Optimality of (s, S) policies with nonlinear processes

It is observed empirically that mean-reverting processes are more realistic in modeling the inventory level of a company. In a typical mean-reverting process, the inventory level is assumed to be linearly dependent on the deviation of the inventory level from the long-term mean. However, when the deviation is large, it is reasonable to assume that the company might want to increase the intensity of interference to the inventory level significantly rather than in a linear manner. In this paper, we attempt to model inventory replenishment as a nonlinear continuous feedback process. We study both infinite horizon discounted cost and the long-run average cost, and derive the corresponding optimal (s, S) policy.