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Optimal boundary surface for irreversible investment with stochastic costs

Author

Listed:
  • Tiziano De Angelis

    (School of Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, United Kingdom)

  • Salvatore Federico

    (Dipartimento di Scienze per l'Economia e l'Impresa, Universita' degli Studi di Firenze, Via delle Pandette 9, 50127 Firenze, Italy)

  • Giorgio Ferrari

    (Center for Mathematical Economics (IMW), Bielefeld University, Universitatsstrasse 25, D-33615 Bielefeld, Germany)

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 reflected diffusion 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.

Suggested Citation

  • Tiziano De Angelis & Salvatore Federico & Giorgio Ferrari, 2015. "Optimal boundary surface for irreversible investment with stochastic costs," Working Papers - Mathematical Economics 2015-03, Universita' degli Studi di Firenze, Dipartimento di Scienze per l'Economia e l'Impresa.
    • Handle: RePEc:flo:wpaper:2015-03
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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. 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.
    3. Ferrari, Giorgio, 2016. "Controlling public debt without forgetting Inflation," Center for Mathematical Economics Working Papers 564, Center for Mathematical Economics, Bielefeld University.
    4. 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.
    5. Giorgio Ferrari & Hanwu Li & Frank Riedel, 2020. "A Knightian Irreversible Investment Problem," Papers 2003.14359, arXiv.org, revised Apr 2020.
    6. Dianetti, Jodi & Ferrari, Giorgio & Xu, Renyuan, 2025. "Exploratory Optimal Stopping: A Singular Control Formulation," Center for Mathematical Economics Working Papers 740, Center for Mathematical Economics, Bielefeld University.
    7. 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.
    8. Ferrari, Giorgio, 2025. "On the Optimal Management of Public Debt: a Singular Stochastic Control Problem," Center for Mathematical Economics Working Papers 709, Center for Mathematical Economics, Bielefeld University.
    9. 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.
    10. Kexin Chen & Kyunghyun Park & Hoi Ying Wong, 2024. "Robust dividend policy: Equivalence of Epstein-Zin and Maenhout preferences," Papers 2406.12305, arXiv.org, revised Oct 2025.
    11. 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.
    12. 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.
    13. 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.
    14. Calvia, Alessandro & Cannerozzi, Federico & Ferrari, Giorgio, 2025. "Optimal Policy Characterization for a Class of Multi-Dimensional Ergodic Singular Stochastic Control Problems," Center for Mathematical Economics Working Papers 754, Center for Mathematical Economics, Bielefeld University.
    15. 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.
    16. 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.
    17. 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.
    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. Andrea Bovo & Tiziano De Angelis & Jan Palczewski, 2023. "Stopper vs. singular-controller games with degenerate diffusions," Papers 2312.00613, arXiv.org, revised Jul 2024.
    20. Lijun Bo & Meng Li & Tingting Zhang, 2023. "Evaluation Timing with Dynamic Information: Optimization and Heuristic," Production and Operations Management, Production and Operations Management Society, vol. 32(12), pages 3931-3950, December.
    21. Jodi Dianetti & Giorgio Ferrari & Renyuan Xu, 2024. "Exploratory Optimal Stopping: A Singular Control Formulation," Papers 2408.09335, arXiv.org, revised Oct 2024.
    22. 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.
    23. 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.
    24. Giorgio Ferrari, 2016. "On the Optimal Management of Public Debt: a Singular Stochastic Control Problem," Papers 1607.04153, arXiv.org, revised Dec 2017.
    25. 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.

    More about this item

    Keywords

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    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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