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 rst derive the equivalent separated problem under full information, with state-space components given by the Brownian motion and the ltering estimate of its unknown drift, and we then completely solve this latter problem. Our approach uses the transition amongst three di erent but equivalent problem formulations, links between two-dimensional bounded-variation stochastic control problems and games of optimal stopping, and probabilistic methods in combination with re ned 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 rst time that such a problem has been solved in the literature.
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