In the present work, we delve into further study of numerical approximations of SDEs with non-globally monotone coefficients. We design and analyze a new family of stopped increment-tamed time discretization schemes of Euler, Milstein and order 1.5 type for such SDEs. By formulating a novel unified framework, the proposed methods are shown to possess the exponential integrability properties, which are crucial to recovering convergence rates in the non-global monotone setting. Armed with such exponential integrability properties and by the arguments of perturbation estimates, we successfully identify the optimal strong convergence rates of the aforementioned methods in the non-global monotone setting. Numerical experiments are finally presented to corroborate the theoretical results.

翻译：本研究深入探讨了具有非全局单调系数的随机微分方程数值逼近问题。针对此类方程，我们设计并分析了一类新型的停时增量驯服时间离散化格式，包括Euler型、Milstein型及1.5阶型方法。通过构建新颖的统一框架，证明了所提方法具有指数可积性，这一特性对于在非全局单调情形下恢复收敛速率至关重要。借助该指数可积性特性并结合摄动估计论证，我们成功确定了上述方法在非全局单调情形下的最优强收敛速率。最后通过数值实验验证了理论结果。