Impulse Control of Brownian Motion

Consider a storage system, such as an inventory or cash fund, whose content fluctuates as a (μ, σ 2 ) Brownian motion in the absence of control. Holding costs are continuously incurred at a rate proportional to the storage level and we may cause the storage level to jump by any desired amount at any time except that the content must be kept nonnegative. Both positive and negative jumps entail fixed plus proportional costs, and our objective is to minimize expected discounted costs over an infinite planning horizon. A control band policy is one that enforces an upward jump to q whenever level zero is hit, and enforces a downward jump to Q whenever level S is hit (0 < q < Q < S). We prove the existence of an optimal control band policy and calculate explicitly the optimal values of the critical numbers (q, Q, S).