Cost Equality and Inequality Results for a Partially Observed Stochastic Optimization Problem

A finite-horizon partially observed stochastic optimization problem is presented, where the underlying (or core) process subject to control is a finite-state discrete-time controlled semi-Markov vector process, the information pattern is classical, and times of control reset and noise corrupted observation occur at times of core process transition. Conditions for optimality are stated. The new problem formulation is shown to generalize several well-known problem formulations. Cost equality and inequality results associated with observation quality are determined, and the subsequent simplified dynamic programming equations obtained. Particular attention is given to cases where not all elements of the vector core process are either completely observed or completely unobserved.