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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
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    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.
    5. Kamien, Morton I & Schwartz, Nancy L, 1972. "Timing of Innovations Under Rivalry," Econometrica, Econometric Society, vol. 40(1), pages 43-60, January.
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