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On the value of a time-inconsistent mean-field zero-sum Dynkin game

Author

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  • Boualem Djehiche

    (KTH Royal Institute of Technology)

Abstract

We study a mean-field zero-sum Dynkin game (MF-ZSDG) with time-inconsistent performance functionals adapted to the Brownian filtration. Despite the time-inconsistency of the MF-ZSDG, we show that it admits a value and that the pair of first times the value process hits the upper and lower obstacles, respectively, is a saddle point for the game. We solve the problem by approximating the associated lower and upper value processes with a sequence of value processes of interacting time-consistent zero-sum Dynkin games for which the saddle point of each of the value processes is the pair of first times each of those value processes hits the associated upper and lower obstacles, respectively. Under mild assumptions, we show that this sequence of saddle points converges in probability to the pair of first hitting times of the value process of the upper and lower obstacles, respectively, and that the limit is a saddle point for the time-inconsistent MF-ZSDG.

Suggested Citation

  • Boualem Djehiche, 2024. "On the value of a time-inconsistent mean-field zero-sum Dynkin game," Mathematics and Financial Economics, Springer, volume 18, number 12, December.
    • Handle: RePEc:spr:mathfi:v:18:y:2024:i:2:d:10.1007_s11579-024-00367-x
    DOI: 10.1007/s11579-024-00367-x
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    References listed on IDEAS

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    1. Tomasz Bielecki & Stephane Crepey & Monique Jeanblanc & Marek Rutkowski, 2008. "Arbitrage pricing of defaultable game options with applications to convertible bonds," Quantitative Finance, Taylor & Francis Journals, vol. 8(8), pages 795-810.
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    3. Touzi, N. & Vieille, N., 1999. "Continuous-Time Dynkin Games with Mixed Strategies," Papiers d'Economie Mathématique et Applications 1999.112, Université Panthéon-Sorbonne (Paris 1).
    4. Grenadier, Steven R, 1996. "The Strategic Exercise of Options: Development Cascades and Overbuilding in Real Estate Markets," Journal of Finance, American Finance Association, vol. 51(5), pages 1653-1679, December.
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