On an integral equation for the free-boundary of stochastic, irreversible investment problems

Abstract

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 $b(X(t))=l^{\ast }(t)$, with $l^{\ast }$ 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 $l^{\ast }$ uniquely solves a backward stochastic equation, we find the integral problem for the free-boundary.

Published in at http://dx.doi.org/10.1214/13-AAP991 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (1)

• Continuous-Time Public Good Contribution under Uncertainty: A Stochastic Control Approach