Explicit Solution to an Irreversible Investment Model with a stochastic production capacity

This paper studies the problem of a company which expands its stochastic production capacity in irreversible investments by purchasing capital at a given price. The profit production function is of a very general form satisfying minimal standard assumptions. The objective of the company is to find optimal production decisions to maximize its expected total net profit in an infinite horizon. The resulting dynamic programming principle is a singular stochastic control problem. The value function is analyzed in great detail relying on viscosity solutions of the associated Bellman variational inequation: we state several general properties and in particular regularity results on the value function. We provide a complete solution with explicit expressions of the value function and the optimal control: the firm invests in capital so as to maintain its capacity above a certain threshold. This boundary may be computed quite explicitly.