I.The Basic Information of the Course
Course Number:202012420029
The English Name of the Course：Stochastic Differential Equation Theory and Its Numerical Solution
The Chinese Name of the Course：随机微分方程理论及其数值解法
In-class Hours and Allocation: total class hours:32, classroom teaching: 24 class hours, classroom discussion: 8 class hours
Credit(s): 2
Semester: 2
Applied Discipline(Professional Degree Category)：Mathematics
Course Object Oriented：Sample: Academic Master, Academic Doctor
Evaluation Mode: Sample: Process Evaluation
Teaching Method:Seminar-style Teaching, Case Teaching
Course Opening Department:College of Mathematical Sciences
II.Prerequisite Course
Probability and Mathematical Statistics,Stochastic Processes,Numerical Calculation
III. The Objectives and Requirements of the Course
This course aims to introduce the basic theory and basic concepts of stochastic differential equations, introduces the basic concepts of stability of stochastic differential equations, introduces the construction method of stochastic differential equation numerical format and the current common numerical format, and introduces the commonly used stochastic differential equations in finance and Some applications in biomathematics and other fields.
By learning this course, the students should master the basic concepts of stochastic differential equation theory and numerical calculation methods, grasp the concept of stochastic differential equation stability, understand the practical application of stochastic differential equations in finance and biological mathematics, and understand the current research hotspots and frontier issues in this field.
IV. The Content of the Course
Chapter 1 Introduction of Probability and Mathematical Statistics (3 class hours)
The concepts of one-dimensional and multi-dimensional random variables and their distribution, the numerical characteristics of random variables, the concepts of mathematical statistics, sampling distribution, and parameter estimation.
Chapter 2 Stochastic Processes（6 class hours）
Random process concept, stationary process, autocorrelation function, spectral density, Brownian motion, Markov process, levy process, Gauss noise, colored noise, narrow-band random process, Monte Carlo method and its application.
Chapter 3 Numerical Calculation（3 class hours）
Euler formula, Taylor formula, Runge-Kutta method
Chapter 4Stochastic Differential Equation（8 class hours）
Ito type stochastic differential equation, Stratonivich type stochastic differential equation, strong convergence, weak convergence concept, stochastic differential equation stability concept
Chapter 5 Numerical Solution of Stochastic Differential Equations（12 class hours）
Stochastic Taylor expansion, Euler-Maruyama method, stationary Euler-Maruyama method, Milstein method, stationary Milstein method, stochastic Runge-Kutta method, predictive correction method, numerical methods for high-dimensional stochastic differential equations, analysis methods for the stability of stochastic differential equations And its applications.
Ideological and Political Case 1: Dice and Probability: Use the probability distribution theory to point out the relationship between dice roll and probability, use the data and results obtained by probability, point out that there is no winner in gambling, guide students away from gambling hazards, and establish a correct outlook on life and values.
Ideological and Political Case 2: Stochastic Differential Equations and Finance: In practice, stochastic differential equations can be used to model and study financial issues such as stock prices and stock options, thereby introducing the independent relationship between the US stock market and the Chinese stock market And pointed out the importance of maintaining domestic financial stability to the development of the country under the complex and ever-changing international situation.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1. Eckhard Platen, Nicola Bruti-Liberati. Numerical Solution of Stochastic Differential Equations with Jumps in Finance. World Publishing Corporation, 2018.
2. Peter E. Kloeden, Eckhard Platen. Numerical Solution of Stochastic Differential Equations. Springer-Verlag Berlin Heidelberg. 1992.
3. Hu Shigeng, Huang Chengming, Wu Fuke. Stochastic Differential Equation. Science Press, 2018.
B. Learning Resources (Time New Rome 12 points)
1. https://next.xuetangx.com/course/NWPU08079000602/1511667
2. http://www.icourses.cn/sCourse/course_3737.html
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