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.Comment: 43 page