A Maximum Principle For Partial Information Stochastic Optimal Control With Terminal State Constraints And Its Applications

Linear backward stochastic differential equations(BSDEs) were firstly introduced by Bismut [1] while he solved stochastic optimal control prob-lems, and then Pardoux and Peng [2] introduced the general nonlinear BS-DEs. A BSDE coupled with a forward stochastic differential equation(SDE) formulates a forward-backward stochastic differential equation(FBSDE). One kind of these equations is the so-called Hamiltonian system arising from maximum principles for stochastic optimal control problems. As we all know, generally speaking, for a forward-backward stochastic differen-tial equation only partial information can be obtained in most cases and there are some constraints imposed on the state. So it is very natural to study a partial information stochastic optimal control problem with state constraints.This paper is concerned with a partial information stochastic optimal control problem where the controlled system is described by a forward-backward stochastic differential equation(FBSDE), while at the terminal time, the forward state is adapted with partial information and constrained in a convex set. We introduce an equivalent backward control problem to solve it. By using Ekeland's variational principle, a necessary condition of the stochastic optimal control, i.e., a stochastic maximum principle is obtained. At the end of this paper, applications to a partial information stochastic linear control model with state constraints and recursive utility optimization problems are investigated by filtering equations.