This paper examines a Markovian model for the optimal irreversible investment

problem of a firm aiming at minimizing total expected costs of production. We model market

uncertainty and the cost of investment per unit of production capacity as two independent

one-dimensional regular diffusions, and we consider a general convex running cost function.

The optimization problem is set as a three-dimensional degenerate singular stochastic control

problem.

We provide the optimal control as the solution of a Skorohod reflection problem at a suitable

free-boundary surface. Such boundary arises from the analysis of a family of two-dimensional

parameter-dependent optimal stopping problems and it is characterized in terms of the family of

unique continuous solutions to parameter-dependent nonlinear integral equations of Fredholm

type.