  In this paper, we derive a new handy integral equation for the free-boundary
of infinite time horizon, continuous time, stochastic, irreversible investment
problems with uncertainty modeled as a one-dimensional, regular diffusion $X$.
The new integral equation allows to explicitly find the free-boundary
$b(\cdot)$ in some so far unsolved cases, as when the operating profit function
is not multiplicatively separable and $X$ is a three-dimensional Bessel process
or a CEV process. Our result follows from purely probabilistic arguments.
Indeed, we first show that $b(X(t))=l^*(t)$, with $l^*$ the unique optional
solution of a representation problem in the spirit of Bank-El Karoui [Ann.
Probab. 32 (2004) 1030-1067]; then, thanks to such an identification and the
fact that $l^*$ uniquely solves a backward stochastic equation, we find the
integral problem for the free-boundary.
