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Mean field games with common noise and degenerate idiosyncratic noise

Pierre Cardaliaguet, Benjamin Seeger, Panagiotis Souganidis

TL;DR

This paper develops a rigorous framework for mean field games with common noise and potentially degenerate idiosyncratic noise, addressing the lack of smooth solutions by introducing a novel notion of weak solutions for backward stochastic HJB equations. The authors prove the existence of probabilistically weak solutions to the transformed MFG system via a discretization- and fixed-point-based strategy, and establish a Lasry-Lions–type monotonicity result that yields pathwise uniqueness and strong solvability. A key technical contribution is the propagation of semiconcavity for deterministic HJB equations, which underpins well-posedness in the stochastic setting and enables distributional interpretations of second-order terms. The analysis combines PDE techniques (semiconcavity, distributional solutions) with probabilistic constructions (weak solutions, Kakutani fixed points, discretization in probability space) to extend MFG theory to degenerate, common-noise models on the whole space. The results advance the mathematical understanding of equilibria in large populations under shared randomness and broaden the applicability of MFGs to more general diffusion structures.

Abstract

We study the forward-backward system of stochastic partial differential equations describing a mean field game for a large population of small players subject to both idiosyncratic and common noise. The unique feature of the problem is that the idiosyncratic noise coefficient may be degenerate, so that the system does not admit smooth solutions in general. We develop a new notion of weak solutions for backward stochastic Hamilton-Jacobi-Bellman equations, and use this to build probabilistically weak solutions of the mean field game system.

Mean field games with common noise and degenerate idiosyncratic noise

TL;DR

This paper develops a rigorous framework for mean field games with common noise and potentially degenerate idiosyncratic noise, addressing the lack of smooth solutions by introducing a novel notion of weak solutions for backward stochastic HJB equations. The authors prove the existence of probabilistically weak solutions to the transformed MFG system via a discretization- and fixed-point-based strategy, and establish a Lasry-Lions–type monotonicity result that yields pathwise uniqueness and strong solvability. A key technical contribution is the propagation of semiconcavity for deterministic HJB equations, which underpins well-posedness in the stochastic setting and enables distributional interpretations of second-order terms. The analysis combines PDE techniques (semiconcavity, distributional solutions) with probabilistic constructions (weak solutions, Kakutani fixed points, discretization in probability space) to extend MFG theory to degenerate, common-noise models on the whole space. The results advance the mathematical understanding of equilibria in large populations under shared randomness and broaden the applicability of MFGs to more general diffusion structures.

Abstract

We study the forward-backward system of stochastic partial differential equations describing a mean field game for a large population of small players subject to both idiosyncratic and common noise. The unique feature of the problem is that the idiosyncratic noise coefficient may be degenerate, so that the system does not admit smooth solutions in general. We develop a new notion of weak solutions for backward stochastic Hamilton-Jacobi-Bellman equations, and use this to build probabilistically weak solutions of the mean field game system.
Paper Structure (18 sections, 20 theorems, 231 equations)

This paper contains 18 sections, 20 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Informal discussion of the main results
  3. Previous work on MFG
  4. Main assumptions
  5. Organization of the paper
  6. Notation
  7. Propagation of semiconcavity for deterministic HJB equations
  8. The backward stochastic HJB equation
  9. The definition of solutions
  10. Existence
  11. Sub/supersolutions and a comparison principle
  12. Stability
  13. Optimal control representation
  14. The MFG system
  15. Existence of a weak solution
  16. ...and 3 more sections

Key Result

Proposition 2.1

Assume $a$, $G$, and $H$ satisfy A:MAINdiffusion, A:MAINterminal, A:MAINHamiltonian, and A:MAINHcondition, and let $u$ be the unique viscosity solution of E:dethjb. Then, for all $t\in [0,T]$, Moreover, $u$ is semiconcave and Lipschitz in the $x$-variable, uniformly on $[0,T] \times {\mathbb R}^d$, and there exists a constant $C > 0$ depending only on $T$ and the various data in those assumptions

Theorems & Definitions (49)

  • Remark 1.1
  • Proposition 2.1
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof : Proof of Proposition \ref{['P:dethjb']}
  • proof : Proof of Lemma \ref{['L:epshjb']}
  • Remark 3.1
  • Lemma 3.1
  • proof
  • ...and 39 more