In this talk, I will introduce a partially observed recursive

optimization problem which is time inconsistent and  does not admit

the Bellman optimality principle.

 To obtain the desired optimization results, we first establish the Kalman-Bucy filtering equations for a family of parameterized forward and

backward stochastic differential equations, which is a kind of Hamiltonian system derived from the general maximum principle for the fully observed time inconsistency recursive optimization problem. And then, by means of the backward separation technique, we give the equilibrium control for the partially observed time inconsistency recursive optimization problem, which is the feedback representation of the state filtering estimation.

At last, to illustrate the applications of the theoretical results, we present an insurance premium policy problem under partial information and derive a closed-form solution of the observable equilibrium policy.