The modeling and approximation of stochastic differential equations driven by semi-martingales with both jump and continuous components are considered. A model, which is a generalization of Mcshane's canonical extension and the stochastic differential equations of Fisk-Stratonovich, is defined and analyzed by means of the theory of semimartingales. It is proved that this generalized canonical extension possesses many of the same desirable properties as that of Mcshane, but it is applicable to a much wider class of noise processes. In particular, several approximation (or continuity or stability) results are proved; these show (under various sets of hypotheses and in various topologies) that if a sequence of noise processeszmconverges to a noise processz, then the solutions of the canonical extension corresponding tozmconverge to the solution corresponding toz.