On an Integral Equation for the Free Boundary of Stochastic, Irreversible Investment Problems

In this paper we derive a new handy integral equation for the free boundary of innite time horizon, continuous time, stochastic, irreversible investment problems with un- certainty modeled as a one-dimensional, regular difusion X0;x. The new integral equation allows to explicitly find the free boundary b() in some so far unsolved cases, as when X^(0;x) is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we frst show that b(X_0;x(t)) = l^*(t), with l^*(t) unique optional solution of a representation problem in the spirit of Bank-El Karoui [4]; then, thanks to such identification and the fact that l^* uniquely solves a backward stochastic equation, we nd the integral problem for the free boundary.