Functional differential equations driven by c\`adl\`ag rough paths

The existence of unique solutions is established for rough differential equations (RDEs) with path-dependent coefficients and driven by c\`adl\`ag rough paths. Moreover, it is shown that the associated solution map, also known as It\^o-Lyons map, is locally Lipschitz continuous. These results are then applied to various classes of rough differential equations, such as controlled RDEs and RDEs with delay, as well as stochastic differential equations with delay. To that end, a joint rough path is constructed for a c\`adl\`ag martingale and its delayed version, that corresponds to stochastic It\^o integration.Comment: 30 page

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