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

Cong Wu, Jianfeng Zhang

Published 2018-05-07Version 1

Master equation is a powerful tool for studying McKean-Vlasov dynamics where the distribution of the state process enters the coefficients directly, with particular applications including mean field games and stochastic control problems with partial information. In this paper we propose an intrinsic notion of viscosity solution for master equations and establish its wellposedness. In particular we prove the comparison principle in some special cases, which seems new in the literature. Our main innovation is to restrict the involved measures to certain set of semimartingale measures which satisfies the desired compactness. As one 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 Wasserstein derivatives, 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.