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Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications

Author

Listed:
  • René Aïd

    (FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CREST - EDF R&D - EDF R&D - EDF - EDF, LEDa - Laboratoire d'Economie de Dauphine - IRD - Institut de Recherche pour le Développement - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique)

  • Matteo Basei
  • Giorgia Callegaro

    (Dipartimento di Matematica Pura e Applicata [Padova] - Unipd - Università degli Studi di Padova = University of Padua)

  • Luciano Campi

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique, CREST - Centre de Recherche en Économie et Statistique - ENSAI - Ecole Nationale de la Statistique et de l'Analyse de l'Information [Bruz] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - CNRS - Centre National de la Recherche Scientifique, LSE - Department of Statistics - London School of Economics - LSE - London School of Economics and Political Science, FiME Lab - Laboratoire de Finance des Marchés d'Energie - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CREST - EDF R&D - EDF R&D - EDF - EDF, Department of Mathematics "Federigo Enriques" - UNIMI - Università degli Studi di Milano = University of Milan)

  • Tiziano Vargiolu

    (Dipartimento di Matematica Pura e Applicata [Padova] - Unipd - Università degli Studi di Padova = University of Padua)

Abstract
We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of this paper is a verification theorem that provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the payoffs and the strategies of the two players at some Nash equilibrium. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a family of Nash equilibria and provide explicit expressions for the corresponding equilibrium strategies and payoffs. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further nonsymmetric examples where a Nash equilibrium is found numerically.

Suggested Citation

  • René Aïd & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2020. "Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications," Post-Print hal-03355609, HAL.
    • Handle: RePEc:hal:journl:hal-03355609
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    References listed on IDEAS

    as
    1. Paul Joskow & Jean Tirole, 2006. "Retail electricity competition," RAND Journal of Economics, RAND Corporation, vol. 37(4), pages 799-815, December.
    2. Abel Cadenillas & Fernando Zapatero, 2000. "Classical and Impulse Stochastic Control of the Exchange Rate Using Interest Rates and Reserves," Mathematical Finance, Wiley Blackwell, vol. 10(2), pages 141-156, April.
    3. Paul Joskow & Jean Tirole, 2006. "Retail electricity competition," RAND Journal of Economics, The RAND Corporation, vol. 37(4), pages 799-815, December.
    4. Abel Cadenillas & Peter Lakner & Michael Pinedo, 2010. "Optimal Control of a Mean-Reverting Inventory," Operations Research, INFORMS, vol. 58(6), pages 1697-1710, December.
    5. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, January.
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    Cited by:

    1. Giorgio Ferrari & Torben Koch, 2019. "On a strategic model of pollution control," Annals of Operations Research, Springer, vol. 275(2), pages 297-319, April.
    2. Diego Zabaljauregui, 2019. "A fixed-point policy-iteration-type algorithm for symmetric nonzero-sum stochastic impulse control games," Papers 1909.03574, arXiv.org, revised Jun 2020.
    3. Liangchen Li & Michael Ludkovski, 2018. "Stochastic Switching Games," Papers 1807.03893, arXiv.org.
    4. Cao, Haoyang & Guo, Xin, 2022. "MFGs for partially reversible investment," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 995-1014.
    5. Matteo Basei, 2019. "Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(3), pages 355-383, June.
    6. Francesco Giuseppe Cordoni & Luca Di Persio & Yilun Jiang, 2020. "A Bank Salvage Model by Impulse Stochastic Controls," Risks, MDPI, vol. 8(2), pages 1-31, June.
    7. Zhang, Nan & Jin, Zhuo & Qian, Linyi & Fan, Kun, 2019. "Stochastic differential reinsurance games with capital injections," Insurance: Mathematics and Economics, Elsevier, vol. 88(C), pages 7-18.
    8. Francesco Cordoni & Luca Di Persio & Yilun Jiang, 2019. "A bank salvage model by impulse stochastic controls," Papers 1910.03056, arXiv.org.
    9. Ren'e Aid & Lamia Ben Ajmia & M'hamed Gaigi & Mohamed Mnif, 2021. "Nonzero-sum stochastic impulse games with an application in competitive retail energy markets," Papers 2112.10213, arXiv.org.
    10. 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.
    11. Ferrari, Giorgio & Koch, Torben, 2018. "On a Strategic Model of Pollution Control," Center for Mathematical Economics Working Papers 586, Center for Mathematical Economics, Bielefeld University.
    12. Giorgio Ferrari & Tiziano Vargiolu, 2020. "On the singular control of exchange rates," Annals of Operations Research, Springer, vol. 292(2), pages 795-832, September.
    13. Ren'e Aid & Luciano Campi & Liangchen Li & Mike Ludkovski, 2020. "An Impulse-Regime Switching Game Model of Vertical Competition," Papers 2006.04382, arXiv.org.
    14. René Aïd & Luciano Campi & Liangchen Li & Mike Ludkovski, 2021. "An Impulse-Regime Switching Game Model of Vertical Competition," Dynamic Games and Applications, Springer, vol. 11(4), pages 631-669, December.
    15. 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.
    16. 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.
    17. Diego Zabaljauregui, 2020. "Optimal market making under partial information and numerical methods for impulse control games with applications," Papers 2009.06521, arXiv.org.
    18. Joffrey Derchu & Philippe Guillot & Thibaut Mastrolia & Mathieu Rosenbaum, 2020. "AHEAD : Ad-Hoc Electronic Auction Design," Papers 2010.02827, arXiv.org.

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    More about this item

    Keywords

    Stochastic differential game; impulse control; Nash equilibrium; quasi-variational inequality;
    All these keywords.

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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