A Class of Degenerate Mean Field Games, Associated FBSDEs and Master Equations

TL;DR

This paper investigates a class of degenerate mean field games with complex diffusion coefficients, establishing well-posedness of associated FBSDEs and the regularity of the value function, which solves the master equation.

Contribution

It introduces a probabilistic approach to analyze degenerate MFGs with unbounded coefficients, proving well-posedness and regularity of the value functional and master equation solutions.

Findings

Proved well-posedness of FBSDEs for degenerate MFGs.

Established regularity of the value functional under smoothness conditions.

Showed the value functional uniquely solves the master equation.

Abstract

In this paper, we study a class of degenerate mean field games (MFGs) with state-distribution dependent and unbounded functional diffusion coefficients. With a probabilistic method, we study the well-posedness of the forward-backward stochastic differential equations (FBSDEs) associated with the MFG and arising from the maximum principle, and estimate the corresponding Jacobian and Hessian flows. We further establish the classical regularity of the value functional $V$; in particular, we show that when the cost function is $C^3$ in the spatial and control variables and $C^2$ in the distribution argument, then the value functional is $C^1$ in time and $C^2$ in the spatial and distribution variables. As a consequence, the value functional $V$ is the unique classical solution of the degenerate MFG master equation.