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Mean Field Games with Reflected Dynamics

Imane Jarni, Ayoub Laayoun, Badr Missaoui

TL;DR

This paper proves the existence of equilibria for Mean Field Games with reflected dynamics by formulating the representative-player problem in a relaxed control framework and recasting it as a controlled martingale problem. It establishes a fixed-point existence via Kakutani–Fan–Glicksberg for bounded coefficients and extends to unbounded coefficients through a truncation and limiting argument, ensuring the limit solves the RSDE with reflection and matches the mean-field law. The main contributions include extending Lacker's relaxed-control approach to reflected SDEs, formalizing control rules and their cost in a mean-field setting, and providing a rigorous existence result under Assumption (A) with both bounded and unbounded coefficient regimes. The results enable rigorous analysis of approximate Nash equilibria in large symmetric $N$-player games with boundary constraints and have potential applications in queueing networks, constrained stochastic control, and related areas where reflected dynamics arise.

Abstract

This paper establishes an equilibrium existence result for a class of Mean Field Games involving Reflected Stochastic Differential Equations. The proof relies on the framework of relaxed controls and martingale problems.

Mean Field Games with Reflected Dynamics

TL;DR

This paper proves the existence of equilibria for Mean Field Games with reflected dynamics by formulating the representative-player problem in a relaxed control framework and recasting it as a controlled martingale problem. It establishes a fixed-point existence via Kakutani–Fan–Glicksberg for bounded coefficients and extends to unbounded coefficients through a truncation and limiting argument, ensuring the limit solves the RSDE with reflection and matches the mean-field law. The main contributions include extending Lacker's relaxed-control approach to reflected SDEs, formalizing control rules and their cost in a mean-field setting, and providing a rigorous existence result under Assumption (A) with both bounded and unbounded coefficient regimes. The results enable rigorous analysis of approximate Nash equilibria in large symmetric -player games with boundary constraints and have potential applications in queueing networks, constrained stochastic control, and related areas where reflected dynamics arise.

Abstract

This paper establishes an equilibrium existence result for a class of Mean Field Games involving Reflected Stochastic Differential Equations. The proof relies on the framework of relaxed controls and martingale problems.

Paper Structure

This paper contains 9 sections, 11 theorems, 71 equations.

Table of Contents

  1. Introduction :
  2. Assumptions and the main result
  3. Stochastic control problem
  4. Relaxed control
  5. Control rules
  6. The relaxed MFG with controlled RSDE Dynamics
  7. Proof of the main result
  8. The Case of Bounded Coefficients
  9. The Case of Unbounded Coefficients

Key Result

Proposition 2.1

The existence of a solution $\mathbb{P}$ to the martingale problem equ02.5 is equivalent to the existence of a weak solution to the following reflected SDEs: where $\tilde{X}$, $\tilde{M}$ and $\tilde{K}$ are defined on some extension $(\tilde{\Omega},\tilde{\mathcal{F}}, \tilde{\mathbb{Q}})$ and $\tilde{M}$ is a martingale measure with intensity $\tilde{Q}$. Moreover, the two solutions are relat

Theorems & Definitions (29)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.2
  • Definition 2.3
  • Remark 2.3
  • Proposition 2.1
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 19 more