Instructor: Jianfeng Zhang, KAP 248E, (213)7409805
Email: jianfenz@usc.edu Homepage: https://dornsife.usc.edu/profile/jianfeng-zhang/
Time and location: W 12:55-1:55, F 12:30-2 in KAP 245
Office hours: W 10:30-12:00, F 2-5 in KAP 248E or via Zoom
Textbooks: Some reference books will be provided during lectures
Prerequisites: Math 509 (or approval from the instructor). In particular, students should be familiar with the following topics: Filtrations, Brownian motion, stochastic integrals, Ito’s formula, Girsanov theorem, martingale representation, and basic theory on martingales, Markov processes, and parabolic PDEs.
Course Description:
Stochastic controls/games has been a major branch of stochastic analysis, and is also one of the central topics in economics (typically in discrete models). The mean field theory considers the case with infinitely many players, and has received very strong attention in the past decade.
In the first part of the course, we will study stochastic controls with one player and introduce the HJB equation approach. We then introduce various types of games with finitely many players, including zero sum game, nonzero sum game, principal agent problems. One major tool in this case is the Backward SDE.
In the second part of the course, we will study mean field controls and mean field games. In particular, we will introduce the PDE approach, namely the HJB equation for the control problem and the master equation for the game problem. The main feature in this case is that the probability measure (law of the state process) becomes a variable of the value function, and thus we will consider infinite dimensional PDE on the Wasserstein space of probability measures.
The rigorous treatment of the materials is highly technical. However, in order to have time to introduce the state of the art in the mean field theory, for most parts we will focus on the ideas and provide only heuristic arguments. Students are encouraged to dig deeper and carry out some arguments rigorously outside of class.
A tentative schedule is as follows (most probably we will have to skip some parts):
Chapter 0: Preliminaries, and basic theory on SDEs and Backward SDEs
Chapter 1: Stochastic Controls
Section 1.1: Stochastic optimization with open loop controls: the dynamic programming principle
Section 1.2: HJB equations and viscosity solutions
Section 1.3: Stochastic optimization with open loop controls: the stochastic maximum principle
Section 1.4: Stochastic optimization with closed loop controls: the BSDE approach
Chapter 2: Stochastic Differential Games
Section 2.1: Nash equilibria for nonzero-sum games
Section 2.2: Bellman Isaacs equations for zero-sum games
Section 2.3: Set values for non-zero sum games
Section 2.4: Stackelberg games and principal-agent problems
Chapter 3: Stochastic analysis on the Wasserstein space of probability measures
Section 3.1: Propagation of chaos and introduction of mean field problems
Section 3.2: The Wasserstein space of probability measures
Section 3.3: Derivatives of functions on Wassersterin space and Ito’s formula
Chapter 4: The mean field control problems
Section 4.1: The mean field control problems
Section 4.2: Viscosity solution of HJB equations on the Wasserstein space
Section 4.3: Convergence of the $N$-player optimization problems
Chapter 5: The mean field game problems
Section 5.1: The mean field game problems
Section 5.2: Monotonciity condition and uniqueness of mean field equilibrium
Section 5.3: The master equation and the convergence of the $N$-player games
Section 5.4: Convergence of the $N$-player games without monotinicity conditions
Section 5.5: Set values for mean field games
Grading and Examination Policies
40% of the grade will be based on homework assignments, 60% on class participation, including the final presentation.
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.
All students (individually or in groups) will be required to give a presentation on related topics. Details will be discussed in class.
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 schedule one-to-one Zoom meeting with me.
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.
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, as described in Scampus.