On the mean field games with common noise and the McKean-Vlasov SPDEs

TL;DR

This paper develops a new framework for mean field games with common noise using infinite-dimensional PDEs, establishing Nash equilibrium profiles and analyzing convergence and regularity properties of related stochastic equations.

Contribution

It introduces a novel PDE approach to MFG with common noise, linking it to McKean-Vlasov SPDEs and providing convergence rates and regularity results.

Findings

Solution of the PDE yields $1/N$-Nash equilibria.

Established regularity and sensitivity analysis for McKean-Vlasov SPDEs.

Proved $1/N$ convergence rate for propagation of chaos.

Abstract

We formulate the MFG limit for $N$ interacting agents with a common noise as a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. We prove that any its (regular enough) solution provides an $1/N$-Nash-equilibrium profile for the initial $N$-player game. We use the method of stochastic characteristics to provide the link with the basic models of MFG with a major player. We develop two auxiliary theories of independent interest: sensitivity and regularity analysis for the McKean-Vlasov SPDEs and the $1/N$-convergence rate for the propagation of chaos property of interacting diffusions.