Dynamic programming principle and Hamilton-Jacobi-Bellman equation under nonlinear expectation

Mingshang Hu,Shaolin Ji,Xiaojuan Li

doi:10.1051/cocv/2022019

Abstract

In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under G-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution to a type of fully nonlinear HJB equation.

Highlights

Soner et al [28] obtained an existence and uniqueness theorem for a new type of backward stochastic differential equation (BSDE) (2BSDE) under a family of nondominated probability measures
As pointed out in [11], the value function defined in (1.3) is a inf sup problem, which is known as the robust optimal control problem
The nonlinear part with respect to ∂x2xV in the HJB equation related to the optimal control problem (1.1)

Summary

Introduction

Motivated by the model uncertainty in finance, Peng [21–23] established the theory of G-expectation which is a consistent sublinear expectation and does not require a probability space. The representation of G-expectation as the supremum of expectations over a set of nondominated probability measures was obtained in [4, 15]. Due to this set of nondominated probability measures, the backward stochastic differential equation (BSDE for short) is completely different from the classical one. Soner et al [28] obtained an existence and uniqueness theorem for a new type of BSDE (2BSDE) under a family of nondominated probability measures. Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China

HU ET AL

Preliminaries

Stochastic optimal control problem

Dynamic programming principle

The viscosity solution to the HJB equation

Hoverever, by the proof of