Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations

In this paper we study the optimal stochastic control problem for a path-dependent stochastic system under a recursive path-dependent cost functional, whose associated Bellman equation from dynamic programming principle is a path-dependent fully nonlinear partial differential equation of second order. A novel notion of viscosity solutions is introduced. Using Dupire's functional It\^o calculus, we characterize the value functional of the optimal stochastic control problem as the unique viscosity solution to the associated path-dependent Bellman equation.