Storage model with discontinuous holding cost

We consider a Brownian storage system with stepwise holding cost and linear cost of disposal. There are no limits on the rate of disposal. We seek a policy which minimizes total discounted cost on an infinite interval. It is proved that the optimal policy is characterized by two points: one is a reflecting barrier, the second is a point of instantaneous displacement. When the process reaches the second point it must be instantly moved to the first point and kept below the first point with minimal efforts thereafter.