Optimal stochastic control with recursive cost functionals of stochastic differential systems reflected in a domain

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.