Math 509, Stochastic Differential Equations, Fall 2011
Time and location: MWF 12:00pm - 12:50pm, KAP 159.
Instructor: Jianfeng Zhang, KAP 248E, (213)7409805
Email: firstname.lastname@example.org Homepage: http://almaak.usc.edu/~jianfenz
Office hours: W. 11:00-12:00, 12:50-1:50, F: 12:50-1:50 in KAP 248 E
Textbook: No textbook is required. Lecture notes will be provided by the instructor.
The course will cover the basic materials of Ito's stochastic calculus, and the standard theory of Stochastic Differential Equations. Moreover, I will introduce the basic materials of Backward Stochastic Differential Equations, which is closely related to standard SDEs and has been proved more and more important in applications. A tentative list of contents is as follows:
Chapter 1. Preliminaries
Chapter 2. Basics of stochastic calculus
2.1 Brownian motion
2.2 Stochastic integration
2.3 Ito's formula
2.4 Martingale representation theorem
2.5 Girsanov theorem
Chapter 3. Stochastic differential equations
3.1 Linear SDEs
3.2 A priori estimates for SDEs
3.3 Wellposedness of SDEs
3.4 Basic properties of SDEs
3.5 Weak solutions of SDEs
Chapter 4. Backward SDEs
4.2 Linear BSDEs
4.3 A priori estimates of BSDEs
4.4 Wellposedness of BSDEs
4.5 Basic properties of BSDEs
Chapter 5. Markovian BSDEs, PDEs, and Probabilistic numerical methods
5.1 Nonlinear Feynman-Kac formula
5.2 Regularity of solutions
5.3 Time discretization of SDEs and BSDEs
5.4 Monte Carlo methods for BSDEs and PDEs
Grading and Examination Policies
40% of the grade will be based on homework assignments, 20% on the midterm exam, and 40% on the final exam.
Homework will be assigned in class approximately every two weeks. You are encouraged to discuss homework problems with classmates. However, you are not allowed to copy other people's work. Solutions to homework problems can be provided upon students' request.
The (50 minutes) midterm exam will be given in regular class time on Oct. 7, Friday. It will be open book, open notes, but noncooperative.
The final exam will be take-home, which will be handed out two weeks before the semester ends. You are not permitted to discuss the problems with others.
Feedback and Questions
It is very useful to get feedback and questions, both inside and outside class. You are very welcome to visit me during my office hours. You can also make appointments to see me at other time.