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We consider a mean field game describing the limit of a stochasticdifferential game of $N$-players whose state dynamics are subject toidiosyncratic and common noise and that can be absorbed when they hit aprescribed region of the state space. We provide a general result for theexistence of weak mean field equilibria which, due to the absorption and thecommon noise, are given by random flow of sub-probabilities. We first use afixed point argument to find solutions to the mean field problem in a reducedsetting resulting from a discretization procedure and then we prove convergenceof such equilibria to the desired solution. We exploit these ideas also toconstruct $\varepsilon$-Nash equilibria for the $N$-player game. Since theapproximation is two-fold, one given by the mean field limit and one given bythe discretization, some suitable convergence results are needed. We alsointroduce and discuss a novel model of bank run that can be studied within thisframework.

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