Regular conditional distributions for semimartingale SDEs

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 c\`adl\`ag 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.