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

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Listed:
  • Ren'e Aid
  • Matteo Basei
  • Giorgia Callegaro
  • Luciano Campi
  • Tiziano Vargiolu

Abstract

We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the value functions and the optimal strategies of the two players. 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 Nash equilibrium and provide explicit expressions for the optimal strategies and the value functions. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically.

Suggested Citation

  • Ren'e Aid & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2016. "Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications," Papers 1605.00039, arXiv.org, revised Nov 2018.
  • Handle: RePEc:arx:papers:1605.00039
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Giorgio Ferrari & Tiziano Vargiolu, 2020. "On the singular control of exchange rates," Annals of Operations Research, Springer, vol. 292(2), pages 795-832, September.
    2. 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.
    3. Francesco Giuseppe Cordoni & Luca Di Persio & Yilun Jiang, 2020. "A Bank Salvage Model by Impulse Stochastic Controls," Risks, MDPI, Open Access Journal, vol. 8(2), pages 1-31, June.
    4. 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.
    5. 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.
    6. Francesco Cordoni & Luca Di Persio & Yilun Jiang, 2019. "A bank salvage model by impulse stochastic controls," Papers 1910.03056, arXiv.org.
    7. Diego Zabaljauregui, 2020. "Optimal market making under partial information and numerical methods for impulse control games with applications," Papers 2009.06521, arXiv.org.
    8. Liangchen Li & Michael Ludkovski, 2018. "Stochastic Switching Games," Papers 1807.03893, arXiv.org.
    9. Ren'e Aid & Luciano Campi & Liangchen Li & Mike Ludkovski, 2020. "An Impulse-Regime Switching Game Model of Vertical Competition," Papers 2006.04382, arXiv.org.
    10. Joffrey Derchu & Philippe Guillot & Thibaut Mastrolia & Mathieu Rosenbaum, 2020. "AHEAD : Ad-Hoc Electronic Auction Design," Papers 2010.02827, arXiv.org.
    11. 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.
    12. 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.
    13. Giorgio Ferrari & Torben Koch, 2019. "On a strategic model of pollution control," Annals of Operations Research, Springer, vol. 275(2), pages 297-319, April.

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

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