Mean Field Games with Reflected Dynamics: Penalization and Relaxed Control Approach

TL;DR

This paper studies mean field games with reflected stochastic dynamics, establishing existence of equilibria through penalization and relaxed control methods, and demonstrating approximation by non-reflected and strict control models.

Contribution

It introduces a novel approach combining penalization and relaxed controls to analyze reflected mean field games, proving existence and approximation results.

Findings

Existence of equilibrium in reflected MFGs.

Approximation of reflected MFGs by non-reflected SDE equilibria.

Existence of Markovian and strict-Markovian MFG solutions.

Abstract

In this paper, we investigate a class of Mean Field Games (MFGs) in which the state dynamics are governed by multidimensional reflected stochastic differential equations (SDEs). We establish the existence of an equilibrium and show that it can be approximated by the equilibrium of MFGs with non-reflected SDE. This approximation is constructed via a penalization method combined with the relaxed control approach introduced in [21]. Under a uniform ellipticity condition, and by applying the penalization method together with the mimicking theorem, we prove the existence of a Markovian MFG. Furthermore, under an additional convexity assumption, we demonstrate the existence of a strict-Markovian MFG. In the general case, we prove that relaxed MFG solutions with reflected dynamics can be approximated by strict controls whose dynamics are governed by penalized SDEs.