An optimal control problem for functional forward-backward stochastic systems and related Path-dependent HJB equations

TL;DR

This paper investigates a stochastic recursive optimal control problem governed by functional forward-backward stochastic differential equations, establishing the dynamic programming principle and connecting the value function to a Path-dependent HJB equation.

Contribution

It introduces a novel framework for control problems with functional FBSDEs and derives the associated Path-dependent HJB equations using functional Itô calculus.

Findings

Established the dynamic programming principle for the control problem.

Proved the value function is the viscosity solution of the Path-dependent HJB equation.

Developed a stochastic verification theorem for the smooth case.

Abstract

In this paper, we study a stochastic recursive optimal control problem in which the system is governed by a functional forward-backward stochastic differential equation. Under standard assumptions, we establish the dynamic programming principle and the related Path-dependent Hamilton-Jacobi-Bellman (HJB) equation in the framework of functional It\^o calculus. The stochastic verification theorem for the smooth case is proved. Finally, we show that the value function is the viscosity solution of the Path-dependent HJB equation.