Drift Rate Control of a Brownian Processing System

A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate ? that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ=dX-?(Z)?dt+dL-dU, where X is a (0,s) Brownian motion, and L and U are increasing processes that enforce a lower reflecting barrier at Z=0 and an upper reflecting barrier at Z=b, respectively. The cumulative cost process increases according to the differential relationship d?=c(?(Z))?dt+p?dU, where c(