IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v44y2019i2p512-531.html
   My bibliography  Save this article

A Solvable Two-Dimensional Degenerate Singular Stochastic Control Problem with Nonconvex Costs

Author

Listed:
  • Tiziano De Angelis

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

  • Giorgio Ferrari

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

  • John Moriarty

    (School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom)

Abstract

In this paper we provide a complete theoretical analysis of a two-dimensional degenerate nonconvex singular stochastic control problem. The optimisation is motivated by a storage-consumption model in an electricity market, and features a stochastic real-valued spot price modelled by Brownian motion. We find analytical expressions for the value function, the optimal control, and the boundaries of the action and inaction regions. The optimal policy is characterised in terms of two monotone and discontinuous repelling free boundaries, although part of one boundary is constant and the smooth fit condition holds there.

Suggested Citation

  • Tiziano De Angelis & Giorgio Ferrari & John Moriarty, 2019. "A Solvable Two-Dimensional Degenerate Singular Stochastic Control Problem with Nonconvex Costs," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 512-531, May.
    • Handle: RePEc:inm:ormoor:v:44:y:2019:i:2:p:512-531
    DOI: 10.1287/moor.2018.0934
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/moor.2018.0934
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2018.0934?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2020. "Singular Control of the Drift of a Brownian System," Center for Mathematical Economics Working Papers 637, Center for Mathematical Economics, Bielefeld University.
    2. Federico, Salvatore & Ferrari, Giorgio & Schuhmann, Patrick, 2019. "A Model for the Optimal Management of Inflation," Center for Mathematical Economics Working Papers 624, Center for Mathematical Economics, Bielefeld University.
    3. Salvatore Federico & Giorgio Ferrari & Patrick Schuhmann, 2019. "A Model for the Optimal Management of Inflation," Department of Economics University of Siena 812, Department of Economics, University of Siena.
    4. Jodi Dianetti & Giorgio Ferrari & Renyuan Xu, 2024. "Exploratory Optimal Stopping: A Singular Control Formulation," Papers 2408.09335, arXiv.org, revised Oct 2024.
    5. Dianetti, Jodi & Ferrari, Giorgio, 2021. "Multidimensional Singular Control and Related Skorokhod Problem: Suficient Conditions for the Characterization of Optimal Controls," Center for Mathematical Economics Working Papers 645, Center for Mathematical Economics, Bielefeld University.
    6. Andrea Bovo & Tiziano De Angelis & Jan Palczewski, 2023. "Stopper vs. singular-controller games with degenerate diffusions," Papers 2312.00613, arXiv.org, revised Jul 2024.
    7. 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.
    8. Dianetti, Jodi & Ferrari, Giorgio, 2023. "Multidimensional singular control and related Skorokhod problem: Sufficient conditions for the characterization of optimal controls," Stochastic Processes and their Applications, Elsevier, vol. 162(C), pages 547-592.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. de Angelis, Tiziano & Ferrari, Giorgio & Moriarty, John, 2016. "A solvable two-dimensional singular stochastic control problem with non convex costs," Center for Mathematical Economics Working Papers 561, Center for Mathematical Economics, Bielefeld University.
    2. 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.
    3. 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.
    4. Erhan Bayraktar & Masahiko Egami, 2008. "An Analysis of Monotone Follower Problems for Diffusion Processes," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 336-350, May.
    5. Hongzhong Zhang, 2018. "Stochastic Drawdowns," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 10078, August.
    6. Manuel Guerra & Cláudia Nunes & Carlos Oliveira, 2021. "The optimal stopping problem revisited," Statistical Papers, Springer, vol. 62(1), pages 137-169, February.
    7. Liangchen Li & Michael Ludkovski, 2018. "Stochastic Switching Games," Papers 1807.03893, arXiv.org.
    8. Sabri Boubaker & Zhenya Liu & Yaosong Zhan, 2022. "Risk management for crude oil futures: an optimal stopping-timing approach," Annals of Operations Research, Springer, vol. 313(1), pages 9-27, June.
    9. Li, Lingfei & Linetsky, Vadim, 2014. "Optimal stopping in infinite horizon: An eigenfunction expansion approach," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 122-128.
    10. Bolton, Patrick & Wang, Neng & Yang, Jinqiang, 2019. "Investment under uncertainty with financial constraints," Journal of Economic Theory, Elsevier, vol. 184(C).
    11. S. C. P. Yam & S. P. Yung & W. Zhou, 2014. "Game Call Options Revisited," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 173-206, January.
    12. de Angelis, Tiziano & Ferrari, Giorgio & Moriarty, John, 2016. "Nash equilibria of threshold type for two-player nonzero-sum games of stopping," Center for Mathematical Economics Working Papers 563, Center for Mathematical Economics, Bielefeld University.
    13. Erhan Bayraktar & Masahiko Egami, 2010. "A unified treatment of dividend payment problems under fixed cost and implementation delays," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(2), pages 325-351, April.
    14. Hobson, David, 2021. "The shape of the value function under Poisson optimal stopping," Stochastic Processes and their Applications, Elsevier, vol. 133(C), pages 229-246.
    15. Zbigniew Palmowski & Jos'e Luis P'erez & Kazutoshi Yamazaki, 2020. "Double continuation regions for American options under Poisson exercise opportunities," Papers 2004.03330, arXiv.org.
    16. Katia Colaneri & Tiziano De Angelis, 2019. "A class of recursive optimal stopping problems with applications to stock trading," Papers 1905.02650, arXiv.org, revised Jun 2021.
    17. Liu, Zhenya & Lu, Shanglin & Wang, Shixuan, 2021. "Asymmetry, tail risk and time series momentum," International Review of Financial Analysis, Elsevier, vol. 78(C).
    18. Alex S. L. Tse & Harry Zheng, 2023. "Speculative trading, prospect theory and transaction costs," Finance and Stochastics, Springer, vol. 27(1), pages 49-96, January.
    19. Luis H. R. Alvarez E. & Paavo Salminen, 2017. "Timing in the presence of directional predictability: optimal stopping of skew Brownian motion," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(2), pages 377-400, October.
    20. Savas Dayanik, 2008. "Optimal Stopping of Linear Diffusions with Random Discounting," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 645-661, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormoor:v:44:y:2019:i:2:p:512-531. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.