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

TL;DR

The paper tackles mean-field games with state constraints enforced by reflection and subject to jumps, incorporating a terminal cost. It develops a linear programming formulation over occupation measures and proves its equivalence to the classical weak relaxed-control MFE under mild conditions, using a cost reparametrization that absorbs jump-induced reflection via the continuous reflection component. Existence of LP mean-field equilibria is established in both bounded and unbounded coefficient regimes: first by a fixed-point argument in the bounded case, then via a truncation-and-limit approach for the general case, with a constructive link to Markovian relaxed controls. A numerical example in inventory management demonstrates the practical computability of LPMFEs through LP discretization and a fictitious-play–style procedure, highlighting the method’s potential for large-population systems with state constraints and jumps. Overall, the LP approach offers a tractable, numerically friendly avenue for analyzing and computing MFEs in reflected jump-diffusion MFGs, broadening applicability to constrained stochastic control in economics and operations research.

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.

Figures (3)

• Figure 1: Evolution of the distribution ($m_t$) and the terminal distribution ($\nu$).
• Figure 2: Average control at equilibrium.
• Figure 3: Convergence of the algorithm.

Theorems & Definitions (17)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thm:existence']}