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

## TL;DR

This paper investigates mean field games with controlled jump-diffusion dynamics, establishing existence of solutions and applying the results to a model of an illiquid interbank market with numerical analysis.

## Contribution

It extends mean field game theory to jump-diffusion processes with controlled coefficients and provides existence results along with a practical interbank market model.

## Key findings

- Existence of solutions in a relaxed mean field game with jump-diffusion dynamics.
- Conditions for Markovian optimal strategies in jump-diffusion setting.
- Numerical results for an illiquid interbank market model.

## Abstract

We study a family of mean field games with a state variable evolving as a multivariate jump diffusion process. The jump component is driven by a Poisson process with a time-dependent intensity function. All coefficients, i.e. drift, volatility and jump size, are controlled. Under fairly general conditions, we establish existence of a solution in a relaxed version of the mean field game and 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 martingale problems. Finally, to complement the abstract existence results, we study a simple illiquid inter-bank market model, where the banks can change their reserves only at the jump times of some exogenous Poisson processes with a common constant intensity, and provide some numerical results.