It is well known that the Euler method for 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. This order is known to increase to $1$ in some special cases. Here, it is proved that, in many more typical cases, further structures on the noise can be exploited so that the strong convergence is of order 1. In fact, we prove so for any semi-martingale noise. This includes It\^o diffusion processes, point-process noises, transport-type processes with sample paths of bounded variation, and time-changed Brownian motion. The result follows from estimating the global error as an iterated integral over both large and small mesh scales, and by switching the order of integration to move the critical regularity to the large scale. The work is complemented with numerical simulations showing the optimality of the strong order 1 convergence in those cases, and with an example with fractional Brownian motion noise with Hurst parameter $0 < H < 1/2,$ which is not a semi-martingale and for which the order of convergence is $H + 1/2$, hence lower than the attained order 1 in the semi-martingale case, but still higher than the order $H$ of convergence expected from previous works.

翻译：众所周知，对于由样本路径具有θ-Hölder连续性的随机过程{Y_t}_t驱动的随机常微分方程dX_t/dt = f(t, X_t, Y_t)，若f = f(t, x, y)具有足够的正则性及适当的界，则欧拉方法关于时间步长的强收敛阶估计为θ。在某些特殊情况下，已知该收敛阶可提升至1。本文证明，在许多更典型的情形中，可以进一步利用噪声的结构特性，使强收敛阶达到1。事实上，我们证明该结论对任意半鞅噪声均成立。这包括Itô扩散过程、点过程噪声、具有有界变差样本路径的传输型过程，以及时间变换的布朗运动。该结果通过将全局误差估计为在大尺度和小尺度网格上的迭代积分，并通过交换积分顺序将关键正则性转移至大尺度而获得。研究辅以数值模拟，展示了这些情况下强一阶收敛的最优性，并给出了一个赫斯特参数0 < H < 1/2的分式布朗运动噪声的示例——该噪声并非半鞅，其收敛阶为H + 1/2，虽低于半鞅情形下达到的1阶，但仍高于先前研究中预期的H阶收敛。