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On an Integral Equation for the Free-Boundary of Stochastic, Irreversible Investment Problems

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  • Giorgio Ferrari
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    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 mo-deled as a one-dimensional, regular diffusion X. The new integral equation allows to explicitly find the free-boundary 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 (2004); then, thanks to such 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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    File URL: http://arxiv.org/pdf/1211.0412
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1211.0412.

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    Date of creation: Nov 2012
    Date of revision: Jul 2013
    Handle: RePEc:arx:papers:1211.0412

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    1. Jan-Henrik Steg, 2012. "Irreversible investment in oligopoly," Finance and Stochastics, Springer, vol. 16(2), pages 207-224, April.
    2. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
    3. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
    4. Anders ûksendal, 2000. "Irreversible investment problems," Finance and Stochastics, Springer, vol. 4(2), pages 223-250.
    5. Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
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    Cited by:
    1. Ren\'e Aid & Salvatore Federico & Huy\^en Pham & Bertrand Villeneuve, 2014. "Explicit investment rules with time-to-build and uncertainty," Papers 1406.0055, arXiv.org.

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