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

Pathwise convergence of the Euler scheme for rough and stochastic differential equations