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

TL;DR

This paper develops a framework for mean field games with absorption and common noise, providing existence results, convergence analysis, and a new bank run model within this setting.

Contribution

It introduces a general existence theory for weak mean field equilibria with absorption and common noise, and constructs approximate Nash equilibria for large N-player games.

Findings

Existence of weak mean field equilibria with absorption and common noise.

Convergence of discretized solutions to the mean field limit.

A novel bank run model analyzed within this framework.

Abstract

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. Since the approximation is two-fold, one given by the mean field limit and one given by the discretization, some suitable convergence results are needed. We also introduce and discuss a novel model of bank run that can be studied within this framework.