Optimal control of path-dependent McKean–Vlasov SDEs in infinite-dimension

We study the optimal control of path-dependent McKean–Vlasov equations valued in Hilbert spaces motivated by non-Markovian mean-field models driven by stochastic PDEs. We first establish the well-posedness of the state equation, and then we prove the dynamic programming principle (DPP) in such a general framework. The crucial law invariance property of the value function V is rigorously obtained, which means that V can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space. We then define a notion of pathwise measure derivative, which extends the Wasserstein derivative due to Lions (Lions (Audio Conference, 2006–2012)), and prove a related functional Itô formula in the spirit of Dupire ((2009), Functional Itô Calculus, Bloomberg Portfolio Research Paper No. 2009-04-FRONTIERS) and Wu and Zhang (Ann. Appl. Probab. 30 (2020) 936–986). The Master Bellman equation is derived from the DPP by means of a suitable notion of viscosity solution. We provide different formulations and simplifications of such a Bellman equation notably in the special case when there is no dependence on the law of the control.