Optimal Stochastic Control with Recursive Cost Functionals of Stochastic Differential Systems Reflected in a Domain

TL;DR

This paper investigates optimal control of reflected stochastic differential systems with recursive cost functionals, establishing the value function as the unique viscosity solution to a nonlinear PDE with boundary conditions.

Contribution

It introduces a novel analysis of recursive cost functionals for reflected stochastic systems and proves the uniqueness of the viscosity solution to the associated HJB equation.

Findings

Value function characterized as unique viscosity solution.

New estimates for reflected stochastic differential systems.

Connection between control problems and nonlinear PDEs.

Abstract

In this paper we study the optimal stochastic control problem for stochastic differential systems reflected in a domain. The cost functional is a recursive one, which is defined via generalized backward stochastic differential equations developed by Pardoux and Zhang [20]. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. For this, we also prove some new estimates for stochastic differential systems reflected in a domain.