课程详述

COURSE SPECIFICATION

以下课程信息可能根据实际授课需要或在课程检讨之后产生变动。如对课程有任何疑问，请联

系授课教师。

The course information as follows may be subject to change, either during the session because of unforeseen

circumstances, or following review of the course at the end of the session. Queries about the course should be

directed to the course instructor.

课程名称 Course Title

现代概率论 Advanced Probability

授课院系

Originating Department

数学系 Department of Mathematics

课程编号

Course Code

MAT8011

课程学分 Credit Value

课程类别

Course Type

专业选修课 Major Elective Courses

授课学期

Semester

秋季 Fall

授课语言

Teaching Language

英文 English

授课教师、所属学系、联系方

式（如属团队授课，请列明其

他授课教师）

Instructor(s), Affiliation&

Contact

（For team teaching, please list

all instructors）

熊捷，讲座教授，数学系

慧园 3 栋 527

Jie Xiong, Chair Professor, Department of Mathematics

Block 3 Room.527, Wisdom Valley

实验员/助教、所属学系、联系

方式

Tutor/TA(s), Contact

待公布 To be announced

10.

选课人数限额(可不填)

Maximum Enrolment

（Optional）

授课方式

Delivery Method

讲授

Lectures

习题/辅导/讨论

Tutorials

实验/实习

Lab/Practical

其它(请具体注明)

Other（Please specify）

总学时

Total

11.

学时数

Credit Hours

12.

先修课程、其它学习要求

Pre-requisites or Other

Academic Requirements

MA215 Probability Theory， MA301 Theory of Functions of a Real

Variable；MA215 概率论，MA301 实变函数

13.

后续课程、其它学习规划

Courses for which this course

is a pre-requisite

14.

其它要求修读本课程的学系

Cross-listing Dept.

教学大纲及教学日历 SYLLABUS

15.

教学目标 Course Objectives

Modern probability theory is the basis of Stochastic Processes Theory, Stochastic Analysis and Financial Mathematics.

This course is designed to introduce the basic concepts and methods in modern probability theory, which will lay a solid

foundation for further study. We will discuss several important kinds of convergences and their theory, conditional

expectation and martingale method. After the study of this course, students should not only deeply understand and

master concepts and theorems in modern probability theory, but also have the ability to use them in many different

problems.

现代概率论是随机过程论，随即分析和金融数学的基础。本课程旨在介绍现代概率论的基本概念与方法，为

今后的学习打下坚实的基础。本课程重点介绍了几种重要的收敛概念和理论，条件期望与鞅方法。通过本课程

的学习，学生不仅可以深刻理解与掌握各种概念与定理，还应具有把他们运用到不同问题中的能力。

16.

预达学习成果 Learning Outcomes

After learning this course, students should be able

1.to deeply understand and master the basic concepts and conclusions of modern probability theory, not only

to remember these basic concepts and the basic probability laws including conditions and conclusions, but

also deeply to understand the basic principles and ideas of modern probability;

2．to fully master the four basic convergence theorems (Monotone Convergence Theorem, Fatou Lemma,

Dominated Convergence Theorem, and Bounded Convergence Theorems) and be able to apply them in

many important topics and different problems;

3．to clearly understand the probability meaning, difference, and relationships of several kind of

convergence concepts (almost everywhere convergence; convergence in measure/probability; Convergence

in Lp Norm; Weak Convergence) and be able to apply them in different problems;

4．to fully master the very important concepts of conditional expectations and conditional probabilities and

to improve the ability of solving practical problems by applying the basic probability methods of

“conditioning”.

5. to clearly understand and master the basic concepts regarding martingales including the existence,

uniqueness, properties and applications of martingales, super- and sub-martingales and be able to apply the

important martingale method in the study of modern theory of stochastic processes, stochastic analysis and

financial mathematics.

完成本课程后，学生应能够：

1. 深入了解和掌握现代概率论的基本概念和结论，不仅需要记住基本的定理中的条件和结论，还需

要深入理解现代概率论的基本概念和想法。

2. 完全掌握四种基本的收敛定理（单调收敛定理，Fatou 引理，控制收敛定理，有界收敛定理），

并且可以在许多重要的问题上运用他们。

3. 清晰地理解几种不同收敛概念（几乎处处收敛，依概率收敛，Lp 收敛，弱收敛）的概率意义、

区别以及关系，并且可以在许多不同的问题上运用他们

4. 完全掌握条件期望、条件概率的概念，提高在实际问题中使用“条件期望”的能力

5. 清晰地理解与掌握鞅的基本概念，包括存在性、唯一性、性质与应用、超鞅与半鞅，能够在随机过程、随

即分析与金融数学中运用鞅方法。

17.

课程内容及教学日历（如授课语言以英文为主，则课程内容介绍可以用英文；如团队教学或模块教学，教学日历须注明

主讲人）

Course Contents (in Parts/Chapters/Sections/Weeks. Please notify name of instructor for course section(s), if

this is a team teaching or module course.)

Section 1: Independence, Expectation and Convergence (15h): Borel-Cantelli Lemma, Convergence

Theorems, Jensen’ s Inequality for Convex Functions, The Schwarz Inequality, Orthogonal Projection,

Holder from Jensen; Convergence in Probability, Weak Convergence, Convergence in Distributions,

Characteristic Functions.

Section 2: Conditional probability and conditional expectation (3h): Definition of conditional

expectation and conditional probability, Existence & Uniqueness, Properties of conditional expectation and

conditional probability, Tower Property, Some Important Inequalities.

Section 3: Martingales (12h): Definition of martingales, Properties of martingales, Super-martingales, Sub-

martingales, Examples, Convergence of martingales, Stopping times, Optional Sampling Theorem.

Section 4: Super-martingales and Sub-martingales (8h): Definitions and Properties of Super-martingales

and Sub-martingales, Examples, Doob Decomposition. Convergence of martingales, Stopping times,

Optional Sampling Theorem.

Section 5: Martingale Inequality and Martingale Convergence Theorems (8h): Uniform Integrability;

UI Martingales; Martingale Convergence Theorems; Backwards Martingale Convergence Theorems; Strong

Law of Large Numbers; Martingale Central Limit Theorem.

Section 6：Brownian motion: Basic properties. (2h)

18.

教材及其它参考资料 Textbook and Supplementary Readings

Textbook: Jean Jacod & Philip Protter，《Probability Essentials》，Springer-Verlag, Berlin Heidelberg.

Supplementary Readings:

1. David Williams, ，《Probability with Martingales》，Cambridge University Press, Cambridge, 1991.

2. 严士健，王隽骧，刘秀芳，《概率论基础》，科学出版社

课程评估 ASSESSMENT

19.

评估形式

评估时间

占考试总成绩百分比

违纪处罚

备注

Type of

Assessment

Time

% of final

score

Penalty

Notes

出勤 Attendance

课堂表现

Class

Performance

小测验

Quiz

课程项目 Projects

平时作业

Assignments

20%

期中考试

Mid-Term Test

30%

期末考试

Final Exam

50%

期末报告

Final

Presentation

其它（可根据需要

改写以上评估方

式）

Others (The

above may be

modified as

necessary)

20.

记分方式 GRADING SYSTEM

☑ A. 十三级等级制 Letter Grading

 B. 二级记分制（通过/不通过） Pass/Fail Grading

课程审批 REVIEW AND APPROVAL

21.

本课程设置已经过以下责任人/委员会审议通过

This Course has been approved by the following person or committee of authority

数学系课程规划与审核委员会

Curriculum Planning and Review Committee, Department of Mathematics