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 Itô calculus, we build the dynamic programming principle and the related path‐dependent Hamilton–Jacobi–Bellman equation. We prove that the value function is the viscosity solution of the path‐dependent Hamilton–Jacobi–Bellman equation.