On first order mean field game systems with a common noise

TL;DR

This paper establishes existence and uniqueness of solutions for first-order mean field game systems with common noise, and uses these solutions to derive approximate Nash equilibria in N-player games.

Contribution

It introduces a new solution concept for MFG systems with common noise and proves the first general well-posedness result in this setting.

Findings

Proved existence and uniqueness of solutions under monotone coupling.

Derived approximate Nash equilibria for N-player games.

Analyzed well-posedness of stochastic backward Hamilton-Jacobi equations.

Abstract

We consider Mean Field Games without idiosyncratic but with Brownian type common noise. We introduce a notion of solutions of the associated backward-forward system of stochastic partial differential equations. We show that the solution exists and is unique for monotone coupling functions. This the first general result for solutions of the Mean Field Games system with common and no idiosynctratic noise. We also use the solution to find approximate optimal strategies (Nash equilibria) for N-player differential games with common but no idiosyncratic noise. An important step in the analysis is the study of the well-posedness of a stochastic backward Hamilton-Jacobi equation.