Finite state mean field games with Wright-Fisher common noise

We force uniqueness in finite state mean field games by adding a Wright-Fisher common noise. We achieve this by analyzing the master equation of this game, which is a degenerate parabolic second-order partial differential equation set on the simplex whose characteristics solve the stochastic forward-backward system associated with the mean field game; see Cardaliaguet et al. [10]. We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo [28], has a unique smooth solution whenever the normal component of the drift at the boundary is strong enough. Among others, this requires a priori estimates of Holder type for the corresponding Kimura operator when the drift therein is merely continuous. (C) 2021 Elsevier Masson SAS. All rights reserved.

Automated summary

In this study, uniqueness is enforced in finite state mean field games by introducing a Wright-Fisher common noise. The analysis of the master equation reveals a non-linear version of the Kimura type equation, with a unique smooth solution when the drift at the boundary is strong enough. This requires a priori estimates of Holder type for the corresponding Kimura operator with merely continuous drift.