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A Stochastic Reversible Investment Problem on a Finite-Time Horizon: Free Boundary Analysis

Author

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  • de Angelis, Tiziano

    (Center for Mathematical Economics, Bielefeld University)

  • Ferrari, Giorgio

    (Center for Mathematical Economics, Bielefeld University)

Abstract

We study a continuous-time, finite horizon optimal stochastic reversible investment problem for a firm producing a single good. The production capacity is modeled as a onedimensional,time-homogeneous, linear diffusion controlled by a bounded variation process which represents the cumulative investment-disinvestment strategy. We associate to the investment-disinvestment problem a zero-sum optimal stopping game and characterize its value function through a free boundary problem with two moving boundaries. These are continuous, bounded and monotone curves that solve a system of non-linear integral equations of Volterra type. The optimal investment-disinvestment strategy is then shown to be a diffusion reflected at the two boundaries.

Suggested Citation

  • 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.
    • Handle: RePEc:bie:wpaper:477
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    Cited by:

    1. Tiziano De Angelis & Fabien Gensbittel & St'ephane Villeneuve, 2017. "A Dynkin game on assets with incomplete information on the return," Papers 1705.07352, arXiv.org, revised May 2019.
    2. Federico, Salvatore & Ferrari, Giorgio & Rodosthenous, Neofytos, 2021. "Two-Sided Singular Control of an Inventory with Unknown Demand Trend," Center for Mathematical Economics Working Papers 643, Center for Mathematical Economics, Bielefeld University.
    3. de Angelis, Tiziano & Federico, Salvatore & Ferrari, Giorgio, 2016. "On the Optimal Boundary of a Three-Dimensional Singular Stochastic Control Problem Arising in Irreversible Investment," Center for Mathematical Economics Working Papers 509, Center for Mathematical Economics, Bielefeld University.
    4. de Angelis, Tiziano & Ferrari, Giorgio & Moriarty, John, 2016. "A solvable two-dimensional degenerate singular stochastic control problem with non convex costs," Center for Mathematical Economics Working Papers 531, Center for Mathematical Economics, Bielefeld University.
    5. Dammann, Felix & Rodosthenous, Néofytos & Villeneuve, Stéphane, 2023. "Debt management game and debt ceiling," TSE Working Papers 23-1430, Toulouse School of Economics (TSE).
    6. Tiziano De Angelis & Fabien Gensbittel & Stephane Villeneuve, 2021. "A Dynkin Game on Assets with Incomplete Information on the Return," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 28-60, February.
    7. Tiziano De Angelis & Erik Ekstrom, 2016. "The dividend problem with a finite horizon," Papers 1609.01655, arXiv.org, revised Nov 2017.
    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. 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.
    10. Randall Martyr, 2014. "Solving finite time horizon Dynkin games by optimal switching," Papers 1411.4438, arXiv.org, revised Jan 2016.
    11. de Angelis, Tiziano & Ferrari, Giorgio, 2016. "Stochastic nonzero-sum games: a new connection between singular control and optimal stopping," Center for Mathematical Economics Working Papers 565, Center for Mathematical Economics, Bielefeld University.
    12. Tiziano De Angelis & Salvatore Federico & Giorgio Ferrari, 2014. "Optimal Boundary Surface for Irreversible Investment with Stochastic Costs," Papers 1406.4297, arXiv.org, revised Jan 2017.
    13. de Angelis, Tiziano & Ferrari, Giorgio & Martyr, Randall & Moriarty, John, 2016. "Optimal entry to an irreversible investment plan with non convex costs," Center for Mathematical Economics Working Papers 566, Center for Mathematical Economics, Bielefeld University.
    14. Salvatore Federico & Giorgio Ferrari & Neofytos Rodosthenous, 2021. "Two-sided Singular Control of an Inventory with Unknown Demand Trend (Extended Version)," Papers 2102.11555, arXiv.org, revised Nov 2022.
    15. Giorgio Ferrari & Tiziano Vargiolu, 2020. "On the singular control of exchange rates," Annals of Operations Research, Springer, vol. 292(2), pages 795-832, September.

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    Keywords

    zero-sum optimal stoppinggames; reversible investment; free boundary problems; singular stochastic control; Skorokhod reflection problem.;
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