Instantaneous Control of Brownian Motion

Abstract

A controller continuously monitors a storage system, such as an inventory or bank account, whose content Z = {Zt,t≥0} fluctuates as a (μ, σ2) Brownian motion in the absence of control. Holding costs are incurred continuously at rate h(Zt). At any time, the controller may instantaneously increase the content of the system, incurring a proportional cost of r times the size of the increase, or decrease the content at a cost of l times the size of the decrease. We consider the case where h is convex on a finite interval [α, β] and h = ∞ outside this interval. The objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon. It is shown that there exists an optimal control limit policy, characterized by two parameters a and b (α ≤ a < b ≤ β). Roughly speaking, this policy exerts the minimum amounts of control sufficient to keep Zt ∈ [a, b] for all t ≥ 0. Put another way, the optimal control limit policy imposes on Z a lower reflecting barrier at a and an upper reflecting barrier at b. The optimality of a particular control limit policy is proved directly, with heavy reliance on the change of variable formula for semimartingales. We do not give a full-blown algorithm for construction of the optimal control limits, but a computational scheme could easily be developed from our constructive proof of existence.

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