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Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper

Author

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  • Luciano Campi

    (London School of Economics and Political Science
    University of Milan)

  • Davide Santis

    (London School of Economics and Political Science)

Abstract

We study a two-player nonzero-sum stochastic differential game, where one player controls the state variable via additive impulses, while the other player can stop the game at any time. The main goal of this work is to characterize Nash equilibria through a verification theorem, which identifies a new system of quasivariational inequalities, whose solution gives equilibrium payoffs with the correspondent strategies. Moreover, we apply the verification theorem to a game with a one-dimensional state variable, evolving as a scaled Brownian motion, and with linear payoff and costs for both players. Two types of Nash equilibrium are fully characterized, i.e. semi-explicit expressions for the equilibrium strategies and associated payoffs are provided. Both equilibria are of threshold type: in one equilibrium players’ intervention are not simultaneous, while in the other one the first player induces her competitor to stop the game. Finally, we provide some numerical results describing the qualitative properties of both types of equilibrium.

Suggested Citation

  • Luciano Campi & Davide Santis, 2020. "Nonzero-Sum Stochastic Differential Games Between an Impulse Controller and a Stopper," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 688-724, August.
    • Handle: RePEc:spr:joptap:v:186:y:2020:i:2:d:10.1007_s10957-020-01718-6
    DOI: 10.1007/s10957-020-01718-6
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    References listed on IDEAS

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    Cited by:

    1. Cao, Haoyang & Guo, Xin, 2022. "MFGs for partially reversible investment," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 995-1014.
    2. Utsav Sadana & Puduru Viswanadha Reddy & Tamer Başar & Georges Zaccour, 2021. "Sampled-Data Nash Equilibria in Differential Games with Impulse Controls," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 999-1022, September.
    3. Sadana, Utsav & Reddy, Puduru Viswanadha & Zaccour, Georges, 2021. "Nash equilibria in nonzero-sum differential games with impulse control," European Journal of Operational Research, Elsevier, vol. 295(2), pages 792-805.

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