**Math 509, Stochastic Differential Equations, Fall 2017**

**Time and location: **WF 12:00pm - 1:15pm, KAP 159.

**Instructor: **Jianfeng Zhang, KAP 248E, (213)7409805

Email: *jianfenz@usc.edu *Homepage: *http://almaak.usc.edu/~jianfenz*

**Grader: **Jian Wang, wang945@usc.edu

**Office hours: **WF 10:00am-11:30pm, or by appointment, in KAP 248 E

**Textbook** (recommended): *Backward Stochastic Differential Equations --- from linear to fully nonlinear theory*, by Jianfeng Zhang.

**Additional reading** (optional):

*Brownian Motion and Stochastic Calculus*, by Ioannis Karatzas and Steven Shreve

* Continuous Martingales and Brownian Motion*, by Daniel Revuz and Marc Yor

**Course Contents: **

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

4.1 Introduction

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 (75 minutes) midterm exam will be given in regular class time on **Oct. 13, 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.

**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/.