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-赫尔德连续样本路径的随机过程{Yₜ}ₜ驱动的随机常微分方程dXₜ/dt = f(t, Xₜ, Yₜ)，其解近似中欧拉方法的强收敛阶关于时间步长估计为θ，前提是函数f=f(t,x,y)满足充分正则性及适当有界条件。本文证明，在许多典型情形下，可进一步利用噪声的额外条件，使得强收敛阶实际达到1，与样本路径的赫尔德正则性无关。该结论适用于加性或乘性伊藤过程噪声（如维纳过程、奥恩斯坦-乌伦贝克过程及几何布朗运动）、点过程噪声（如泊松点过程及霍克斯自激过程，后者甚至具有跳跃型不连续性），以及样本路径有界变差的输运型过程。证明基于一种新方法：将全局误差表示为大网格与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至大尺度。本文辅以数值模拟，展示上述情形下强收敛阶为1的数值验证；同时给出赫斯特参数0<H<1/2的分数布朗运动噪声示例，其收敛阶为H+1/2，虽低于前述示例达到的阶数1，但仍高于既往研究预期的收敛阶H。