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Mean Field Game with Delay: A Toy Model

Author

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  • Jean-Pierre Fouque

    (Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA
    Work supported by NSF grants DMS-1409434 and DMS-1814091.)

  • Zhaoyu Zhang

    (Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA)

Abstract

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

Suggested Citation

  • Jean-Pierre Fouque & Zhaoyu Zhang, 2018. "Mean Field Game with Delay: A Toy Model," Risks, MDPI, vol. 6(3), pages 1-17, September.
    • Handle: RePEc:gam:jrisks:v:6:y:2018:i:3:p:90-:d:167248
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    References listed on IDEAS

    as
    1. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 2018. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 366-399, November.
    2. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    3. Vassili Kolokoltsov & Marianna Troeva & Wei Yang, 2014. "On the Rate of Convergence for the Mean-Field Approximation of Controlled Diffusions with Large Number of Players," Dynamic Games and Applications, Springer, vol. 4(2), pages 208-230, June.
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    Cited by:

    1. Zachary Feinstein & Andreas Sojmark, 2019. "A Dynamic Default Contagion Model: From Eisenberg-Noe to the Mean Field," Papers 1912.08695, arXiv.org.
    2. Li-Hsien Sun, 2022. "Mean Field Games with Heterogeneous Groups: Application to Banking Systems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 130-167, January.
    3. Li-Hsien Sun, 2019. "Systemic Risk and Heterogeneous Mean Field Type Interbank Network," Papers 1907.03082, arXiv.org, revised Sep 2019.
    4. Hanchao Liu & Dena Firoozi, 2024. "Hilbert Space-Valued LQG Mean Field Games: An Infinite-Dimensional Analysis," Papers 2403.01012, arXiv.org.
    5. Eduardo Abi Jaber & Eyal Neuman & Moritz Vo{ss}, 2023. "Equilibrium in Functional Stochastic Games with Mean-Field Interaction," Papers 2306.05433, arXiv.org, revised Feb 2024.
    6. Djehiche, Boualem & Gozzi, Fausto & Zanco, Giovanni & Zanella, Margherita, 2022. "Optimal portfolio choice with path dependent benchmarked labor income: A mean field model," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 48-85.

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