Partially Observed Time-Inconsistency Recursive Optimization Problem and Application

In this paper, we study a partially observed recursive optimization problem, which is time inconsistent in the sense that it does not admit the Bellman optimality principle. To obtain the desired results, we establish the Kalman–Bucy filtering equations for a family of parameterized forward and backward stochastic differential equations, which is a Hamiltonian system derived from the general maximum principle for the fully observed time‐inconsistency recursive optimization problem. By means of the backward separation technique, the equilibrium control for the partially observed time‐inconsistency recursive optimization problem is obtained, which is a feedback of the state filtering estimation. To illustrate the applications of theoretical results, an insurance premium policy problem under partial information is presented, and the observable equilibrium policy is derived explicitly.