INSTANTANEOUS CONTROL OF BROWNIAN-MOTION

A controller continuously monitors a storage system, such as an inventory or bank account, whose content Z equals left brace Z//t,t greater than equivalent to 0 right brace fluctuates as a ( mu , sigma **2) Brownian motion in the absence of control. Holding costs are incurred continuously at rate h(Z//t). At any time, the controller may instantaneously increase the content of the system, incurring a proportional cost of r times the size of the increase, or decrease the content at a cost of l times the size of the decrease. The authors consider the case where h is convex on a finite interval left bracket alpha , beta right bracket and h equals infinity outside this interval. The objective is to minimize the expected discounted sum of holding costs and control costs over an infinite planning horizon. It is shown that there exists an optimal control limit policy, characterized by two parameters a and b ( alpha less than equivalent to a less than b less than equivalent to beta ). Roughly speaking, this policy exerts the minimum amounts of control sufficient to keep Z//t an element of left bracket a,b right bracket for all t greater than equivalent to 0. Put another way, the optimal control limit policy imposes on Z a lower reflecting barrier at a and an upper reflecting barrier at b. 12 refs.