The Annals of Applied Probability

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

Giorgio Ferrari

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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.

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

First available in Project Euclid: 16 December 2014

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 91B70: Stochastic models 60H25: Random operators and equations [See also 47B80]

Integral equation free-boundary irreversible investment singular stochastic control optimal stopping one-dimensional diffusion Bank and El Karoui’s representation theorem base capacity


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.

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