  We consider an inventory system whose state is modeled by a L\'{e}vy process.
There are two types of costs--the running costs and the inventory control
costs. The running costs (also known as the holding/penalty costs) are incurred
continuously at some rate as a function of the inventory state. The inventory
control costs, incurred only when interventions of the inventory state are
placed, have both a fixed and a variable component. The objective is to
minimize the expectation of the infinite horizon discounted costs. We formulate
this as a stochastic impulse control problem. In our setting, we obtain
analytical results that are of significant implications. Specifically, we
establish the existence of the optimal control, and we provide the solution in
closed-form. More importantly, we prove the optimality of the simple control
band policy. Furthermore, we investigate the transient and the steady-state
behavior of the controlled process and the stochastic decomposition property.
