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

• Published in: Stochastic processes and their applications Vol. 164; pp. 206 - 241
• Main Authors: Burzoni, Matteo, Campi, Luciano
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2023
• Subjects: Absorbing boundary; Bank runs; Common noise; Mean field games; Nash equilibria; Common noise; Absorbing boundary; Mean field games; Nash equilibria; Bank runs
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2023.07.007

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 ɛ-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.