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

TL;DR

This paper studies multidimensional McKean-Vlasov stochastic differential equations with mean reflection, establishing well-posedness and the existence of optimal controls, advancing the understanding of stochastic systems with law-dependent reflection.

Contribution

It introduces a framework for well-posedness of multidimensional McKean-Vlasov SDEs with mean reflection and proves the existence of optimal relaxed controls.

Findings

Existence and uniqueness of solutions for multidimensional McKean-Vlasov SDEs with mean reflection.

Development of a multidimensional Skorokhod problem with law-dependent minimal reflection.

Proof of existence of optimal relaxed controls for these stochastic systems.

Abstract

In this work, we investigate the multidimensional Skorokhod problem for c\`adl\`ag 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.