Mean-field games with rough common noise: the compactification approach

TL;DR

This paper introduces a new framework for mean-field games with rough common noise, utilizing compactification techniques to prove the existence of equilibria and relate them to classical models.

Contribution

It develops a canonical weak formulation for MFGs with rough noise and introduces novel compactification tools for proving equilibrium existence.

Findings

Proved existence of pathwise mean-field equilibria with rough common noise.

Established a relationship between pathwise and classical MFG problems.

Provided an alternative proof of strong equilibrium without pathwise uniqueness.

Abstract

We study mean-field game (MFG) problems with rough common noise, in which the representative state dynamics are governed by a controlled rough stochastic differential equation driven by an idiosyncratic Brownian motion and a deterministic rough-path signal that affects the whole population. Within this new framework, we introduce a canonical weak formulation based on relaxed controls and rough martingale problems. We prove the existence of a pathwise mean-field equilibrium by developing new compactification tools that accommodate rough integration and differ substantially from classical compactification arguments in the literature. Finally, we discuss the relationship between the pathwise problem and the classical MFG problem with randomized Brownian common noise. Using the notion of a pathwise admissible set, we recast mean-field game problems with common noise as optimization problems over an extended space of probability measures. We establish an equivalent characterization of Carmona-Delarue-Lacker's weak equilibrium and, as an application, give an alternative proof of strong equilibrium without first establishing pathwise uniqueness.