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We study a family of mean field games with a state variable evolving as amultivariate jump diffusion process. The jump component is driven by a Poissonprocess with a time-dependent intensity function. All coefficients, i.e. drift,volatility and jump size, are controlled. Under fairly general conditions, weestablish existence of a solution in a relaxed version of the mean field gameand give conditions under which the optimal strategies are in fact Markovian,hence extending to a jump-diffusion setting previous results established in[30]. The proofs rely upon the notions of relaxed controls and martingaleproblems. Finally, to complement the abstract existence results, we study asimple illiquid inter-bank market model, where the banks can change theirreserves only at the jump times of some exogenous Poisson processes with acommon constant intensity, and provide some numerical results.

Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model