Mean field games with common noise via Malliavin calculus

TL;DR

This paper introduces a simplified proof for the existence of strong equilibria in mean field games with common noise, leveraging Malliavin calculus to handle conditional laws and optimal controls adapted to the common noise.

Contribution

It extends a Malliavin calculus-based compactness criterion to processes, enabling a more straightforward existence proof for equilibria in mean field games with common noise.

Findings

Existence of strong equilibria established using Malliavin calculus.

Approach requires only measurability of drift and cost functionals.

Equilibria are adapted to the common noise filtration.

Abstract

We present a simpler proof of the existence of equilibria for a class of mean field games with common noise, where players interact through the conditional law given the current value of the common noise rather than its entire path. By extending a compactness criterion for Malliavin-differentiable random variables to processes, we establish existence of strong equilibria, where the conditional law and optimal control are adapted to the common noise filtration and defined on the original probability space. Notably, our approach only requires measurability of the drift and cost functionals with respect to the state variable.