A description based on optimal transport for a class of stochastic McKean-Vlasov control problems

TL;DR

This paper investigates the convergence of controlled particle systems to stochastic McKean-Vlasov problems, using optimal transport techniques to analyze value functions and probability distributions.

Contribution

It introduces a novel approach employing optimal transport reformulations to establish convergence results for stochastic McKean-Vlasov control problems.

Findings

Proves convergence of value functions and distributions.

Establishes entropy convergence in path-space laws.

Utilizes Benamou-Brenier reformulation and superposition principle.

Abstract

We study the convergence of an $N$-particle Markovian controlled system to the solution of a family of stochastic McKean-Vlasov control problems, either with a finite horizon or Schr\"odinger type cost functional. Specifically, under suitable assumptions, we prove the convergence of the value functions, the fixed-time probability distributions, and the relative entropy of their path-space probability laws. These proofs are based on a Benamou-Brenier type reformulation of the problem and a superposition principle, both of which are tools from the theory of optimal transport.