Dynamic programming principle for delayed stochastic recursive optimal control problem and HJB equation with non-Lipschitz generator

TL;DR

This paper develops a dynamic programming framework for delayed stochastic recursive control problems with non-Lipschitz generators, linking the value function to viscosity solutions of HJB equations and applying it to a financial utility optimization.

Contribution

It establishes the dynamic programming principle and connects the value function to viscosity solutions for a class of delayed stochastic control problems with non-Lipschitz generators.

Findings

Established the dynamic programming principle for the problem.

Proved the value function is a viscosity solution of the HJB equation.

Applied the theory to a consumption-investment model with Epstein-Zin utility.

Abstract

In this paper, we study the delayed stochastic recursive optimal control problem with a non-Lipschitz generator, in which both the dynamics of the control system and the recursive cost functional depend on the past path segment of the state process in a general form. First, the dynamic programming principle for this control problem is obtained. Then, by the generalized comparison theorem of backward stochastic differential equations and the stability of viscosity solutions, we establish the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Finally, an application to the consumption-investment problem under the delayed continuous-time Epstein-Zin utility with a non-Lipschitz generator is presented.