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Probability and Computational Finance Seminar
Daniel Lacker
Princeton University
Title: A general characterization of the mean field limit for stochastic differential games

Abstract: Mean field game (MFG) theory generalizes classical models of interacting particle systems by replacing the particles with rational agents, making the theory applicable in economics and other social sciences. Intuitively, (stochastic differential) MFGs are continuum limits of large-population stochastic differential games of a certain symmetric type, and a solution of an MFG is analogous to a Nash equilibrium. Most research so far addresses the problem of solving the continuum model, typically by way of forward-backward systems of PDEs or McKean-Vlasov SDEs, and the solution is then used to construct approximate equilibria for corresponding finite-population games.

This talk discusses some new results in this direction, particularly for MFGs with common noise, but more attention is payed to recent progress on a less well-understood problem: Given for each $n$ a Nash equilibrium for the $n$-player game, in what sense if any do these equilibria converge as $n$ tends to infinity? The answer is somewhat unexpected, and certain forms of randomness can prevail in the limit which are well beyond the scope of the usual notion of MFG solution. A new notion of weak MFG solutions is shown to precisely characterize the set of possible limits of approximate Nash equilibria of $n$-player games, for a large class of models. Unlike for the related (uncontrolled) particle systems, genuinely stochastic limits exist which are not simply mixtures of (non-unique) deterministic equilibria.

Date: Monday, January 12, 2015
Time: 4:30 pm
Location: Wean Hall 8220
Submitted by:  Steve Shreve