**A Multiperiod Newsvendor Problem with Partially Observed
Demand**

**Abstract: **We consider a newsvendor
problem with partially observed Markovian demand. Demand is observed if it is less than
the inventory. Otherwise, only the event that it is larger than or equal to the inventory
is observed. These observations are used to update the demand distribution from one period
to the next. The state of the resulting dynamic programming equation is the current demand
distribution, which is generally infinite dimensional. We use unnormalized probabilities to
convert the nonlinear state transition equation to a linear one. This helps in proving the
existence of an optimal feedback ordering policy. So as to learn more about the demand,
the optimal order is set to exceed the myopic optimal order. The optimal cost decreases
as the demand distribution decreases in the hazard rate order. In a special case with
finitely many demand values, we characterize a near-optimal solution by establishing that
the value function is piecewise linear.