#### Partially Observed Inventory Systems: The Case of
Zero-Balance Walk

**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.