A Linear-Quadratic Optimal Control Problem of Forward-Backward Stochastic Differential Equations With Partial Information

This paper studies a linear-quadratic optimal control problem derived by forward-backward stochastic differential equations, where the drift coefficient of the observation equation is linear with respect to the state $x$, and the observation noise is correlated with the state noise, in the sense that the cross-variation of the state and the observation is nonzero. A backward separation approach is introduced. Combining it with variational method and stochastic filtering, two optimality conditions and a feedback representation of optimal control are derived. Closed-form optimal solutions are obtained in some particular cases. As an application of the optimality conditions, a generalized recursive utility problem from financial markets is solved explicitly.