Optimization of Observations: a Stochastic Control Approach

We study a stochastic control problem for the optimization of observations in a partially observable stochastic system. Using a method of discontinuous time transformation, we associate with the original problem with unbounded controls a problem that has bounded controls. This latter problem allows us to construct nearly optimal nonanticipative Lipschitz Markov controls with finite observation power for the original problem. Since the controlled observation equation may degenerate, we also derive a corresponding filtering result and show a separation property of the optimal controls.