A Class of Solvable Stochastic Investment Problems Involving Singular Controls

We consider a simple mathematical model for the following economic problem: A company wants to expand its production capacity in an uncertain market The investments needed are irreversible in the sense that if the market later drops, the company can not get the invested capital back by reducing the capacity Which strategy should the company follow in order to maximize the long term expected profit? In our model, the state of the market is described by a one dimensional, geometric Brownian motion We show that under some monotonocity conditions the optimal strategy can be described as a deflection off a “forbidden region”, and that the boundary of this region can be computed quite explicitly