Existence and Uniqueness of Solutions for a Partially Observed Stochastic Control Problem

We develop a general methodology for a partially observed stochastic control problem. The dynamics is governed by a discrete-time Markov process. We describe an application to an inventory system with possibility of shrinkage, and introduce unnormalized conditional probabilities to transform the nonlinear state evolution into a linear one. We then prove the existence and uniqueness of the solution for the Bellman equation between the floor and the ceiling functions in the case of unbounded costs.