10.1214/13-AAP991
Ferrari, Giorgio
Giorgio
Ferrari
On an integral equation for the free-boundary of stochastic, irreversible investment problems
Institute Of Mathematical Statistics
2015
2015-02-16T10:39:54Z
2018-07-24T13:01:24Z
journal_article
https://pub.uni-bielefeld.de/record/2718995
https://pub.uni-bielefeld.de/record/2718995.json
In this paper, we derive a new handy integral equation for the freeboundary 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 freeboundary b(.) 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.