IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i2p462-d1036545.html
   My bibliography  Save this article

The Hamilton–Jacobi–Bellman Equation for Differential Games with Composite Distribution of Random Time Horizon

Author

Listed:
  • Tatyana Balas

    (Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, 199034 St. Petersburg, Russia
    These authors contributed equally to this work.)

  • Anna Tur

    (Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, 199034 St. Petersburg, Russia
    These authors contributed equally to this work.)

Abstract

A differential game with random duration is considered. The terminal time of the game is a random variable settled using a composite distribution function. Such a scenario occurs when the operating mode of the system changes over time at the appropriate switching points. On each interval between switchings, the distribution of the terminal time is characterized by its own distribution function. A method for solving such games using dynamic programming is proposed. An example of a non-renewable resource extraction model is given, where a solution of the problem of maximizing the total payoff in closed-loop strategies is found. An analytical view of the optimal control of each player and the optimal trajectory depending on the parameters of the described model is obtained.

Suggested Citation

  • Tatyana Balas & Anna Tur, 2023. "The Hamilton–Jacobi–Bellman Equation for Differential Games with Composite Distribution of Random Time Horizon," Mathematics, MDPI, vol. 11(2), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:2:p:462-:d:1036545
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/2/462/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/2/462/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Menahem E. Yaari, 1965. "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 32(2), pages 137-150.
    2. Leung, Siu Fai, 2009. "Cake eating, exhaustible resource extraction, life-cycle saving, and non-atomic games: Existence theorems for a class of optimal allocation problems," Journal of Economic Dynamics and Control, Elsevier, vol. 33(6), pages 1345-1360, June.
    3. Anastasiia Zaremba & Ekaterina Gromova & Anna Tur, 2020. "A Differential Game with Random Time Horizon and Discontinuous Distribution," Mathematics, MDPI, vol. 8(12), pages 1-21, December.
    4. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329, September.
    5. Kamien, Morton I & Schwartz, Nancy L, 1972. "Timing of Innovations Under Rivalry," Econometrica, Econometric Society, vol. 40(1), pages 43-60, January.
    Full references (including those not matched with items on IDEAS)

    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. José Daniel López-Barrientos & Ekaterina Viktorovna Gromova & Ekaterina Sergeevna Miroshnichenko, 2020. "Resource Exploitation in a Stochastic Horizon under Two Parametric Interpretations," Mathematics, MDPI, vol. 8(7), pages 1-29, July.
    2. Ekaterina Gromova & Anastasiia Zaremba & Nahid Masoudi, 2022. "Reclamation of a Resource Extraction Site Model with Random Components," Mathematics, MDPI, vol. 10(24), pages 1-15, December.
    3. Fouad Ouardighi & Matan Shnaiderman & Federico Pasin, 2014. "Research and Development with Stock-Dependent Spillovers and Price Competition in a Duopoly," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 626-647, May.
    4. Ekaterina Gromova & Anastasiya Malakhova & Arsen Palestini, 2018. "Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration," Mathematics, MDPI, vol. 6(9), pages 1-17, September.
    5. Anastasiia Zaremba & Ekaterina Gromova & Anna Tur, 2020. "A Differential Game with Random Time Horizon and Discontinuous Distribution," Mathematics, MDPI, vol. 8(12), pages 1-21, December.
    6. S. Kostyunin & A. Palestini & E. Shevkoplyas, 2014. "On a Nonrenewable Resource Extraction Game Played by Asymmetric Firms," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 660-673, November.
    7. Mengyuan Zhou, 2022. "Does the Source of Inheritance Matter in Bequest Attitudes? Evidence from Japan," Journal of Family and Economic Issues, Springer, vol. 43(4), pages 867-887, December.
    8. Markku Ollikainen, 1998. "Sustainable Forestry: Timber Bequests, Future Generations and Optimal Tax Policy," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 12(3), pages 255-273, October.
    9. An Chen & Thai Nguyen & Thorsten Sehner, 2022. "Unit-Linked Tontine: Utility-Based Design, Pricing and Performance," Risks, MDPI, vol. 10(4), pages 1-27, April.
    10. Jiang Cheng & Lu Yu, 2019. "Life and health insurance consumption in China: demographic and environmental risks," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 44(1), pages 67-101, January.
    11. Anikó Bíró, 2013. "Subjective mortality hazard shocks and the adjustment of consumption expenditures," Journal of Population Economics, Springer;European Society for Population Economics, vol. 26(4), pages 1379-1408, October.
    12. Masahiko Hattori & Yasuhito Tanaka, 2019. "General analysis of dynamic oligopoly with sticky price," Economics Bulletin, AccessEcon, vol. 39(4), pages 2990-2998.
    13. Liliana Meza-González & Jaime Marie Sepulveda, 2019. "The impact of competition with China in the US market on innovation in Mexican manufacturing firms," Latin American Economic Review, Springer;Centro de Investigaciòn y Docencia Económica (CIDE), vol. 28(1), pages 1-21, December.
    14. Frédéric Gannon & Vincent Touzé, 2006. "Insurance and Optimal Growth," Post-Print halshs-00085181, HAL.
    15. Robert Gazzale & Julian Jamison & Alexander Karlan & Dean Karlan, 2013. "Ambiguous Solicitation: Ambiguous Prescription," Economic Inquiry, Western Economic Association International, vol. 51(1), pages 1002-1011, January.
    16. Drouhin, Nicolas, 2015. "A rank-dependent utility model of uncertain lifetime," Journal of Economic Dynamics and Control, Elsevier, vol. 53(C), pages 208-224.
    17. Bommier, Antoine & Lanz, Bruno & Zuber, Stéphane, 2015. "Models-as-usual for unusual risks? On the value of catastrophic climate change," Journal of Environmental Economics and Management, Elsevier, vol. 74(C), pages 1-22.
    18. Rajnish Mehra & Facundo Piguillem & Edward C. Prescott, 2011. "Costly financial intermediation in neoclassical growth theory," Quantitative Economics, Econometric Society, vol. 2(1), pages 1-36, March.
    19. Beshears, John & Choi, James J. & Laibson, David & Madrian, Brigitte C., 2011. "Behavioral economics perspectives on public sector pension plans," Journal of Pension Economics and Finance, Cambridge University Press, vol. 10(2), pages 315-336, April.
    20. Luca Lambertini & Arsen Palestini & Alessandro Tampieri, 2016. "CSR in an Asymmetric Duopoly with Environmental Externality," Southern Economic Journal, John Wiley & Sons, vol. 83(1), pages 236-252, July.

    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:gam:jmathe:v:11:y:2023:i:2:p:462-:d:1036545. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.