Mean field games with common noise and degenerate idiosyncratic noise

TL;DR

This paper investigates mean field games influenced by both common and potentially degenerate idiosyncratic noise, introducing a new weak solution concept to handle the lack of smoothness in the system.

Contribution

It develops a novel weak solution framework for mean field games with degenerate noise, expanding the analytical tools available for such stochastic systems.

Findings

Established a new notion of weak solutions for backward stochastic HJB equations

Constructed probabilistically weak solutions for the mean field game system

Addressed the challenge of degeneracy in idiosyncratic noise coefficients

Abstract

We study the forward-backward system of stochastic partial differential equations describing a mean field game for a large population of small players subject to both idiosyncratic and common noise. The unique feature of the problem is that the idiosyncratic noise coefficient may be degenerate, so that the system does not admit smooth solutions in general. We develop a new notion of weak solutions for backward stochastic Hamilton-Jacobi-Bellman equations, and use this to build probabilistically weak solutions of the mean field game system.