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.
Citation
Giorgio Ferrari. "On an integral equation for the free-boundary of stochastic, irreversible investment problems." Ann. Appl. Probab. 25 (1) 150 - 176, February 2015. https://doi.org/10.1214/13-AAP991
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