IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v37y2024i4d10.1007_s10959-024-01344-2.html
   My bibliography  Save this article

Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments

Author

Listed:
  • Alexander Kalinin

    (LMU Munich)

  • Thilo Meyer-Brandis

    (LMU Munich)

  • Frank Proske

    (University of Oslo)

Abstract

We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itô’s formula and leads to pth moment and pathwise exponential stability for $$p\ge 2$$ p ≥ 2 with explicit Lyapunov exponents.

Suggested Citation

  • Alexander Kalinin & Thilo Meyer-Brandis & Frank Proske, 2024. "Stability, Uniqueness and Existence of Solutions to McKean–Vlasov Stochastic Differential Equations in Arbitrary Moments," Journal of Theoretical Probability, Springer, vol. 37(4), pages 2941-2989, November.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01344-2
    DOI: 10.1007/s10959-024-01344-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-024-01344-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-024-01344-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Xiaojie Ding & Huijie Qiao, 2021. "Euler–Maruyama Approximations for Stochastic McKean–Vlasov Equations with Non-Lipschitz Coefficients," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1408-1425, September.
    2. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Tiago Roux Oliveira & Victor Hugo Pereira Rodrigues & Miroslav Krstić & Tamer Başar, 2021. "Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes," Journal of Optimization Theory and Applications, Springer, vol. 191(2), pages 700-735, December.
    2. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.
    3. Li, Hanwu, 2024. "Backward stochastic differential equations with double mean reflections," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    4. Arvind Shrivats & Dena Firoozi & Sebastian Jaimungal, 2020. "A Mean-Field Game Approach to Equilibrium Pricing in Solar Renewable Energy Certificate Markets," Papers 2003.04938, arXiv.org, revised Aug 2021.
    5. Qun Shi, 2021. "Generalized Mean-Field Fractional BSDEs With Non-Lipschitz Coefficients," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(3), pages 1-77, June.
    6. Kaitong Hu & Zhenjie Ren & Junjian Yang, 2019. "Principal-agent problem with multiple principals," Working Papers hal-02088486, HAL.
    7. Yan, Tingjin & Chiu, Mei Choi & Wong, Hoi Ying, 2023. "Portfolio liquidation with delayed information," Economic Modelling, Elsevier, vol. 126(C).
    8. Kamal Boukhetala & Jean-François Dupuy, 2019. "Modélisation Stochastique et Statistique Book of Proceedings," Post-Print hal-02593238, HAL.
    9. Douissi, Soukaina & Wen, Jiaqiang & Shi, Yufeng, 2019. "Mean-field anticipated BSDEs driven by fractional Brownian motion and related stochastic control problem," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 282-298.
    10. Bender, Christian, 2014. "Backward SDEs driven by Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2892-2916.
    11. Salah Eddine Choutri & Tembine Hamidou, 2018. "A Stochastic Maximum Principle for Markov Chains of Mean-Field Type," Games, MDPI, vol. 9(4), pages 1-21, October.
    12. 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.
    13. Wu, Zhen & Xu, Ruimin, 2019. "Probabilistic interpretation for Sobolev solutions of McKean–Vlasov partial differential equations," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 273-283.
    14. Sin, Myong-Guk & Ri, Kyong-Il & Kim, Kyong-Hui, 2022. "Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 190(C).
    15. Romuald Elie & Thibaut Mastrolia & Dylan Possamaï, 2019. "A Tale of a Principal and Many, Many Agents," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 440-467, May.
    16. Klimsiak, Tomasz & Rzymowski, Maurycy, 2023. "Nonlinear BSDEs on a general filtration with drivers depending on the martingale part of the solution," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 424-450.
    17. Rene Carmona, 2020. "Applications of Mean Field Games in Financial Engineering and Economic Theory," Papers 2012.05237, arXiv.org.
    18. Briand, Philippe & Cardaliaguet, Pierre & Chaudru de Raynal, Paul-Éric & Hu, Ying, 2020. "Forward and backward stochastic differential equations with normal constraints in law," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7021-7097.
    19. Yu-Jui Huang & Li-Hsien Sun, 2023. "Partial Information in a Mean-Variance Portfolio Selection Game," Papers 2312.04045, arXiv.org, revised Jan 2025.
    20. Cai, Yujie & Huang, Jianhui & Maroulas, Vasileios, 2015. "Large deviations of mean-field stochastic differential equations with jumps," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 1-9.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:37:y:2024:i:4:d:10.1007_s10959-024-01344-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.