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

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  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

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 X0;x. The new integral equation allows to explicitly find the free boundary b(.) in some so far unsolved cases, as when X0;x is a three-dimensional Bessel process or a CEV process. Our result follows from purely probabilistic arguments. Indeed, we first show that b(X0;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 find the integral problem for the free boundary.

Suggested Citation

  • Ferrari, Giorgio, 2014. "On an integral equation for the free boundary of stochastic, irreversible investment problems," Center for Mathematical Economics Working Papers 471, Center for Mathematical Economics, Bielefeld University.
  • Handle: RePEc:bie:wpaper:471
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    References listed on IDEAS

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    1. Maria B. Chiarolla & Giorgio Ferrari, 2011. "Identifying the Free Boundary of a Stochastic, Irreversible Investment Problem via the Bank-El Karoui Representation Theorem," Papers 1108.4886, arXiv.org, revised Dec 2013.
    2. Anders ûksendal, 2000. "Irreversible investment problems," Finance and Stochastics, Springer, vol. 4(2), pages 223-250.
    3. Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
    4. Jan-Henrik Steg, 2012. "Irreversible investment in oligopoly," Finance and Stochastics, Springer, vol. 16(2), pages 207-224, April.
    5. Ioannis Karatzas & Fridrik M. Baldursson, 1996. "Irreversible investment and industry equilibrium (*)," Finance and Stochastics, Springer, vol. 1(1), pages 69-89.
    6. Maria B. Chiarolla & Giorgio Ferrari & Frank Riedel, 2012. "Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources," Papers 1203.3757, arXiv.org, revised Aug 2013.
    7. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
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    Citations

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    Cited by:

    1. Ferrari, Giorgio & Salminen, Paavo, 2016. "Irreversible Investment under Lévy Uncertainty: an Equation for the Optimal Boundary," Center for Mathematical Economics Working Papers 530, Center for Mathematical Economics, Bielefeld University.
    2. Aïd, René & Federico, Salvatore & Pham, Huyên & Villeneuve, Bertrand, 2015. "Explicit investment rules with time-to-build and uncertainty," Journal of Economic Dynamics and Control, Elsevier, vol. 51(C), pages 240-256.
    3. de Angelis, Tiziano & Ferrari, Giorgio, 2014. "A Stochastic Reversible Investment Problem on a Finite-Time Horizon: Free Boundary Analysis," Center for Mathematical Economics Working Papers 477, Center for Mathematical Economics, Bielefeld University.
    4. Giorgio Ferrari & Paavo Salminen, 2014. "Irreversible Investment under L\'evy Uncertainty: an Equation for the Optimal Boundary," Papers 1411.2395, arXiv.org.
    5. Ferrari, Giorgio, 2016. "Controlling public debt without forgetting Inflation," Center for Mathematical Economics Working Papers 564, Center for Mathematical Economics, Bielefeld University.
    6. Tiziano De Angelis & Giorgio Ferrari & John Moriarty, 2014. "A Non Convex Singular Stochastic Control Problem and its Related Optimal Stopping Boundaries," Papers 1405.2442, arXiv.org, revised Nov 2014.
    7. Giorgio Ferrari, 2016. "On the Optimal Management of Public Debt: a Singular Stochastic Control Problem," Papers 1607.04153, arXiv.org, revised Dec 2017.
    8. Chiarolla, Maria B. & Ferrari, Giorgio & Stabile, Gabriele, 2015. "Optimal dynamic procurement policies for a storable commodity with Lévy prices and convex holding costs," European Journal of Operational Research, Elsevier, vol. 247(3), pages 847-858.
    9. Giorgio Ferrari & Frank Riedel & Jan-Henrik Steg, 2013. "Continuous-Time Public Good Contribution under Uncertainty: A Stochastic Control Approach," Papers 1307.2849, arXiv.org, revised Oct 2015.
    10. Salvatore Federico & Mauro Rosestolato & Elisa Tacconi, 2018. "Irreversible investment with fixed adjustment costs: a stochastic impulse control approach," Papers 1801.04491, arXiv.org.
    11. Ferrari, Giorgio & Riedel, Frank & Steg, Jan-Henrik, 2016. "Continuous-Time Public Good Contribution under Uncertainty," Center for Mathematical Economics Working Papers 485, Center for Mathematical Economics, Bielefeld University.

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