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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
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    References listed on IDEAS

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    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.
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