On Lipschitz solutions of mean field games master equations

TL;DR

This paper establishes existence and uniqueness results for Lipschitz continuous solutions of mean field game master equations across different settings, without structural assumptions, and identifies maximal existence times.

Contribution

It develops a general theory for Lipschitz solutions of MFG master equations, covering finite state, Hilbert space, and probability measure cases with common noise, without requiring monotonicity.

Findings

Proves uniqueness of Lipschitz solutions given Lipschitz initial conditions.

Identifies maximal time of existence for solutions without structural assumptions.

Analyzes solutions in finite state, Hilbert space, and measure spaces with common noise.

Abstract

We develop a theory of existence and uniqueness of solutions of MFG master equations when the initial condition is Lipschitz continuous. Namely, we show that as long as the solution of the master equation is Lipschitz continuous in space, it is uniquely defined. Because we do not impose any structural assumptions, such as monotonicity for instance, there is a maximal time of existence for the notion of solution we provide. We analyze three cases: the case of a finite state space, the case of master equation set on a Hilbert space, and finally on the set of probability measures, all in cases involving common noises. In the last case, the Lipschitz continuity we refer to is on the gradient of the value function with respect to the state variable of the player.