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.
