Inventory Control for Spectrally Positive Lévy Demand Processes

A new approach to solve the continuous‐time stochastic inventory problem using the fluctuation theory of Lévy processes is developed. This approach involves the recent developments of the scale function that is capable of expressing many fluctuation identities of spectrally one‐sided Lévy processes. For the case with a fixed cost and a general spectrally positive Lévy demand process, we show the optimality of an (s, S)‐policy. The optimal policy and the value function are concisely expressed via the scale function. Numerical examples under a Lévy process in the β‐family with jumps of infinite activity are provided to confirm the analytical results. Furthermore, the case with no fixed ordering costs is studied.