Instructor: Jianfeng Zhang, KAP 248E, (213)740-9805 jianfenz@usc.edu http://almaak.usc.edu/~jianfenz
Time and location: WF 12:30 – 1:45pm, KAP 427
Office hours: W F 11:00am-12:30pm, or by appointment, in KAP 248 E
Course Description:
Pathwise stochastic analysis is a challenging subject in probability theory and is important in applications. One typical example is the pathwise stochastic integration. The recent development on nonlinear expectation can also be viewed as a pathwise conditional expectation in certain sense. The main idea is to view all paths as the state variable and consider all possible distributions on the paths. The analysis of these problems provides a convenient and powerful mathematical tool for path dependent (or Non-Markovian) problems.
In this course we will introduce the recent development on the subject related to the instructor’s own research. Some topics include: Functional Ito’s formula, G-expectation, Second Order Backward Stochastic Differential Equations, and Path Dependent Partial Differential Equations. If times allows, we may also introduce some other approaches such as Rough Path Theory and Malliavin Calculus. The lectures will be based on related research papers. No textbook is required.
The course will be in PhD level. Although the materials will be self-contained (based on the basic theory of stochastic analysis), a good understanding on stochastic analysis and some graduate level of Functional Analysis and Partial Differential Equations will be helpful.
Prerequisites: Math 507 or Math 509 (can be waived by the instructor in special cases)
Tentative list of contents:
Chapter 1: Preliminaries
Chapter 2: Canonical space and weak formulation
Chapter 3: $G$-expectation and $G$-matingales
Chapter 4: Backward SDEs and Second Order Backward SDEs
Chapter 5: Stochastic calculus on canonical space
Chapter 6: Viscosity solution of path dependent PDEs: definitions and basic properties
Chapter 7: Viscosity solution of path dependent PDEs: comparison principle
Chapter 8: Additional topics
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 an important part of the course. Students are (strongly) encouraged to discuss the problems together, but each one should write his/her 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.
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.
Grading Policies: 10% on class participation, 40% on homework, 40% on final take-home exam, and 10% on final presentation.
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.
Statement for Students with Disabilities: Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester. A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me (or to TA) as early in the semester as possible. DSP is located in STU 301 and is open 8:30 a.m.-5:00 p.m., Monday through Friday. The phone number for DSP is (213) 740-0776.
Statement on Academic Integrity: USC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect one’s own academic work from misuse by others as well as to avoid using another’s work as one’s own. All students are expected to understand and abide by these principles. Scampus, the Student Guidebook, contains the Student Conduct Code in Section 11.00, while the recommended sanctions are located in Appendix A: http://www.usc.edu/dept/publications/SCAMPUS/gov/. Students will be referred to the Office of Student Judicial Affairs and Community Standards for further review, should there be any suspicion of academic dishonesty. The Review process can be found at: http://www.usc.edu/student-affairs/SJACS/.