Mean field games with absorption and common noise with a model of bank run

Understand

We consider a mean field game describing the limit of a stochastic differential game of $N$-players whose state dynamics are subject to idiosyncratic and common noise and that can be absorbed when they hit a prescribed region of the state space.

• We provide a general result for the existence of weak mean field equilibria which, due to the absorption and the common noise, are given by random flow of sub-probabilities.
• We first use a fixed point argument to find solutions to the mean field problem in a reduced setting resulting from a discretization procedure and then we prove convergence of such equilibria to the desired solution.
• We exploit these ideas also to construct $\varepsilon$-Nash equilibria for the $N$-player game.