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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. Boetius, Frederik & Kohlmann, Michael, 1998. "Connections between optimal stopping and singular stochastic control," Stochastic Processes and their Applications, Elsevier, vol. 77(2), pages 253-281, September.
    2. Frank Riedel & Xia Su, 2011. "On irreversible investment," Finance and Stochastics, Springer, vol. 15(4), pages 607-633, December.
    3. Jan-Henrik Steg, 2012. "Irreversible investment in oligopoly," Finance and Stochastics, Springer, vol. 16(2), pages 207-224, April.
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    5. 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.
    6. 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.
    7. Avinash K. Dixit & Robert S. Pindyck, 1994. "Investment under Uncertainty," Economics Books, Princeton University Press, edition 1, number 5474.
    8. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14, April.
    9. Maria B. Chiarolla & Ulrich G. Haussmann, 2005. "Explicit Solution of a Stochastic, Irreversible Investment Problem and Its Moving Threshold," Mathematics of Operations Research, INFORMS, vol. 30(1), pages 91-108, February.
    10. Anders ûksendal, 2000. "Irreversible investment problems," Finance and Stochastics, Springer, vol. 4(2), pages 223-250.
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    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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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. 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.
    3. Ferrari, Giorgio, 2016. "Controlling public debt without forgetting Inflation," Center for Mathematical Economics Working Papers 564, Center for Mathematical Economics, Bielefeld University.
    4. Torben Koch & Tiziano Vargiolu, 2019. "Optimal Installation of Solar Panels with Price Impact: a Solvable Singular Stochastic Control Problem," Papers 1911.04223, arXiv.org.
    5. 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.
    6. 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.
    7. 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.
    8. Tiziano De Angelis & Salvatore Federico & Giorgio Ferrari, 2017. "Optimal Boundary Surface for Irreversible Investment with Stochastic Costs," Mathematics of Operations Research, INFORMS, vol. 42(4), pages 1135-1161, November.
    9. 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.
    10. Salvatore Federico & Mauro Rosestolato & Elisa Tacconi, 2018. "Irreversible investment with fixed adjustment costs: a stochastic impulse control approach," Papers 1801.04491, arXiv.org, revised Feb 2019.
    11. Aïd, René & Basei, Matteo & Ferrari, Giorgio, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Center for Mathematical Economics Working Papers 679, Center for Mathematical Economics, Bielefeld University.
    12. Peter Bank & Yan Dolinsky, 2018. "Continuous-time Duality for Super-replication with Transient Price Impact," Papers 1808.09807, arXiv.org, revised May 2019.
    13. Giorgio Ferrari & Hanwu Li & Frank Riedel, 2020. "A Knightian Irreversible Investment Problem," Papers 2003.14359, arXiv.org, revised Apr 2020.
    14. Peter Bank & David Besslich, 2018. "Modelling information flows by Meyer-$\sigma$-fields in the singular stochastic control problem of irreversible investment," Papers 1810.08495, arXiv.org, revised Mar 2020.
    15. Junkee Jeon & Geonwoo Kim, 2020. "An Integral Equation Approach to the Irreversible Investment Problem with a Finite Horizon," Mathematics, MDPI, vol. 8(11), pages 1-10, November.
    16. De Angelis, Tiziano & Ferrari, Giorgio, 2014. "A stochastic partially reversible investment problem on a finite time-horizon: Free-boundary analysis," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4080-4119.
    17. Giorgio Ferrari & Paavo Salminen, 2014. "Irreversible Investment under L\'evy Uncertainty: an Equation for the Optimal Boundary," Papers 1411.2395, arXiv.org.
    18. Giorgio Ferrari, 2016. "On the Optimal Management of Public Debt: a Singular Stochastic Control Problem," Papers 1607.04153, arXiv.org, revised Dec 2017.
    19. Andrea Bovo & Tiziano De Angelis & Jan Palczewski, 2023. "Stopper vs. singular-controller games with degenerate diffusions," Papers 2312.00613, arXiv.org.
    20. Andrea Bovo & Tiziano De Angelis & Jan Palczewski, 2023. "Zero-sum stopper vs. singular-controller games with constrained control directions," Papers 2306.05113, arXiv.org, revised Feb 2024.
    21. Alain Bensoussan & Benoît Chevalier-Roignant, 2019. "Sequential Capacity Expansion Options," Operations Research, INFORMS, vol. 67(1), pages 33-57, January.
    22. 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.
    23. Ren'e Aid & Matteo Basei & Giorgio Ferrari, 2023. "A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy," Papers 2305.00541, arXiv.org.
    24. Dianetti, Jodi, 2023. "Linear-Quadratic-Singular Stochastic Differential Games and Applications," Center for Mathematical Economics Working Papers 678, Center for Mathematical Economics, Bielefeld University.

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