McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations

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