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

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  • Giorgio Ferrari

    (Bielefeld University)

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    Abstract

    In this paper we derive a new handy integral equation for the free boundary of in nite time horizon, continuous time, stochastic, irreversible investment problems with un- certainty modeled as a one-dimensional, regular difusion X0;x. The new integral equation allows to explicitly find the free boundary b() in some so far unsolved cases, as when X^(0;x) is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we frst show that b(X_0;x(t)) = l^*(t), with l^*(t) unique optional solution of a representation problem in the spirit of Bank-El Karoui [4]; then, thanks to such identification and the fact that l^* uniquely solves a backward stochastic equation, we nd the integral problem for the free boundary.

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    File URL: http://www.imw.uni-bielefeld.de/n/upload/paper/115f89503138416a242f40fb7d7f338e.pdf
    File Function: First version, 2012
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    Bibliographic Info

    Paper provided by Bielefeld University, Center for Mathematical Economics in its series Working Papers with number 471.

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    Length: 20 pages
    Date of creation: Nov 2012
    Date of revision:
    Handle: RePEc:bie:wpaper:471

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    1. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
    2. Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
    3. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
    4. Anders ├╗ksendal, 2000. "Irreversible investment problems," Finance and Stochastics, Springer, vol. 4(2), pages 223-250.
    5. Jan-Henrik Steg, 2012. "Irreversible investment in oligopoly," Finance and Stochastics, Springer, vol. 16(2), pages 207-224, April.
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