Path‐dependent Hamilton–Jacobi–Bellman equations related to controlled stochastic functional differential systems

Abstract

SummaryIn 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. Copyright © 2013 John Wiley & Sons, Ltd.

Stochastic Control and Differential Games with Path-Dependent Influence of Controls on Dynamics and Running Cost

“…Stochastic control has been already extended to consider path-dependence of dynamics and running cost with respect to the state variable x, see, for example, Fournié [2010], Xu [2013] and Ji et al [2015]. We say the control problem in this case exihbits path-depndence in the state variable.…”