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-Hölder样本路径的随机过程$\{Y_t\}_t$驱动的随机常微分方程$\mathrm{d}X_t/\mathrm{d}t = f(t, X_t, Y_t)$，若函数$f=f(t,x,y)$满足充分的正则性及适当有界条件，则其解逼近的欧拉方法关于时间步长具有强收敛阶θ的估计。本文证明，在许多典型情形下，可进一步利用噪声的结构特性使得强收敛阶实际上达到1阶，而与样本路径的Hölder正则性无关。该结论适用于加性或乘性伊藤过程噪声（如Wiener过程、Ornstein-Uhlenbeck过程及几何Brown运动过程）、点过程噪声（如泊松点过程和Hawkes自激过程——后者甚至具有跳跃型间断性），以及具有有界变差样本路径的输运型过程。研究基于一种新方法：将全局误差表示为粗细网格尺度上的迭代积分，并通过交换积分次序将关键正则性转移至粗尺度。文中辅以数值模拟验证了这些情形下的强1阶收敛性，并给出分数阶Brown运动噪声（Hurst参数$0<H<1/2$）的算例——其收敛阶为$H+1/2$，虽低于前述算例的1阶收敛阶，但仍高于先前工作中预期的H阶收敛阶。