Monotone solutions for mean field games master equations : continuous state space and common noise

TL;DR

This paper introduces the concept of monotone solutions for mean field games master equations in continuous state spaces, proving their existence, uniqueness, and stability, especially under common noise scenarios with jumps or correlated randomness.

Contribution

It develops a new framework for monotone solutions in continuous spaces, extending the analysis of mean field games master equations to include complex noise structures.

Findings

Established existence and uniqueness of monotone solutions

Analyzed stability under standard assumptions

Extended to cases with jumps and correlated noise

Abstract

We present the notion of monotone solution of mean field games master equations in the case of a continuous state space. We establish the existence, uniqueness and stability of such solutions under standard assumptions. This notion allows us to work with solutions which are merely continuous in the measure argument, in the case of first order master equations. We study several structures of common noises, in particular ones in which common jumps (or aggregate shocks) can happen randomly, and ones in which the correlation of randomness is carried by an additional parameter.