Two-sided Singular Control of an Inventory with Unknown Demand Trend (Extended Version)

TL;DR

This paper addresses the complex problem of optimally controlling an inventory with an unknown demand trend by developing a comprehensive stochastic control framework and solving it using advanced probabilistic and viscosity methods.

Contribution

It introduces the first complete solution to a two-sided singular control problem with partial observation and unknown demand trend, using novel problem reformulations and mathematical techniques.

Findings

Derived the separated full-information problem with filtering estimates.

Proved regularity and Lipschitz continuity of the value function boundaries.

Constructed an explicit optimal control strategy.

Abstract

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.