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

Abstract: We study mean-field game (MFG) problems with rough common noise where the representative state dynamics is governed by a controlled rough stochastic differential equation driven by an idiosyncratic Brownian motion and a deterministic rough path noise affecting 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 in this context by developing new technical tools for compactification to accommodate rough integration, which deviate 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: conditioning yields the pathwise problem almost surely; and conversely, under a suitable causality/measurable-selection requirement, pathwise mean-field equilibria can be aggregated to produce randomized mean-field equilibria in the classical problem.