To obtain strong convergence rates of numerical schemes, an overwhelming majority of existing works impose a global monotonicity condition on coefficients of SDEs. On the contrary, a majority of SDEs from applications do not have globally monotone coefficients. As a recent breakthrough, the authors of [Hutzenthaler, Jentzen, Ann. Probab., 2020] originally presented a perturbation theory for stochastic differential equations (SDEs), which is crucial to recovering strong convergence rates of numerical schemes in a non-globally monotone setting. However, only a convergence rate of order $1/2$ was obtained there for time-stepping schemes such as a stopped increment-tamed Euler-Maruyama (SITEM) method. As an open problem, a natural question was raised by the aforementioned work as to whether higher convergence rate than $1/2$ can be obtained when higher order schemes are used. The present work attempts to solve the tough problem. To this end, we develop some new perturbation estimates that are able to reveal the order-one strong convergence of numerical methods. As the first application of the newly developed estimates, we identify the expected order-one pathwise uniformly strong convergence of the SITEM method for additive noise driven SDEs and multiplicative noise driven second order SDEs with non-globally monotone coefficients. As the other application, we propose and analyze a positivity preserving explicit Milstein-type method for Lotka-Volterra competition model driven by multi-dimensional noise, with a pathwise uniformly strong convergence rate of order one recovered under mild assumptions. These obtained results are completely new and significantly improve the existing theory. Numerical experiments are also provided to confirm the theoretical findings.

翻译：为获得数值格式的强收敛阶，现有绝大多数研究均对随机微分方程的系数施加全局单调性条件。然而，实际应用中的大多数随机微分方程并不具有全局单调系数。作为近期重大突破，[Hutzenthaler, Jentzen, Ann. Probab., 2020] 的作者原创性地提出了随机微分方程的摄动理论，这对在非全局单调背景下恢复数值格式的强收敛阶至关重要。然而，对于如停步增量驯化欧拉-丸山（SITEM）法等时间步进格式，该研究仅获得了 $1/2$ 阶收敛率。前述工作提出一个自然问题作为公开难题：当采用高阶格式时，能否获得高于 $1/2$ 阶的收敛率？本文尝试解决这一棘手问题。为此，我们发展了若干全新的摄动估计，这些估计能够揭示数值方法的一阶强收敛性。作为新估计的首个应用，我们证实了SITEM法在加性噪声驱动的随机微分方程及非全局单调系数下乘性噪声驱动二阶随机微分方程中具有预期的一阶路径一致强收敛性。作为另一应用，我们提出并分析了一类保正的显式米尔斯坦型方法，用于多维噪声驱动的洛特卡-沃尔泰拉竞争模型，并在温和假设下恢复了其路径一致强收敛的一阶收敛率。这些结果具有全新性，显著改进了现有理论。同时提供数值实验以验证理论发现。