1
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.
1.
课程名称 Course Title
随机分析及其在金融中的应用
Stochastic calculus and their applications in finance
2.
授课院系
Originating Department
数学系 Department of Mathematics
3.
课程编号
Course Code
MAT7030
4.
课程学分 Credit Value
3
5.
课程类别
Course Type
专业选修课 Major Elective Courses
6.
授课学期
Semester
春季 Spring
7.
授课语言
Teaching Language
英文 English
8.
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
9.
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）
2
授课方式
Delivery Method
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
48
12.
先修课程、其它学习要求
Pre-requisites or Other
Academic Requirements
MA301 实 变 函 数 ， MA215 概 率 论 ； MA301Theory of functions of a real
variable，MA215 Probability Theory
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
With the fast development of mathematical finance in recent years, stochastic calculus has been
widely used in finance. This course is designed as the first courses in financial calculus for students
having a good background in mathematics. After learning this course, students should understand
the key concepts in stochastic analysis such as martingales and change of measure and some deep
properties of Brownian motion process. The students should also be able to apply the basic methods
and tools learning from this course such as the Ito’s formula and the Black-Scholes pricing formula
in practical problems in finance.
随着金融数学的迅速发展，随机分析在金融中有了越来越多的应用。本课程是针对具有良好
数学背景的学生设计的金融数学里面的第一门课程。通过本课程的学习，学生应该了解在随
机分析中的一些关键概念，例如鞅和测度变换和布朗运动过程中的一些深层次的性质。学生
还应能运用本课程学习中的基本方法和工具，如伊藤公式和布莱克-斯科尔斯定价公式，来
解决金融实际问题。
16.
预达学习成果 Learning Outcomes
After completing this course, students should master the basic concepts and methods in stochastic analysis. After
learning this course, the students should be familiar with a range of methods and techniques for solving real life
problems in finance. In particular, after learning this course, the students should be able
1．to master the basic knowledge, deeply to understand and master the nature of the definitions, theorems,
principles and formulae. After the study, the students should be able not only to remember the above concepts and the
basic laws, but also deeply to understand some difficult theoretic conclusion with fully applications and examples.
Students can know how to use these theoretic conclusions first and then know how to make the strict proofs
2．to understand the exact probabilistic meaning of these concepts and could fully master the related conclusions and
then could apply them in many different problems.
3．to train the ability of thinking and to enhance the ability to Pay attention to the newly and recently obtained
conclusions ；
4．To improve the ability of solving practical problems. After learning this course, students should also be able to
apply the basic methods and tools learning from this course such as the Ito’s formula and the Black-Scholes pricing
3
formula in practical problems in finance.
完成本课程后,学生应掌握随机分析的基本概念和方法，熟悉各种在金融中的相关应用，并能解决现
实生活提出的问题。特别是,在学习本课程后,学生应该能够
1.掌握基本知识，深刻理解和掌握定义、定理、原理和公式的性质。经过研究，学生不仅要记住上
述概念和基本规律，而且通过课堂的充分应用和举例说明，学生要充分理解一些疑难理论结论。学
生首先知道如何使用这些理论结论，然后知道如何做严格的证明。
2. 能理解这些概念的确切概率意义，能充分掌握相关结论，并将其应用到许多不同的问题中。
3.培养思维能力，提高对最新的、最前沿的知识结论学习的能力。
4.提高解决实际问题的能力。学习本课程后, 学生应能运用本课程学习中的基本方法和工具，如伊
藤公式和布莱克-斯科尔斯定价公式，来解决金融实际问题。
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.)
Section1
Brownian motion (16 hours)
Definition of the process；Levy’s construction of Brownian motion；The reflection principle and scaling；
Martingales in continuous time
Section 2
Stochastic calculus(16 hours)
Stochastic integration； Ito’s formula； Integration by parts; Stochastic Fubini Theorem ； The Girsanov
Theorem；The Brownian Martingale Representation Theorem；The Feynman–Kac representation
Section 3
The Black–Scholes model(16 hours)
The basic Black–Scholes model; Black–Scholes price and hedge for European options ;Foreign exchange
;Dividends; Bonds ;Market price of risk.
18.
教材及其它参考资料 Textbook and Supplementary Readings
Textbook:
Alison Etheridge, A course in financial calculus. Cambridge University Press 2002
Supplementary Readings:
1. Ioannis Karatzas, Steven Shreve, Brownian Motion and Stochastic Calculus
4
2 Rene Schilling, Brownian Motion: An Introduction to Stochastic Processes.
课程评估 ASSESSMENT
19.
评估形式
Type of
Assessment
评估时间
Time
占考试总成绩百分比
% of final
score
违纪处罚
Penalty
备注
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