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A Class of Degenerate Mean Field Games, Associated FBSDEs and Master Equations

Alain Bensoussan, Ziyu Huang, Shanjian Tang, Sheung Chi Phillip Yam

TL;DR

The paper develops a probabilistic framework for a class of degenerate mean field games with state-distribution dependent diffusion, proving well-posedness for the associated forward–backward systems and establishing strong regularity for the value functional. Through a systematic study of Gâteaux derivatives in the initial condition and in the state-measure variables, the authors construct Jacobian and Hessian flows and show that the value function $V$ is a classical solution to the degenerate master equation, with $D_x V$ and $D_x^2 V$ characterized by first- and second-order adjoint processes. The results rely on a continuation method for FBSDEs in Hilbert spaces under monotonicity, convexity, and small mean-field interaction conditions, plus stronger regularity assumptions to obtain Hessian-type derivatives. Consequently, $V$ serves as the decoupling field for the MFG HJB–FP system, and the master equation is shown to admit a unique classical solution under the stated assumptions. The framework accommodates degenerate diffusion, permits nonseparable costs, and yields explicit regularity and growth properties for $V$, enabling a rigorous master-equation analysis in a broader MFG setting with distribution-dependent dynamics and unbounded diffusion coefficients.

Abstract

In this paper, we study a class of degenerate mean field games (MFGs) with state-distribution dependent and unbounded functional diffusion coefficients. With a probabilistic method, we study the well-posedness of the forward-backward stochastic differential equations (FBSDEs) associated with the MFG and arising from the maximum principle, and estimate the corresponding Jacobian and Hessian flows. We further establish the classical regularity of the value functional $V$; in particular, we show that when the cost function is $C^3$ in the spatial and control variables and $C^2$ in the distribution argument, then the value functional is $C^1$ in time and $C^2$ in the spatial and distribution variables. As a consequence, the value functional $V$ is the unique classical solution of the degenerate MFG master equation.

A Class of Degenerate Mean Field Games, Associated FBSDEs and Master Equations

TL;DR

The paper develops a probabilistic framework for a class of degenerate mean field games with state-distribution dependent diffusion, proving well-posedness for the associated forward–backward systems and establishing strong regularity for the value functional. Through a systematic study of Gâteaux derivatives in the initial condition and in the state-measure variables, the authors construct Jacobian and Hessian flows and show that the value function is a classical solution to the degenerate master equation, with and characterized by first- and second-order adjoint processes. The results rely on a continuation method for FBSDEs in Hilbert spaces under monotonicity, convexity, and small mean-field interaction conditions, plus stronger regularity assumptions to obtain Hessian-type derivatives. Consequently, serves as the decoupling field for the MFG HJB–FP system, and the master equation is shown to admit a unique classical solution under the stated assumptions. The framework accommodates degenerate diffusion, permits nonseparable costs, and yields explicit regularity and growth properties for , enabling a rigorous master-equation analysis in a broader MFG setting with distribution-dependent dynamics and unbounded diffusion coefficients.

Abstract

In this paper, we study a class of degenerate mean field games (MFGs) with state-distribution dependent and unbounded functional diffusion coefficients. With a probabilistic method, we study the well-posedness of the forward-backward stochastic differential equations (FBSDEs) associated with the MFG and arising from the maximum principle, and estimate the corresponding Jacobian and Hessian flows. We further establish the classical regularity of the value functional ; in particular, we show that when the cost function is in the spatial and control variables and in the distribution argument, then the value functional is in time and in the spatial and distribution variables. As a consequence, the value functional is the unique classical solution of the degenerate MFG master equation.

Paper Structure

This paper contains 28 sections, 17 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Forward-backward systems
  4. Well-posedness of MFG \ref{['intro_1']}
  5. Well-posedness of Problem \ref{["intro_1'"]}
  6. Derivatives of $\left(Y_{t\xi}(s),p_{t\xi}(s),q_{t\xi}(s)\right)$ in $\xi$
  7. Derivatives of $(Y_{tx\mu}(s),p_{tx\mu}(s),q_{tx\mu}(s))$ in $(x,\mu)$
  8. Jacobian flow of first-order derivatives
  9. Hessian flows of second-order derivatives
  10. Regularity of the value functional
  11. Master equation
  12. Proof of Statements in Section \ref{['sec:MP']}
  13. Proof of Lemma \ref{['lem:MP1']}
  14. Proof of Lemma \ref{['lem:2']}
  15. Proof of Lemma \ref{['lem:5']}
  16. ...and 13 more sections

Key Result

Lemma 2.2

Under Assumptions (A1)-(A3), suppose that the following FBSDEs has a unique solution $(Y_{t\xi},p_{t\xi},q_{t\xi})\in \mathcal{S}^2_\mathscr{F}(t,T)\times\mathcal{L}^2_{\mathscr{F}}(t,T)\times\left(\mathcal{L}^2_{\mathscr{F}}(t,T)\right)^n$. Then, there exists a constant $C(L,T)$ depending only on $(L,T)$, such that when $\lambda_v\geq C(L,T)$, MFG intro_1 has Furthermore, if Assumption (A3') is

Theorems & Definitions (25)

  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • Lemma 3.1
  • ...and 15 more