Affiliations: [a] Department of Mathematics, Stanford University, Sloan Hall, Stanford, CA 94305, USA. E-mails: [email protected], [email protected] | [b] School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA. E-mail: [email protected]
Abstract: In this paper, we consider a mean field game (MFG) model perturbed by small common noise. Our goal is to give an approximation of the Nash equilibrium strategy of this game using a solution from the original no common noise MFG whose solution can be obtained through a coupled system of partial differential equations. We characterize the first order approximation via linear mean-field forward-backward stochastic differential equations whose solution is a centered Gaussian process with respect to the common noise. The first order approximate strategy can be described as follows: at time t∈[0,T], applying the original MFG optimal strategy for a sub game over [t,T] with the initial being the current state and distribution. We then show that this strategy gives an approximate Nash equilibrium of order ϵ2.
Keywords: Mean field games, common noise, asymptotic, forward-backward stochastic differential equations
