Path-dependent Hamilton-Jacobi-Bellman equations related to controlled   stochastic functional differential systems
  In this paper, a stochastic optimal control problem is investigated in which the system is governed by a stochastic functional differential equation. In the framework of functional It\^o calculus, we build the dynamic programming principle and the related Path-dependent Hamilton-Jacobi-Bellman (HJB) equation. We prove that the value function is the viscosity solution of the Path-dependent HJB equation. 
Shaolin Ji and Shuzhen Yang
math.OC math.PR