Viscosity Solutions to Parabolic Master Equations and McKean-Vlasov SDEs with Closed-loop Controls

The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean-Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire \cite{Dupire}'s functional It\^{o} formula. This It\^{o} formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions \cite{Lions4} in the state dependent case. We provided an elementary proof for this well known result in the short note \cite{WZ}, and the same arguments work in the path dependent setting here.