McKean-Vlasov optimal control: limit theory and equivalence between different formulations

TL;DR

This paper develops a comprehensive limit theory for McKean-Vlasov optimal control problems with common noise, establishing the equivalence of strong, weak, and relaxed formulations, and connecting them to large population control limits.

Contribution

It introduces a unified framework for different formulations of McKean-Vlasov control problems and proves their equivalence under mild conditions, advancing the theoretical understanding.

Findings

Proved the set of relaxed controls is the closure of strong controls.

Established the equivalence between strong, weak, and relaxed formulations.

Demonstrated the limit of large population control problems with common noise.

Abstract

We study a McKean-Vlasov optimal control problem with common noise, in order to establish the corresponding limit theory, as well as the equivalence between different formulations, including the strong, weak and relaxed formulation. In contrast to the strong formulation, where the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an $(H)$-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classical optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls, when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem, under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean-Vlasov control problem, that is to say proving that it consists in the limit of a large population control problem with common noise.