Nonzero-sum stochastic impulse games with an application in competitive retail energy markets

René Aïd, L. Ben Ajmia, M’hamed Gaïgi and Mohamed Mnif
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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
L. Ben Ajmia: LR-LAMSIN-ENIT - Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis] - ENIT - Ecole Nationale d'Ingénieurs de Tunis - UTM - Tunis El Manar University [University of Tunis El Manar] [Tunisia] = Université de Tunis El Manar [Tunisie] = جامعة تونس المنار (ar)
M’hamed Gaïgi: LR-LAMSIN-ENIT - Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis] - ENIT - Ecole Nationale d'Ingénieurs de Tunis - UTM - Tunis El Manar University [University of Tunis El Manar] [Tunisia] = Université de Tunis El Manar [Tunisie] = جامعة تونس المنار (ar)
Mohamed Mnif: LR-LAMSIN-ENIT - Laboratoire de Modélisation Mathématique et Numérique dans les Sciences de l'Ingénieur [Tunis] - ENIT - Ecole Nationale d'Ingénieurs de Tunis - UTM - Tunis El Manar University [University of Tunis El Manar] [Tunisia] = Université de Tunis El Manar [Tunisie] = جامعة تونس المنار (ar)

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Abstract: We study a nonzero-sum stochastic differential game with both players adopting impulse controls, on a finite time horizon. The objective of each player is to maximize her total expected discounted profits. The resolution methodology relies on the connection between Nash equilibrium and the corresponding system of quasi-variational inequalities (QVIs in short). We prove, by means of the weak dynamic programming principle for the stochastic differential game, that the equilibrium expected payoff of each player is a constrained viscosity solution to the associated QVIs system in the class of linear growth functions. We also introduce a family of equilibrium expected payoffs converging to our equilibrium expected payoff of each player, and which is characterized as the unique constrained viscosity solutions of an approximation of our QVIs system. This convergence result is useful for numerical purpose. We apply a probabilistic numerical scheme which approximates the solution of the QVIs system to the case of the competition between two electricity retailers. We show how our model reproduces the qualitative behavior of electricity retail competition.

Keywords: Stochastic impulse games; Nash equilibrium; viscosity solution (search for similar items in EconPapers)
Date: 2024
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Published in ESAIM: Control, Optimisation and Calculus of Variations, 2024, 30

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-04594074

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