Title
Mean Field Games, their FBSDEs and Master Equations
Abstract
Modeling collective behaviors of individuals in account of their mutual interactions arisen in various physical or sociological dynamical systems have been one of the major problems in the history of mankind. To resolve this matter, a completely different macroscopic approach inspired from statistical physics had been gradually developed in the last decade, which eventually leads to the primitive notion of mean field game theory. In this talk, we shall introduce a theory of global-in-time well-posedness for a general class of mean field game problems, which include as an example settings with quasi-convex payoff functions as long as the mean field sensitivity is not too large. Through the stochastic maximum principle, we adopt the forward backward stochastic differential equation (FBSDE) approach to investigate the unique existence of the corresponding equilibrium strategies. This FBSDE is first solved locally in time, then by controlling the sensitivity with respect to the initial condition of the solution to the backward equation via studying its Jacobian flow, the global-in-time solution is warranted. Further analysis of the Jacobian flow of the solution to the FBSDE will be discussed so as to establish the regularities of the value function, including its linear functional differentiability, that also leads to the classical well-posedness of the complicated one-directional master equations on R^n. In contrast to the recent approach with an emphasis on the well-posedness of the master equations, we solve the whole problem by tackling the mean field game equilibrium problems directly; indeed, we extend the well-posedness result in [1], which founds their theory on a torus in a H¨older space, to the whole unbounded domain R^n via the Sobolev space language.
[1] P. Caradaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions. The master equation and the convergence problem in mean field games:(ams-201).
Princeton University Press, 2019
Bio
Phillip Yam received his BSc in Actuarial Science with first class honours and MPhil from the University of Hong Kong. He obtained an MASt (Master of Advanced Study) degree, Part III of the Mathematical Tripos, with Distinction in Mathematics from Universityof Cambridge and a DPhil in Mathematics from University of Oxford. He is currently the Co- Director of the Interdisciplinary Major Program in QuantitativeFinance and Risk Management Science, and a full Professor at the Department of Statistics of CUHK. He is also Assistant Dean (Education) of CUHK Faculty of Science, and Fellow of the Centre for Promoting Science Education in the Faculty. He was a VisitingProfessor in the Department of Statistics of Columbia University in the City of New York. He has about a hundred journal articles in actuarial science and financial mathematics, applied mathematics, engineering, and statistics, and has also been serving ineditorial boards of several journals in these fields. Together with Alain Bensoussan and Jens Frehse, he wrote up the first monograph on mean field games and mean field type control theory. Recently, his research project with the title “CIBer: A Robust andEffective FinTech and InsurTech Tool” was awarded a Silver Medal in the 48th International Exhibition of Inventions Geneva.