Time: W F 9:30am-10:45am
Location: KAP 245
Office hours: W F 10:45am-12:15pm in KAP 248 E
Instructor: Jianfeng Zhang
Office: KAP 248E, (213)740-9805
Email: jianfenz@usc.edu
Homepage: http://math.usc.edu/~jianfenz
Course Description: Pathwise stochastic analysis is a challenging subject in probability theory and is important in applications. Two typical examples are pathwise stochastic integration and pathwise conditional expectation. A powerful tool for the former one is the rough path theory, which allows one to study SDEs and SPDEs in a pathwise manner and the solution could be continuous in terms of the underlying paths. The latter one can be viewed as a viscosity solution to certain path dependent PDEs with terminal conditions, which provides a convenient framework for many stochastic control problems and stochastic differential games in non-Markovian (or path dependent) setting.
The functional Ito calculus of Dupire has proven to be very convenient for studying backward problem in pathwise manner. In this course we shall extend the functional Ito calculus to pathwise Ito calculus. With the help of this, we will present the basic idea of the rough path theory in a similar manner as standard stochastic calculus. To certain extent, this unifies the language of pathwise analysis for forward equations and backward equations. We will study the pathwise solutions of (forward) SDEs and SPDEs, and if time allows, we will also study backward path dependent PDEs (with examples including backward SDEs and SPDEs). The lectures will be based on related research papers. No textbook is required.
The course will be at PhD level. A good understanding on stochastic analysis and some graduate level of Functional Analysis and Partial Differential Equations will be helpful.
Prerequisites: Math 509 (can be waived by the instructor in special cases)
Suggested reading:
[1] Keller, C. and Zhang, J. “Pathwise Ito Calculus for Rough Paths and Rough PDEs with Path Dependent Coefficients”, preprint, arXiv:1412.7464.
[2] Ekren, I., Touzi, N., and Zhang, J. “Viscosity Solutions of Fully Nonlinear Parabolic Path Dependent PDEs: Part I”, preprint, arXiv:1210.0006.
[3] Friz, P. and Hairer, M. (2014) “A Course on Rough Paths: With an Introduction to Regularity Structures”, Springer Universitext, September 2014. ISBN 978-3-319-08331-5. (Also available at: http://www.hairer.org/notes/RoughPaths.pdf)
[4] Zhang, J. “Backward Stochastic Differential Equations”, Springer, in preparation.
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/.