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Mean Field Games with Reflected Dynamics: Penalization and Relaxed Control Approach

Ayoub Laayoun, Badr Missaoui

Abstract

In this paper, we investigate a class of Mean Field Games (MFGs) in which the state dynamics are governed by multidimensional reflected stochastic differential equations (SDEs). We establish the existence of an equilibrium and show that it can be approximated by the equilibrium of MFGs with non-reflected SDE. This approximation is constructed via a penalization method combined with the relaxed control approach introduced in [21]. Under a uniform ellipticity condition, and by applying the penalization method together with the mimicking theorem, we prove the existence of a Markovian MFG. Furthermore, under an additional convexity assumption, we demonstrate the existence of a strict-Markovian MFG. In the general case, we prove that relaxed MFG solutions with reflected dynamics can be approximated by strict controls whose dynamics are governed by penalized SDEs.

Mean Field Games with Reflected Dynamics: Penalization and Relaxed Control Approach

Abstract

In this paper, we investigate a class of Mean Field Games (MFGs) in which the state dynamics are governed by multidimensional reflected stochastic differential equations (SDEs). We establish the existence of an equilibrium and show that it can be approximated by the equilibrium of MFGs with non-reflected SDE. This approximation is constructed via a penalization method combined with the relaxed control approach introduced in [21]. Under a uniform ellipticity condition, and by applying the penalization method together with the mimicking theorem, we prove the existence of a Markovian MFG. Furthermore, under an additional convexity assumption, we demonstrate the existence of a strict-Markovian MFG. In the general case, we prove that relaxed MFG solutions with reflected dynamics can be approximated by strict controls whose dynamics are governed by penalized SDEs.
Paper Structure (9 sections, 12 theorems, 142 equations)

This paper contains 9 sections, 12 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Assumptions and Existence of MFG Solutions
  3. Relaxed Formulation of the MFG with Penalized Dynamics
  4. Relaxed MFGs with Reflected Dynamics
  5. Proof of Theorem \ref{['MFGR']}
  6. Relative Compactness of the Penalized Processes $(X^n, K^n, Q^n, M^n)$
  7. Admissibility and optimality of the limit of $(P^n)_{n\ge1}$
  8. Existence of Markovian and Strict Markovian MFG Solutions
  9. Approximation of Relaxed MFGs with Reflected Dynamics by Strict Controls with Penalized Dynamics

Key Result

Proposition 2.1

A probability measure $P^n \in \mathcal{R}_n(\mu^n)$ if and only if there exists a filtered probability space $(\Omega^n, \mathcal{F}^n_t, \mathbb{Q}^n)$ supporting: such that $\mathbb{Q}^n \circ (X^n, Q^n)^{-1} = P^n$ , and the following state equation is satisfied: Under standard Lipschitz and growth assumptions, this equation admits a unique strong solution on any such filtered probability sp

Theorems & Definitions (28)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Definition 2.3
  • Remark 2.3
  • Proposition 2.2
  • Definition 2.4
  • ...and 18 more