Mean-field games with rough common noise: the linear-quadratic case

TL;DR

This paper develops a well-posedness theory for linear-quadratic mean-field games with rough common noise, introducing a novel Volterra formulation and analyzing the associated rough FBSDEs.

Contribution

It introduces a new Volterra-type formulation for rough MFGs, establishing existence, uniqueness, and stability results in the linear-quadratic setting.

Findings

Established well-posedness for rough LQ MFGs with stability estimates.

Derived a characterization of optimal controls via rough FBSDEs.

Showed solutions coincide with conditioned solutions when common noise is a Stratonovich lift of Brownian motion.

Abstract

Motivated by mean-field games (MFG) with common noise on the one hand and pathwise stochastic control theory on the other, we formulate here a linear-quadratic (LQ) MFG with rough common noise, along with a satisfactory well-posedness theory for the linear-quadratic case. A novel Volterra-type (or mild) formulation allows to keep technical (rough-stochastic) consideration to a minimum. We derive a characterization of the optimal state and optimal control through a rough forward-backward SDE (rough FBSDE), and provide an existence and uniqueness result under the usual assumptions. Our theory is accompanied by stability estimates with respect to initial data and common noise while we also establish continuity of what we call the It\^o-Lions-Lyons map for rough mean-field games. Finally, we discuss randomization of the rough common noise under appropriate conditions on the coefficients. When the latter is given by the Stratonovich lift of a Brownian motion independent of the idiosyncratic noise, we show that solutions of the rough LQ MFG coincide with those obtained by conditioning on the common noise.