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Infinite-Dimensional LQ Mean Field Games with Common Noise: Small and Arbitrary Finite Time Horizons

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Listed:
  • Hanchao Liu
  • Dena Firoozi

Abstract

We extend the results of (Liu and Firoozi, 2025), which develops the theory of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, by incorporating a common noise. This common noise is modeled as an infinite-dimensional Wiener process affecting the dynamics of all agents. In the presence of common noise, the mean-field consistency condition is characterized by a system of coupled forward-backward stochastic evolution equations (FBSEEs) in Hilbert spaces, whereas, in its absence it is represented by coupled forward-backward deterministic evolution equations. We establish the existence and uniqueness of solutions to the coupled linear FBSEEs associated with the LQ MFG framework for small time horizons and prove the $\epsilon$-Nash property of the resulting equilibrium strategy. Furthermore, we establish the well-posedness of these coupled linear FBSEEs for arbitrary finite time horizons. Beyond the specific context of MFGs, our analysis also yields a broader contribution by providing, to the best of our knowledge, the first well-posedness result for a class of infinite-dimensional linear FBSEEs, for which only mild solutions exist, over arbitrary finite time horizons.

Suggested Citation

  • Hanchao Liu & Dena Firoozi, 2026. "Infinite-Dimensional LQ Mean Field Games with Common Noise: Small and Arbitrary Finite Time Horizons," Papers 2601.13493, arXiv.org, revised Jan 2026.
    • Handle: RePEc:arx:papers:2601.13493
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    References listed on IDEAS

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    1. Hanchao Liu & Dena Firoozi, 2024. "Hilbert Space-Valued LQ Mean Field Games: An Infinite-Dimensional Analysis," Papers 2403.01012, arXiv.org, revised Aug 2025.
    2. Diogo A. Gomes & João Saúde, 2021. "A Mean-Field Game Approach to Price Formation," Dynamic Games and Applications, Springer, vol. 11(1), pages 29-53, March.
    3. Eduardo Abi Jaber & Eyal Neuman & Moritz Voss, 2023. "Equilibrium in Functional Stochastic Games with Mean-Field Interaction," Working Papers hal-04119787, HAL.
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    6. Filippo de Feo & Fausto Gozzi & Andrzej 'Swik{e}ch & Lukas Wessels, 2025. "Stochastic Optimal Control of Interacting Particle Systems in Hilbert Spaces and Applications," Papers 2511.21646, arXiv.org.
    7. Fu, Guanxing & Horst, Ulrich & Xia, Xiaonyu, 2022. "Portfolio Liquidation Games with Self-Exciting Order Flow," Rationality and Competition Discussion Paper Series 327, CRC TRR 190 Rationality and Competition.
    8. 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.
    9. Rinel Foguen Tchuendom, 2018. "Uniqueness for Linear-Quadratic Mean Field Games with Common Noise," Dynamic Games and Applications, Springer, vol. 8(1), pages 199-210, March.
    10. Xin Yue Ren & Dena Firoozi, 2024. "Risk-Sensitive Mean Field Games with Common Noise: A Theoretical Study with Applications to Interbank Markets," Papers 2403.03915, arXiv.org.
    11. Guanxing Fu & Paulwin Graewe & Ulrich Horst & Alexandre Popier, 2021. "A Mean Field Game of Optimal Portfolio Liquidation," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1250-1281, November.
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    14. Arvind V. Shrivats & Dena Firoozi & Sebastian Jaimungal, 2022. "A mean‐field game approach to equilibrium pricing in solar renewable energy certificate markets," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 779-824, July.
    15. Philippe Casgrain & Sebastian Jaimungal, 2020. "Mean‐field games with differing beliefs for algorithmic trading," Mathematical Finance, Wiley Blackwell, vol. 30(3), pages 995-1034, July.
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