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

Jiusheng Liu, Jing Zhang

TL;DR

This work develops a probabilistic framework for mean-field games with jump diffusion by linking the master equation to coupled McKean–Vlasov forward–backward SDEs with jumps. It proves well-posedness on a short time horizon and establishes regularity for first- and second-order derivatives with respect to both state and measure arguments, using Lions and linear derivatives and jump-integral estimates. The decoupling field $V(t,x, mu)=Y^{t,x, mu}_t$ is shown to be the unique classical solution to the master equation, with $Z$ and $H$ recovered from derivatives of $V$ via Itô–Lions calculus, extending diffusion results to jump-diffusion settings. The results provide a rigorous probabilistic representation and a foundation for numerical methods for jump-diffusion MFG master equations.

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.

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

TL;DR

This work develops a probabilistic framework for mean-field games with jump diffusion by linking the master equation to coupled McKean–Vlasov forward–backward SDEs with jumps. It proves well-posedness on a short time horizon and establishes regularity for first- and second-order derivatives with respect to both state and measure arguments, using Lions and linear derivatives and jump-integral estimates. The decoupling field is shown to be the unique classical solution to the master equation, with and recovered from derivatives of via Itô–Lions calculus, extending diffusion results to jump-diffusion settings. The results provide a rigorous probabilistic representation and a foundation for numerical methods for jump-diffusion MFG master equations.

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.
Paper Structure (14 sections, 14 theorems, 143 equations)

This paper contains 14 sections, 14 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations
  4. Differentiability of functions w.r.t. probability measures
  5. Poisson random measures
  6. Mean-field games with jump diffusion
  7. The well-posedness of MV-FBSDEs
  8. Assumptions
  9. The well-posedness of MV-FBSDEs on small time duration
  10. The first order derivative of the solution
  11. First-order derivatives of $(X^{t,x,\mu},Y^{t,x,\mu}, Z^{t,x,\mu},H^{t,x,\mu})$ w.r.t. $x$
  12. First-order derivatives of $(X^{t,x,\mu},Y^{t,x,\mu}, Z^{t,x,\mu},H^{t,x,\mu})$ w.r.t. $\mu$
  13. The second-order derivatives of the solutions
  14. Related integral-master equations

Key Result

Proposition 1

(cardaliaguet2019master, Proposition 2.2.3) Assume that $U$ is linear differentiable. Moreover, $\partial_y\frac{\delta U}{\delta\mu}(\mu,y)$ exists and is jointly continuous and bounded on $\mathcal{P}_2(\mathbb{R}^d)\times\mathbb{R}^d$. Then for any Borel measurable map $\phi:\mathbb{R}^d\to\mathb where $id$ denotes the identity map.

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 1
  • proof
  • Proposition 2
  • proof
  • ...and 19 more