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