Optimal control problem of backward stochastic differential delay equation under partial information

In this paper, we study the optimal control problem of backward stochastic differential delay equation under partial information. A class of time-advanced stochastic differential equations (ASDEs) is introduced as the adjoint process via duality relation. By means of ASDEs, we suggest a maximum principle of this problem and obtain necessary and sufficient conditions of optimality. We apply the theoretical results to study linear quadratic optimal control problem and obtain a forward-backward stochastic differential filtering equation (FBSDFE) with time-advanced forward equation that leads to optimal control. At the same time, some auxiliary filtering results and properties about time-advanced SDEs are also presented.