Optimal Control of a Mean-Reverting Inventory

Motivated by empirical observations, we assume that the inventory level of a company follows a mean reverting process. The objective of the management is to keep this inventory level as close as possible to a given target; there is a running cost associated with the difference between the actual inventory level and the target. If inventory deviates too much from the target, management may perform an intervention in the form of either a purchase or a sale of an amount of the goods. There are fixed and proportional costs associated with each intervention. The objective of this paper is to find the optimal inventory levels at which interventions should be performed as well as the magnitudes of the interventions, in order to minimize the total cost. We solve this problem by applying the theory of stochastic impulse control. Our analysis yields the optimal policy which at times exhibits a behavior that is not intuitive.