In this paper, we introduce fully implementable, adaptive Euler–Maruyama schemes for McKean–Vlasov stochastic differential equations (SDEs) assuming only a standard monotonicity condition on the drift and diffusion coefficients but no global Lipschitz continuity in the state variable for either, while global Lipschitz continuity is required for the measure component. We prove moment stability of the discretised processes and a strong convergence rate of $1∕2$. Several numerical examples, centred around a mean-field model for FitzHugh–Nagumo neurons, illustrate that the standard uniform scheme fails and that the adaptive approach shows in most cases superior performance to tamed approximation schemes. In addition, we introduce and analyse an adaptive Milstein scheme for a certain sub-class of McKean–Vlasov SDEs with linear measure-dependence of the drift.

在本文中，我们为 McKean-Vlasov 随机微分方程 (SDE) 引入了完全可实现的自适应 Euler-Maruyama 方案，假设仅在漂移和扩散系数上存在标准单调性条件，但在状态变量中没有全局 Lipschitz 连续性，而全局测量组件需要 Lipschitz 连续性。我们证明了离散过程的矩稳定性和强大的收敛速度$1∕2$. 以 FitzHugh-Nagumo 神经元的平均场模型为中心的几个数值示例说明标准统一方案失败，并且自适应方法在大多数情况下显示出优于驯服近似方案的性能。此外，我们介绍并分析了具有线性测量依赖性漂移的 McKean-Vlasov SDE 的某个子类的自适应 Milstein 方案。