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In this paper we prove existence of the regular conditional distribution of strong solutions to a large class of semimartingale-driven stochastic differential equations. For proving this result, we show for the first time that solutions of these stochastic differential equations can be written as measurable functions of their driving processes into the space of all càdlàg functions equipped with the Borel algebra generated by all open sets with respect to the Skorohod metric. As a corollary, the two theorems prove a Markov property, which is for example relevant in computational stochastics.

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We present the first higher-order approximation scheme for solutions of jump-diffusion stochastic differential equations with discontinuous drift. For this transformation-based jump-adapted quasi-Milstein scheme we prove L p -convergence order 3/4. To obtain this result, we prove that under slightly stronger assumptions (but still weaker than anything known before) a related jump-adapted quasi-Milstein scheme has convergence order 3/4 -in a special case even order 1. Order 3/4 is conjectured to be optimal.