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We study the problem of optimally managing an inventory with unknown demandtrend. Our formulation leads to a stochastic control problem under partialobservation, in which a Brownian motion with non-observable drift can besingularly controlled in both an upward and downward direction. We first derivethe equivalent separated problem under full information, with state-spacecomponents given by the Brownian motion and the filtering estimate of itsunknown drift, and we then completely solve this latter problem. Our approachuses the transition amongst three different but equivalent problemformulations, links between two-dimensional bounded-variation stochasticcontrol problems and games of optimal stopping, and probabilistic methods incombination with refined viscosity theory arguments. We show substantialregularity of (a transformed version of) the value function, we construct anoptimal control rule, and we show that the free boundaries delineating(transformed) action and inaction regions are bounded globally Lipschitzcontinuous functions. To our knowledge this is the first time that such aproblem has been solved in the literature.

Two-sided Singular Control of an Inventory with Unknown Demand Trend (Extended Version)