A general characterization of the mean field limit for stochastic differential games
arXiv:1408.2708
Abstract
The mean field limit of large-population symmetric stochastic differential games is derived in a general setting, with and without common noise, on a finite time horizon. Minimal assumptions are imposed on equilibrium strategies, which may be asymmetric and based on full information. It is shown that approximate Nash equilibria in the -player games admit certain weak limits as tends to infinity, and every limit is a weak solution of the mean field game (MFG). Conversely, every weak MFG solution can be obtained as the limit of a sequence of approximate Nash equilibria in the -player games. Thus, the MFG precisely characterizes the possible limiting equilibrium behavior of the -player games. Even in the setting without common noise, the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of MFG solution.