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