# Mean Field Game with Reflected Jump Diffusion Dynamics: A Linear Programming Approach

## TL;DR

This paper introduces a linear programming method to analyze mean field games with reflected jump-diffusion dynamics, establishing theoretical equivalences and existence results, and demonstrating the approach with a numerical example.

## Contribution

It develops a novel linear programming framework for mean field games with complex dynamics, proving equivalence with weak control formulations and establishing existence of equilibria.

## Key findings

- Proves equivalence between linear programming and weak relaxed control formulations.
- Establishes existence of mean field equilibria under general conditions.
- Provides a numerical example illustrating the computation of equilibria.

## Abstract

This paper develops a linear programming approach for mean field games with reflected jump-diffusion dynamics. We first prove the equivalence between the mean field equilibria in the linear programming formulation and those in the weak relaxed control formulation under some measurability and growth conditions on model coefficients. Building upon the characterization of the occupation measure in the equivalence result, we further establish the existence of linear programming mean field equilibria under fairly general conditions on model coefficients. Finally, a numerical example is presented to illustrate the computation of a mean field equilibrium using the linear programming formulation.

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Source: https://tomesphere.com/paper/2508.20388