  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 the latter. 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.
