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We study the problem of optimally managing an inventory with unknown demand trend. Our

formulation leads to a stochastic control problem under partial observation, in which a Brownian motion

with non-observable drift can be singularly controlled in both an upward and downward direction. We

first derive the equivalent separated problem under full information, with state-space components given

by the Brownian motion and the filtering estimate of its unknown drift, and we then completely solve

this latter problem. Our approach uses the transition amongst three different but equivalent problem

formulations, links between two-dimensional bounded-variation stochastic control problems and games

of optimal stopping, and probabilistic methods in combination with refined viscosity theory arguments.

We show substantial regularity of (a transformed version of) the value function, we construct an optimal

control rule, and we show that the free boundaries delineating (transformed) action and inaction regions

are bounded globally Lipschitz continuous functions. To our knowledge this is the first time that such

a problem has been solved in the literature.


MSC2020 subject classifications: Primary 93E20, 93E11, 91A55, 49J40, 90B05.