A Probabilistic Interpretation of the Master Equation Arising from Mean Field Games with Jump Diffusion

TL;DR

This paper develops a probabilistic framework using coupled McKean-Vlasov stochastic differential equations with jumps to analyze the master equation in mean-field games driven by jump-diffusion processes, establishing existence, regularity, and uniqueness.

Contribution

It extends the probabilistic interpretation of the master equation to jump-diffusion mean-field games, providing well-posedness and regularity results for the associated stochastic equations.

Findings

Proved well-posedness of coupled MV-FBSDEs with jumps.

Established regularity of derivatives of solutions with respect to variables.

Showed the decoupling field satisfies the master equation uniquely.

Abstract

In this paper we study the classical solution to the master equation arising from mean-field games (MFGs) driven by jump-diffusion processes. The master equation, a nonlinear partial differential equation on Wasserstein space, characterizes the value function of MFGs and is challenging to analyze directly due to its measure-valued derivatives. We propose a probabilistic interpretation using coupled McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) with jumps. Under suitable Lipschitz and differentiability assumptions on the coefficients, we first establish the well-posedness of the MV-FBSDEs on a small time interval via a contraction mapping argument. We then prove the existence and regularity of the first- and second-order derivatives of the solutions with respect to the spatial and measure variables, relying on careful estimates involving jump terms and measure derivatives. Finally, we show that the decoupling field of the MV-FBSDEs satisfies the master equation in the classical sense, providing both existence and uniqueness of the solution. Our work extends earlier results on diffusion-driven MFGs to the jump-diffusion setting and offers a probabilistic framework for analyzing and numerically solving such kind of master equations.