Instantaneous control of Brownian motion with a positive lead time

Consider a storage system where the content is driven by a Brownian motion in the absence of control. At any time, one may increase or decrease the content at a cost proportional to the amount of adjustment. A decrease of the content takes effect immediately, while an increase is realized after a fixed lead time ℓ. Holding costs are incurred continuously over time and are a convex function of the content. The objective is to find a control policy that minimizes the expected present value of the total costs. Because of the positive lead time for upward adjustments, one needs to keep track of all of the outstanding upward adjustments as well as the actual content at time t as there may also be downward adjustments during [t, t + ℓ)—that is, the state of the system is a function on [0, ℓ]. We first extend the concept of L^Z-convexity to function spaces and establish the L^Z-convexity of the optimal cost function. We then derive various properties of the cost function and identify the structure of the optimal policy as a state-dependent two-sided reflection mapping making the minimum amount of adjustment necessary to keep the system states within a certain region.