G-BSDEs with non-Lipschitz coefficients and the corresponding stochastic recursive optimal control problem
Wei He, Qiangjun Tang
TL;DR
This work extends stochastic recursive control to $G$-Brownian motion settings with non-Lipschitz aggregators by employing Lipschitz approximations to establish existence, uniqueness, and a comparison principle for $G$-BSDEs. It proves a dynamic programming principle and connects the value function to the unique viscosity solution of a fully nonlinear $G$-HJB equation, using stability results in the viscosity solution framework. The analysis enables robust recursive utilities under model uncertainty, demonstrated via a continuous-time Epstein–Zin utility example. Collectively, the paper advances the theory of stochastic control under $G$-frames and non-Lipschitz generators, with implications for finance under volatility and model ambiguity.
Abstract
In this paper, we study the existence and uniqueness of solutions to a class of non-Lipschitz G-BSDEs and the corresponding stochastic recursive optimal control problem. More precisely, we suppose that the generator of G-BSDE is uniformly continuous and monotonic with respect to the first unknown variable. Using the comparison theorem for G-BSDE and the stability of viscosity solutions, we establish the dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation.We provide an example of continuous time Epstein-Zin utility to demonstrate the application of our study.