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

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.