Wellposedness of Mean Field Games with Common Noise Under a Weak Monotonicity Condition

TL;DR

This paper establishes the existence and uniqueness of solutions for Mean Field Games with common noise under a weak monotonicity condition, using the Stochastic Maximum Principle and fixed point methods.

Contribution

It introduces a novel approach to prove well-posedness of Mean Field Games with common noise under relaxed assumptions.

Findings

Existence of solutions over small time intervals

Extension of solutions over arbitrary finite durations

Uniqueness of solutions under weak monotonicity

Abstract

In this paper, we consider Mean Field Games in the presence of common noise relaxing the usual independence assumption of individual random noise. We assume a simple linear model with terminal cost satisfying a convexity and a weak monotonicity property. Our main result is showing existence and uniqueness of a Mean Field Game solution using the Stochastic Maximum Principle. The uniqueness is a result of a monotonicity property similar to that of Lasry and Lions. We use the Banach Fixed Point Theorem to establish an existence over small time duration and show that it can be extended over an arbitrary finite time duration.