Optimal control of a Brownian storage system

Consider a storage system (such as an inventory or bank account) whose content fluctuates as a Brownian Motion X = {X(t), t [greater-or-equal, slanted] 0} in the absence of any control. Let Y = {Y(t), t [greater-or-equal, slanted] 0} and Z = {Z(t), t [greater-or-equal, slanted] 0} be non-decreasing, non-anticipating functionals representing the cumulative input to the system and cumulative output from the system respectively. Theproblem is to choose Y and Z so as to minimize expected discounted cost subject to the requirement that X(t) + Y(t) - Z(t) [greater-or-equal, slanted] 0 for all t [greater-or-equal, slanted] 0 almost surely. In our first formulation, we assume a proportional input cost, a linear holding cost, and a proportional output reward (or cost). We explicitly compute an optimal policy involving a single critical number. In our second formulation, the cumulative input Y is required to be a step function, and an additional fixed charge is incurred each time that an input jump occurs. We explicitly compute an optimal policy involving two critical numbers. Applications to inventory/production control and stochastic cash management are discussed.