The convergence of the first order Euler scheme and an approximative variant thereof, along with convergence rates, are established for rough differential equations driven by c\`adl\`ag paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with vanishing mesh size. This property is then verified for almost all sample paths of Brownian motion, It\^o processes, L\'evy processes and general c\`adl\`ag semimartingales, as well as the driving signals of both mixed and rough stochastic differential equations, relative to various time discretizations. Consequently, we obtain pathwise convergence in p-variation of the Euler--Maruyama scheme for stochastic differential equations driven by these processes.

翻译：针对由满足适当准则（即所谓(RIE)性质）的càdlàg路径驱动的粗糙微分方程，建立了基于时间离散化且网格尺寸趋于零的一阶Euler格式及其近似变体的收敛性及其收敛速率。随后，针对布朗运动、Itô过程、Lévy过程、一般càdlàg半鞅以及混合与粗糙随机微分方程的驱动信号，验证了该性质在几乎所有样本路径上的成立性，并考虑了多种时间离散化方式。由此，我们获得了由这些过程驱动的随机微分方程的Euler-Maruyama格式在p-变差意义下的逐轨收敛性。