  A new approach to solve the continuous-time stochastic inventory problem
using the fluctuation theory of Levy 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 Levy processes.
For the case with a fixed cost and a general spectrally positive Levy 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 Levy process in the beta-family with jumps of infinite
activity are provided to confirm the analytical results. Furthermore, the case
with no fixed ordering costs is studied.
