Dynamic Programming Principle and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations for Stochastic Recursive Control Problem with Non-Lipschitz Generator

Abstract

In this paper, we consider the stochastic recursive control problem under non-Lipschitz framework. More precisely, we assume that the generator of the backward stochastic differential equation that describes the cost functional is monotonic with respect to the first unknown variable and uniformly continuous in the second unknown variable. A dynamic programming principle is established by making use of a Girsanov transformation argument and the BSDE methods. The value function is then shown to be the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation via truncation methods, approximation techniques and the stability result of viscosity solutions.