Finite state Mean Field Games with Wright-Fisher common noise

TL;DR

This paper introduces a Wright-Fisher common noise to finite state mean field games to ensure uniqueness, analyzing the associated master equation which is a degenerate parabolic PDE on the simplex, and establishing conditions for unique smooth solutions.

Contribution

It demonstrates how adding Wright-Fisher noise guarantees uniqueness in finite state mean field games by analyzing the master equation and deriving conditions for smooth solutions.

Findings

Unique smooth solutions exist under certain drift conditions.

The master equation is a non-linear Kimura-type PDE.

A priori Hölder estimates are established for the Kimura operator.

Abstract

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. (2019). We show that this equation, which is a non-linear version of the Kimura type equation studied in Epstein and Mazzeo (2013), 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 H\"older type for the corresponding Kimura operator when the drift therein is merely continuous.