  We study a stochastic, continuous time model on a finite horizon for a firm
that produces a single good. We model the production capacity as an Ito
diffusion controlled by a nondecreasing process representing the cumulative
investment. The firm aims to maximize its expected total net profit by choosing
the optimal investment process. That is a singular stochastic control problem.
We derive some first order conditions for optimality and we characterize the
optimal solution in terms of the base capacity process, i.e. the unique
solution of a representation problem in the spirit of Bank and El Karoui
(2004). We show that the base capacity is deterministic and it is identified
with the free boundary of the associated optimal stopping problem, when the
coefficients of the controlled diffusion are deterministic functions of time.
This is a novelty in the literature on finite horizon singular stochastic
control problems. As a subproduct this result allows us to obtain an integral
equation for the free boundary, which we explicitly solve in the infinite
horizon case for a Cobb-Douglas production function and constant coefficients
in the controlled capacity process.
