Dynamic Programming Principle and Associated Hamilton-Jacobi-Bellman Equation for Stochastic Recursive Control Problem with Non-Lipschitz Aggregator

TL;DR

This paper develops a dynamic programming framework for stochastic recursive control problems with non-Lipschitz, monotonic aggregators, establishing the connection to Hamilton-Jacobi-Bellman equations and demonstrating applications to Epstein-Zin utility.

Contribution

It extends the theory of stochastic recursive control to cases with non-Lipschitz generators, linking value functions to viscosity solutions of HJB equations.

Findings

Established dynamic programming principle for non-Lipschitz generators.

Proved the connection between value functions and viscosity solutions.

Applied results to Epstein-Zin utility control problem.

Abstract

In this work we study the stochastic recursive control problem, in which the aggregator (or called generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem of backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein-Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.