Average Optimal Singular Control and a Related Stopping Problem

Abstract
We observe a (μ, σ2) Brownian motion Xt. We can increase or decrease the value of the process paying r times the size of increase and l times the size of decrease. Holding costs are incurred continuously at a rate h(Zt) where Zt is the resulting process. We consider a convex holding cost function h which is either finite everywhere or finite inside a finite interval and equal to infinity outside this interval. The objective is to minimize average (per unit of time) expected cost. It is shown that the optimal behavior is characterized by two constants a and b, a < b, such that the optimal policy is to keep the process inside the interval [a, b] with minimal efforts. It is shown that to find the optimal interval [a, b], one must solve a free boundary problem for the second order differential equation. We give a probabilistic solution of the free boundary problem in terms of the value of a special game. This game is a generalization of the optimal stopping problem. There are two players observing the (μ, σ2) Brownian motion, and two regions where respectively the first and the second players have the right to stop the process. The one who stops the process pays a constant fee. In addition the first player pays to the second a certain accumulated amount of money. The existence of optimal minmax strategies is shown and it is shown that the region where neither the first nor the second player should stop the process corresponds to the optimal interval [a, b] in the original control problem.

This publication has 0 references indexed in Scilit: