A Skorohod measurable universal functional representation of solutions to semimartingale SDEs

Abstract

In this paper we show the existence of a universal Skorohod measurable functional representation for a large class of semimartingale-driven stochastic differential equations. For this we prove that paths of the strong solutions of stochastic differential equations can be written as measurable functions of the paths of their driving processes into the space of all c\`adl\`ag functions equipped with the Borel sigma-field generated by all open sets with respect to the Skorohod metric. This result can be applied to calculate Malliavin derivatives for SDEs driven by pure-jump L\'evy processes with drift.