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Optimal Boundary Surface for Irreversible Investment with Stochastic Costs

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  • Tiziano De Angelis
  • Salvatore Federico
  • Giorgio Ferrari

Abstract

This paper examines a Markovian model for the optimal irreversible investment problem of a firm aiming at minimizing total expected costs of production. We model market uncertainty and the cost of investment per unit of production capacity as two independent one-dimensional regular diffusions, and we consider a general convex running cost function. The optimization problem is set as a three-dimensional degenerate singular stochastic control problem. We provide the optimal control as the solution of a Skorohod reflection problem at a suitable boundary surface. Such boundary arises from the analysis of a family of two-dimensional parameter-dependent optimal stopping problems and it is characterized in terms of the family of unique continuous solutions to parameter-dependent nonlinear integral equations of Fredholm type.

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  • 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.
    • Handle: RePEc:arx:papers:1406.4297
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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.
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    4. 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.
    5. 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.
    6. Pindyck, Robert S, 1988. "Irreversible Investment, Capacity Choice, and the Value of the Firm," American Economic Review, American Economic Association, vol. 78(5), pages 969-985, December.
    7. Robert McDonald & Daniel Siegel, 1986. "The Value of Waiting to Invest," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 101(4), pages 707-727.
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    9. Guo, Xin & Pham, Huyên, 2005. "Optimal partially reversible investment with entry decision and general production function," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 705-736, May.
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    Citations

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    Cited by:

    1. Felix Dammann & Giorgio Ferrari, 2023. "Optimal execution with multiplicative price impact and incomplete information on the return," Finance and Stochastics, Springer, vol. 27(3), pages 713-768, July.
    2. Thijssen, Jacco J.J., 2022. "Optimal investment and abandonment decisions for projects with construction uncertainty," European Journal of Operational Research, Elsevier, vol. 298(1), pages 368-379.
    3. Giorgia Callegaro & Claudia Ceci & Giorgio Ferrari, 2019. "Optimal Reduction of Public Debt under Partial Observation of the Economic Growth," Papers 1901.08356, arXiv.org, revised Jan 2019.
    4. Ferrari, Giorgio, 2016. "Controlling public debt without forgetting Inflation," Center for Mathematical Economics Working Papers 564, Center for Mathematical Economics, Bielefeld University.
    5. Dammann, Felix & Ferrari, Giorgio, 2022. "Optimal Execution with Multiplicative Price Impact and Incomplete Information on the Return," Center for Mathematical Economics Working Papers 663, Center for Mathematical Economics, Bielefeld University.
    6. Felix Dammann & Giorgio Ferrari, 2022. "Optimal Execution with Multiplicative Price Impact and Incomplete Information on the Return," Papers 2202.10414, arXiv.org, revised Nov 2022.
    7. Jodi Dianetti & Giorgio Ferrari & Renyuan Xu, 2024. "Exploratory Optimal Stopping: A Singular Control Formulation," Papers 2408.09335, arXiv.org, revised Oct 2024.
    8. 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.
    9. 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.
    10. Christensen, Sören & Crocce, Fabián & Mordecki, Ernesto & Salminen, Paavo, 2019. "On optimal stopping of multidimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2561-2581.
    11. Giorgio Ferrari & Hanwu Li & Frank Riedel, 2020. "A Knightian Irreversible Investment Problem," Papers 2003.14359, arXiv.org, revised Apr 2020.
    12. 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.
    13. Cheng Cai & Tiziano De Angelis, 2021. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Papers 2104.05835, arXiv.org, revised Jul 2023.
    14. Cai, Cheng & De Angelis, Tiziano, 2023. "A change of variable formula with applications to multi-dimensional optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 33-61.
    15. Kexin Chen & Kyunghyun Park & Hoi Ying Wong, 2024. "Robust dividend policy: Equivalence of Epstein-Zin and Maenhout preferences," Papers 2406.12305, arXiv.org.
    16. Giorgio Ferrari, 2016. "On the Optimal Management of Public Debt: a Singular Stochastic Control Problem," Papers 1607.04153, arXiv.org, revised Dec 2017.
    17. Andrea Bovo & Tiziano De Angelis & Jan Palczewski, 2023. "Stopper vs. singular-controller games with degenerate diffusions," Papers 2312.00613, arXiv.org, revised Jul 2024.
    18. 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.
    19. Giorgia Callegaro & Claudia Ceci & Giorgio Ferrari, 2020. "Optimal reduction of public debt under partial observation of the economic growth," Finance and Stochastics, Springer, vol. 24(4), pages 1083-1132, October.
    20. Callegaro, Giorgia & Ceci, Claudia & Ferrari, Giorgio, 2019. "Optimal Reduction of Public Debt under Partial Observation of the Economic Growth," Center for Mathematical Economics Working Papers 608, Center for Mathematical Economics, Bielefeld University.
    21. 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.

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    More about this item

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C73 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Stochastic and Dynamic Games; Evolutionary Games
    • E22 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Investment; Capital; Intangible Capital; Capacity
    • D92 - Microeconomics - - Micro-Based Behavioral Economics - - - Intertemporal Firm Choice, Investment, Capacity, and Financing

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