Abstract: In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to nonobservation of demand, spoilage, misplacement, or theft of inventory. We study a partially observed inventory system where the demand is not observed, inventory level is noticed when it reaches zero, the unmet demand is lost, and replenishment orders must be decided so as to minimize the total discounted costs over a infinite horizon. This problem has an infinite-dimensional state space, and for it we establish the existence of a feedback policy when single-period costs are bounded or when the discount factor is sufficiently small. We also provide an approximately optimal feedback policy that uses a finite state representation.