Optimal replenishment in a Brownian Motion EOQ model with hysteretic parameter changes

We consider an EOQ model in which the content level is modelled by a Brownian Motion (BM) with state-dependent drift and diffusion parameters. There are either one or two prespecified switchover threshold values (cases 1 and 2) such that the drift and the volatility change whenever the critical threshold is reached (in case 1) or whenever the upper threshold is reached from below or the lower one from above (in case 2). The controller places an order of fixed size every time the content level reaches 0. We derive all the relevant cost functionals for the discounted case with infinite horizon and for the long-run average case. These explicit results are used for finding the optimal replenishment level in the long-run average case.