Viscosity Solutions to Parabolic Master Equations

Wednesday, June 13, 2018 - 2:00pm - 3:00pm
Lind 305
Jianfeng Zhang (University of Southern California)
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 talk we propose an intrinsic notion of viscosity solution for parabolic master equations, which corresponds to the control problems, and investigate its wellposedness. Our main innovation is to restrict the involved measures to certain set of semimartingale measures which satisfies the desired compactness. Our results can be easily extended to 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 extend this well known result the path dependent setting. Our arguments are elementary and are new even in the state dependent case. The talk is based on a joint work with Cong Wu.