
Course Credit/Hours
3/48
4.
授课语言
Teaching Language
中英双语
Chinese-English bilingual
5.
授课教师
Instructor(s)
孙景瑞，助理教授
Jingrui Sun, Assistant Professor
6.
是否面向本科生开放
Open to undergraduates
or not
是
Yes
7.
先修要求
Pre-requisites
MA212 概率论与数理统计（或 MA215 概率论），MA208 应用随机过
程，MA201a 或 MA201b 常微分方程，MA411 测度论与积分，MA302
泛函分析
Probability Theory, Applied Stochastic Processes, Ordinary Differential
Equation, Measure Theory and Integration, Functional Analysis.
8.
教学目标
Course Objectives
在概率论和随机过程论基础上，掌握随机分析的基础理论与方法，为进一步研究随机控制、
金融数学、金融工程等学科提供必要的随机分析基础。
The main objectives of this course are, based on the preliminary knowledge of Probability Theory and
Stochastic Processes, to master the basic theory and methods in stochastic analysis and to provide
necessary foundations and background in further learning on stochastic control, financial mathematics
and financial engineering.
9.
教学方法
Teaching Methods
PPT 结合板书授课。
Teach with PPT and blackboards.
10.
教学内容
Course Contents
（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the
difference.）
Section 1 (10 hours)
鞅与布朗运动：停时；连续时间鞅；Doob-Meyer 分解定理；连续平
方可积鞅；布朗运动。
Martingales and Brownian Motion : Stopping Times; Continuous Time
Martingales; The Doob-Meyer Decomposition; Continuous, Square-
Integrable Martingales; Brownian Motion.
Section 2 (16 hours)
伊藤积分：伊藤积分的构造；变量替换公式；鞅表示定理； Girsanov
定理。
Ito Integrals: Construction of the Stochastic Integral; The Change-of-
Variable Formula; Representations of Continuous Martingales in Terms of
Brownian Motion; The Girsanov Theorem.
Section 3 (14 hours)
随机 微 分 方 程 ： 強 解 ； 弱 解 ； 強 解 的 存 在 唯 一 性 ； 线 性 方 程 ；
Feynman-Kac 公式。
Stochastic Differential Equations: Strong Solutions; Weak Solutions;
Existence and Uniqueness of Strong Solutions; Linear Equations;
Feynman-Kac Formula.
Section 4 (8 hours)
倒向随机微分方程：适应解的定义；适应解的存在唯一性；线性方
程。
Backward Stochastic Differential Equations: Definition of an Adapted
Solution; Existence and Uniqueness of Adapted Solutions; Linear
Equations.
11.
课程考核
Course Assessment
10%考勤 + 30%期中测试 + 60%期末测试
10% Attendance + 30% midterm exam + 60% final exam
12.
教材及其它参考资料
Textbook and Supplementary Readings
1. I. Karatzas and S.E. Shreve. Brownian Motion and Stochastic Calculus, 2
nd
ed., Springer-Verlag, New
York, 1998.
2. J. Yong and X. Y. Zhou. Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-
Verlag, New York, 1999.
3. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3
rd
ed., Springer-Verlag, New
York, 1999.
4. N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, North-Holland
Publishing Company, New York, 1981.
