Time and location: MWF 1-1:50pm, KAP 245
Office hours: WF 2-3:30pm in KAP 248E
Instructor: Jianfeng Zhang
Office: KAP 248E,
Email: jianfenz@usc.edu
Homepage: http://math.usc.edu/~jianfenz
Presentation schedule and papers
Course Description: In this course we will introduce the very recent development of the new theory of Second Order Backward Stochastic Differential Equations (2BSDE, for short). The 2BSDE, as the name suggests, is an extension of the Backward SDE. In Markovian case, it provides a probabilistic representation of viscosity solution to fully nonlinear parabolic PDEs, e.g. the Hamilton-Jacobi-Bellman equations. The theory is closely related to and has applications in various areas, including: numerical methods for fully nonlinear PDEs, super-hedging contingent claims under volatility uncertainty, super-hedging contingent claims under liquidity risk, dynamic risk measures under volatility uncertainty, stochastic optimization with volatility control, stochastic target problems with Gamma constraints, and two person repeated games.
The theory is also closely related to the so called G-expectation, a nonlinear expectation proposed by Peng in recent years. Our main tool will be the quasi-sure analysis, which involves a family of mutually singular probability measures. We shall develop most of the theory from scratch. The project is still ongoing, and a lot of issues remain to be understood. Students are encouraged to be creative. There is no textbook, but related research papers will be recommended when the course moves on.
A tentative list of contents are:
Chapter 0: Some prerequisites
Chapter 1: Introduction
Chapter 2: Mutually singular probability measures
Chapter 3: Martingale representation for $G$-expectation
Chapter 4: Review of Backward SDEs
Chapter 5: Wellposedness of Second Order Backward SDEs
Homework: The techniques used in the course are as well important. The homework assignments will provide students good opportunities for practice, and thus will be a very important part of the course. Students are (strongly) encouraged to discuss the problems together, but each one should write down the solutions independently.
Presentation: During the last week of classes, students (maybe in groups) will be asked to give a presentation about a topic related to the subject matter of the course. The students may choose the paper, but should get approved by the instructor before March 31.
Final Exam: A take-home final exam will be handed out two weeks before the class ends, and due in the end of the semester. Students are not permitted to discuss the problems with others.
Prerequisite: Math 507 (or equivalent courses on measure theory based probability), Math 509 is preferred but not required. The prerequisite can be waived by the instructor.
Grading Policies: 10% on class participation, 40% on homework, 40% on final take-home exam, and 10% on final presentation.
The final exam will be graded by the instructor. The homeworks will be collected about every two weeks, but will only be checked for completeness. Students are welcome to discuss homework problems with the instructor, and solutions to some selective problems can be provided (upon students’ request).
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.