## The Annals of Applied Probability

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

Giorgio Ferrari

#### 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.

#### Article information

Source
Ann. Appl. Probab., Volume 25, Number 1 (2015), 150-176.

Dates
First available in Project Euclid: 16 December 2014

https://projecteuclid.org/euclid.aoap/1418740182

Digital Object Identifier
doi:10.1214/13-AAP991

Mathematical Reviews number (MathSciNet)
MR3297769

Zentralblatt MATH identifier
1307.93455

#### Citation

Ferrari, Giorgio. On an integral equation for the free-boundary of stochastic, irreversible investment problems. Ann. Appl. Probab. 25 (2015), no. 1, 150--176. doi:10.1214/13-AAP991. https://projecteuclid.org/euclid.aoap/1418740182

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