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

TL;DR

This paper introduces a new integral equation for the free-boundary in stochastic irreversible investment problems, enabling explicit solutions in complex cases involving non-separable profit functions and specific diffusion processes.

Contribution

It derives a novel integral equation for the free-boundary using probabilistic methods, applicable to cases previously unsolvable, including certain Bessel and CEV processes.

Findings

Explicit free-boundary solutions for complex diffusion processes

New integral equation applicable to non-separable profit functions

Probabilistic approach simplifies free-boundary characterization

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 $b(X(t))=l^*(t)$, 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.