Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations

Abstract

In this paper we study the optimal stochastic control problem fora path-dependent stochastic system under a recursive path-dependentcost functional, whose associated Bellman equation from dynamic programmingprinciple is a path-dependent fully nonlinear partial differentialequation of second order. A novel notion of viscosity solutions isintroduced by restricting the semi-jets on an $\alpha$-Hölder space$\mathbf{C}^{\alpha}$ for $\alpha\in(0,\frac{1}{2})$. Using Dupire'sfunctional Itô calculus, we prove that the value functional of theoptimal stochastic control problem is a viscosity solution to theassociated path-dependent Bellman equation. A state-dependent approximationof the path-dependent value functional is given.