Stochastic control and differential games with path-dependent controls

Abstract

In this paper we consider the functional Itˆo calculus framework to find a path- dependent version of the Hamilton-Jacobi-Bellman equation for stochastic control problems with path-dependence in the controls. We also prove a Dynamic Programming Principle for such problems. We apply our results to path-dependence of the delay type. We further study Stochastic Differential Games in this context.