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Multidimensional McKean-Vlasov SDEs with mean reflection: well-posedness and existence of optimal control

Imane Jarni, Ayoub Laayoun, Badr Missaoui

TL;DR

The paper develops a theory for multidimensional McKean–Vlasov SDEs with mean reflection, formulating a Skorokhod problem in time-dependent convex domains to enforce mean-field constraints on the law of the solution. It establishes existence and uniqueness for strong solutions to the mean-reflected MV-SDEs, and analyzes a particle system approach that proves propagation of chaos under the mean-reflection constraint. The control problem is treated in the relaxed framework to guarantee existence of optimal controls, and under Roxin convexity a strict optimal control is recovered; a chattering argument shows relaxation-to-strict approximation. Collectively, the results provide a rigorous foundation for well-posedness, approximation, and optimal control of constrained mean-field dynamics with jumps, expanding the toolbox for mean-field control under law constraints.

Abstract

In this work, we investigate the multidimensional Skorokhod problem for càdlàg processes, where the reflection is subject to a minimality condition depending on the law of the solution. We then apply these results to establish existence and uniqueness for multidimensional McKean-Vlasov stochastic differential equations with mean reflection. Finally, we address the existence of optimal relaxed controls for such equations.

Multidimensional McKean-Vlasov SDEs with mean reflection: well-posedness and existence of optimal control

TL;DR

The paper develops a theory for multidimensional McKean–Vlasov SDEs with mean reflection, formulating a Skorokhod problem in time-dependent convex domains to enforce mean-field constraints on the law of the solution. It establishes existence and uniqueness for strong solutions to the mean-reflected MV-SDEs, and analyzes a particle system approach that proves propagation of chaos under the mean-reflection constraint. The control problem is treated in the relaxed framework to guarantee existence of optimal controls, and under Roxin convexity a strict optimal control is recovered; a chattering argument shows relaxation-to-strict approximation. Collectively, the results provide a rigorous foundation for well-posedness, approximation, and optimal control of constrained mean-field dynamics with jumps, expanding the toolbox for mean-field control under law constraints.

Abstract

In this work, we investigate the multidimensional Skorokhod problem for càdlàg processes, where the reflection is subject to a minimality condition depending on the law of the solution. We then apply these results to establish existence and uniqueness for multidimensional McKean-Vlasov stochastic differential equations with mean reflection. Finally, we address the existence of optimal relaxed controls for such equations.
Paper Structure (11 sections, 20 theorems, 100 equations)

This paper contains 11 sections, 20 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. The Skorokhod problem with mean reflection
  3. The Skorokhod problem with mean reflection
  4. McKean-Vlasov SDEs with mean reflection
  5. Existence and uniqueness of strong solution
  6. Particle approximation of mean-reflected McKean–Vlasov SDEs
  7. Optimal control problem for McKean SDEs with mean reflection
  8. Formulation of the problem
  9. Existence of optimal controls
  10. Approximation of the relaxed control problem
  11. Appendix

Key Result

Lemma 2.1

Assume $\left(\mathbf{H}\right)$. Let $\mathcal{D}$ be a set in $\mathcal{C}$ and define $D_t:=\Phi^{-1}_{t}(\mathcal{D})$. Then, the following properties hold

Theorems & Definitions (46)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Proposition 2.1
  • proof
  • Corollary 2.1
  • ...and 36 more