A Maximum Principle for Forward-Backward Stochastic Control Problem Under Partial Information

We study the stochastic optimal control problem for forward-backward stochastic systems with correlated state and observation noises. In our problem, we consider a more general situation. Firstly, the observation process depends explicitly on control. Secondly, the coefficients of SDE depend on the expectation of state. Using Girsanov's theorem and spike variation, we derive a maximum principle that is described by conditional mathematical expectation. The adjoint equation is a type of mean-field forward-backward stochastic differential equations and contains additional equations which are not necessary in the case of full information.