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

Jingtao Zhu, Yuying Zhao and Siqing Gan

Mathematics and Computers in Simulation (MATCOM), 2026, vol. 246, issue C, 697-726

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.

Keywords: McKean–Vlasov SDEs; Superlinear growth; Milstein-type schemes; Order of strong convergence (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:246:y:2026:i:c:p:697-726

DOI: 10.1016/j.matcom.2026.02.030

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