Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

¹ School of Finance, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
² School of Mathematical Sciences, Fudan University, Shanghai 200433, China

^a Corresponding author: Qi Zhang - qzh@fudan.edu.cn

Received: 25 September 2015
Revised: 7 October 2016
Accepted: 2 October 2017

Abstract

In this work we study the stochastic recursive control problem, in which the aggregator (or 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 for 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.