A Solvable One-Dimensional Model of a Diffusion Inventory System

We analyse the optimal ordering policy for impulse control of a one-product inventory system subject to a demand modelised by a diffusion process. The purpose is to minimize the expected discounted cost that includes a fixed set-up cost and linear costs of purchase, storage and shortage. The optimal cost is explicitly obtained as the smoothest solution of a Quasi-Variational Inequality derived from the optimal principle of Dynamic Programming. The optimal (s, S) policy is determined as the unique solution of a system of algebraic equations.