  In the spirit of Marcus canonical stochastic differential equations, we study
a similar notion of rough differential equations (RDEs), notably dropping the
assumption of continuity prevalent in the rough path literature. A new metric
is exhibited in which the solution map is a continuous function of the driving
rough path and a so-called path function, which directly models the effect of
the jump on the system. In a second part, we show that general multidimensional
semimartingales admit canonically defined rough path lifts. An extension of
L\'epingle's BDG inequality to this setting is given, and in turn leads to a
number of novel limit theorems for semimartingale driven differential
equations, both in law and in probability, conveniently phrased via
Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of
examples illustrate the scope of our results.
