Existence of Optimal Simple Policies for Discounted-Cost Inventory and Cash Management in Continuous Time

We formulate a continuous-time, infinite-horizon, discounted-cost cash management model with both fixed and proportional transactions costs and with linear holding and penalty costs. We model the cumulative demand for cash by a Wiener process with drift and use the optimal control technique of "impulse control" to find sufficient conditions under which an optimal policy exists. We show that these conditions are always met. Therefore, we prove that there always exists an optimal policy for the cash management problem and that this policy is of a simple form. When the proportional transactions cost of decreasing the cash balance is sufficiently high, it is never optimal to decrease the cash balance. Then the cash management model degenerates to the inventory model. We prove that there always exists an optimal policy for the inventory model and that this policy is of a simple form. Under special cases of the cash management model we obtain analytic expressions for the parameters of the optimal policy.