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

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Listed:
  • Ren'e Aid
  • Lamia Ben Ajmia
  • M'hamed Gaigi
  • Mohamed Mnif

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 value function 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 value functions converging to our value function 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 behaviour of electricity retail competition.

Suggested Citation

  • 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.
    • Handle: RePEc:arx:papers:2112.10213
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    References listed on IDEAS

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    1. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    2. René Aïd & Matteo Basei & Giorgia Callegaro & Luciano Campi & Tiziano Vargiolu, 2020. "Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 205-232, February.
    3. Luschgy, Harald & Pagès, Gilles, 2006. "Functional quantization of a class of Brownian diffusions: A constructive approach," Stochastic Processes and their Applications, Elsevier, vol. 116(2), pages 310-336, February.
    4. Matteo Basei, 2018. "Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates," Papers 1803.08166, arXiv.org, revised Mar 2019.
    5. René Aïd & Francisco Bernal & Mohamed Mnif & Diego Zabaljauregui & Jorge P. Zubelli, 2019. "A policy iteration algorithm for non-zero sum stochastic impulse games," Post-Print hal-02294353, HAL.
    6. Marianne Akian & Agnès Sulem & Michael I. Taksar, 2001. "Dynamic Optimization of Long‐Term Growth Rate for a Portfolio with Transaction Costs and Logarithmic Utility," Mathematical Finance, Wiley Blackwell, vol. 11(2), pages 153-188, April.
    7. 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.
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