It is well known that the Euler method for approximating the solutions of a random ordinary differential equation $\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$ driven by a stochastic process $\{Y_t\}_t$ with $\theta$-H\"older sample paths is estimated to be of strong order $\theta$ with respect to the time step, provided $f=f(t, x, y)$ is sufficiently regular and with suitable bounds. Here, it is proved that, in many typical cases, further conditions on the noise can be exploited so that the strong convergence is actually of order 1, regardless of the H\"older regularity of the sample paths. This applies for instance to additive or multiplicative It\^o process noises (such as Wiener, Ornstein-Uhlenbeck, and geometric Brownian motion processes); to point-process noises (such as Poisson point processes and Hawkes self-exciting processes, which even have jump-type discontinuities); and to transport-type processes with sample paths of bounded variation. The result is based on a novel approach, estimating the global error as an iterated integral over both large and small mesh scales, and switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations illustrating the strong order 1 convergence in those cases, and with an example with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2$ for which the order of convergence is $H + 1/2$, hence lower than the attained order 1 in the examples above, but still higher than the order $H$ of convergence expected from previous works.

翻译：众所周知，对于由具有θ-赫尔德样本路径的随机过程$\{Y_t\}_t$驱动的随机常微分方程$\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$，在$f=f(t, x, y)$充分正则且具有适当有界性的条件下，其欧拉方法近似解的强收敛阶关于时间步长估计为θ。本文证明，在许多典型情形中，可借助噪声的进一步条件使强收敛阶实际达到1，而与样本路径的赫尔德正则性无关。这适用于加性或多乘性伊藤过程噪声（如维纳过程、奥恩斯坦-乌伦贝克过程及几何布朗运动过程）、点过程噪声（如泊松点过程与霍克斯自激过程，甚至包含跳跃型间断），以及具有有界变差样本路径的输运型过程。该结果基于一种新方法：将全局误差估计为大小网格尺度上的迭代积分，并通过交换积分次序将临界正则性转移到大尺度上。本文辅以数值模拟，展示上述情形中强1阶收敛性，并给出赫斯特参数$0 < H < 1/2$的分数布朗运动噪声示例，其收敛阶为$H+1/2$，虽低于上述示例中达到的1阶，但仍高于先前研究预期的收敛阶$H$。