Two-Sided Singular Control of an Inventory with Unknown Demand Trend
Giorgio Ferrari and
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Salvatore Federico: Center for Mathematical Economics, Bielefeld University
Giorgio Ferrari: Center for Mathematical Economics, Bielefeld University
Neofytos Rodosthenous: Center for Mathematical Economics, Bielefeld University
No 643, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University
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.
Keywords: bounded-variation stochastic control; partial observation; inventory management; Dynkin games; free boundaries (search for similar items in EconPapers)
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