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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 [E.D.F.] - EDF – Électricité de France, 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] - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique - Groupe ENSAE-ENSAI - Groupe des Écoles Nationales d'Économie et Statistique - IP Paris - Institut Polytechnique de Paris - 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 [E.D.F.] - EDF – Électricité de France, 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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    Cited by:

    1. Jun Lu & Jinbiao Wu & Bixuan Yang, 2025. "Optimal strategies of regular-singular mean-field delayed stochastic differential games," Annals of Operations Research, Springer, vol. 344(1), pages 175-216, January.
    2. 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.
    3. Guohui Guan & Zongxia Liang & Yi Xia, 2024. "Robust mean-variance stochastic differential reinsurance and investment games under volatility risk and model uncertainty," Papers 2412.09171, arXiv.org.
    4. 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.
    5. Giorgio Ferrari & Torben Koch, 2019. "On a strategic model of pollution control," Annals of Operations Research, Springer, vol. 275(2), pages 297-319, April.
    6. 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.
    7. Liangchen Li & Michael Ludkovski, 2018. "Stochastic Switching Games," Papers 1807.03893, arXiv.org.
    8. Cao, Haoyang & Guo, Xin, 2022. "MFGs for partially reversible investment," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 995-1014.
    9. 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.
    10. 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.
    11. 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.
    12. Francesco Cordoni & Luca Di Persio & Yilun Jiang, 2019. "A bank salvage model by impulse stochastic controls," Papers 1910.03056, arXiv.org.
    13. 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.
    14. 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.
    15. 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.
    16. Diego Zabaljauregui, 2020. "Optimal market making under partial information and numerical methods for impulse control games with applications," Papers 2009.06521, arXiv.org.
    17. Tiziano De Angelis & Caio C'esar Graciani Rodrigues & Peter Tankov, 2024. "A model of strategic sustainable investment," Papers 2412.00986, arXiv.org, revised Apr 2025.
    18. Giorgio Ferrari & Tiziano Vargiolu, 2020. "On the singular control of exchange rates," Annals of Operations Research, Springer, vol. 292(2), pages 795-832, September.
    19. 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.
    20. Ren'e Aid & Luciano Campi & Liangchen Li & Mike Ludkovski, 2020. "An Impulse-Regime Switching Game Model of Vertical Competition," Papers 2006.04382, arXiv.org.
    21. Joffrey Derchu & Philippe Guillot & Thibaut Mastrolia & Mathieu Rosenbaum, 2020. "AHEAD : Ad-Hoc Electronic Auction Design," Papers 2010.02827, arXiv.org.
    22. Konark Jain & Nick Firoozye & Jonathan Kochems & Philip Treleaven, 2025. "An Impulse Control Approach to Market Making in a Hawkes LOB Market," Papers 2510.26438, arXiv.org, revised Oct 2025.

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    JEL classification:

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

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