Ergodic control for a mean reverting inventory model

In this paper, an inventory control problem with a mean reverting inventory model is considered. The demand is assumed to follow a continuous diffusion process and a mean-reverting process which will take into account of the demand dependent of the inventory level. By choosing when and how much to stock, the objective is to minimize the long-run average cost, which consists of transaction cost for each replenishment, holding and shortage costs associated with the inventory level. An approach for deriving the average cost value of infinite time horizon is developed. By applying the theory of stochastic impulse control, we show that a unique (s, S) policy is indeed optimal. The main contribution of this work is to present a method to derive the (s, S) policy and hence the minimal long-run average cost.