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Strong error analysis and first-order convergence of Milstein-type schemes for McKean–Vlasov SDEs with superlinear coefficients

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  • Zhu, Jingtao
  • Zhao, Yuying
  • Gan, Siqing

Abstract

In the study of McKean–Vlasov stochastic differential equations (MV-SDEs), numerical approximation plays a crucial role in understanding the behavior of interacting particle systems (IPS). Classical Milstein schemes provide strong convergence of order one under globally Lipschitz coefficients. Nevertheless, many MV-SDEs arising from applications possess super-linearly growing drift and diffusion terms, where classical methods may diverge and particle corruption can occur. In the present work, we aim to fill this gap by developing a unified class of Milstein-type discretizations handling both super-linear drift and diffusion coefficients. The proposed framework includes the tamed-, tanh-, and sine-Milstein methods as special cases and establishes order-one strong convergence for the associated interacting particle system under mild regularity assumptions, requiring only once differentiable coefficients. In particular, our results complement Chen et al. (2025), where a taming-based Euler scheme was only tested numerically without theoretical guarantees, by providing a rigorous convergence theory within a broader Milstein-type framework. The analysis relies on discrete-time arguments and binomial-type expansions, avoiding the continuous-time Itô approach that is standard in the literature. Numerical experiments are presented to illustrate the convergence behavior and support the theoretical findings.

Suggested Citation

  • Zhu, Jingtao & Zhao, Yuying & Gan, Siqing, 2026. "Strong error analysis and first-order convergence of Milstein-type schemes for McKean–Vlasov SDEs with superlinear coefficients," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 246(C), pages 697-726.
    • Handle: RePEc:eee:matcom:v:246:y:2026:i:c:p:697-726
    DOI: 10.1016/j.matcom.2026.02.030
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    References listed on IDEAS

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    1. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    2. Chen, Xingyuan & dos Reis, Gonçalo, 2022. "A flexible split‐step scheme for solving McKean‐Vlasov stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 427(C).
    3. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    4. 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.
    5. Mike Giles, 2008. "Improved Multilevel Monte Carlo Convergence using the Milstein Scheme," Springer Books, in: Alexander Keller & Stefan Heinrich & Harald Niederreiter (ed.), Monte Carlo and Quasi-Monte Carlo Methods 2006, pages 343-358, Springer.
    6. Jianhai Bao & Xing Huang, 2022. "Approximations of McKean–Vlasov Stochastic Differential Equations with Irregular Coefficients," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1187-1215, June.
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