A Skorohod measurable universal functional representation of solutions to semimartingale SDEs

TL;DR

This paper establishes a universal measurable functional representation for solutions of semimartingale-driven SDEs, enabling pathwise analysis and applications like Malliavin derivatives for jump processes.

Contribution

It proves the existence of a Skorohod measurable functional that represents solutions of a broad class of semimartingale SDEs as functions of their driving processes.

Findings

Paths of solutions are measurable functions of driving processes.

Applicable to Malliavin calculus for jump Lévý processes.

Provides a framework for pathwise solution analysis.

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.