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A Differential Game with Random Time Horizon and Discontinuous Distribution

Author

Listed:
  • Anastasiia Zaremba

    (Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, 199034 St Petersburg, Russia)

  • Ekaterina Gromova

    (Department of Mathematics, St. Petersburg School of Physics, Mathematics, and Computer Science, National Research University Higher School of Economics (HSE), Soyuza Pechatnikov ul. 16, 190008 St. Petersburg, Russia
    Krasovskii Institute of Mathematics and Mechanics (IMM UB RAS), 620108 Yekaterinburg, Russia)

  • Anna Tur

    (Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, 199034 St Petersburg, Russia)

Abstract

One class of differential games with random duration is considered. It is assumed that the duration of the game is a random variable with values from a given finite interval. The cumulative distribution function (CDF) of this random variable is assumed to be discontinuous with two jumps on the interval. It follows that the player’s payoff takes the form of the sum of integrals with different but adjoint time intervals. In addition, the first interval corresponds to the zero probability of the game to be finished, which results in terminal payoff on this interval. The method of construction optimal solution for the cooperative scenario of such games is proposed. The results are illustrated by the example of differential game of investment in the public stock of knowledge.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2185-:d:458745
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    References listed on IDEAS

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    2. Feichtinger, Gustav & Jorgensen, Steffen, 1983. "Differential game models in management science," European Journal of Operational Research, Elsevier, vol. 14(2), pages 137-155, October.
    3. 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.
    4. Steffen Jørgensen & Georges Zaccour, 2007. "Developments in differential game theory and numerical methods: economic and management applications," Computational Management Science, Springer, vol. 4(2), pages 159-181, April.
    5. 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, January.
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    Cited by:

    1. 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.

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