solution to an SDE to existence properties of a related modified SDE. [@RufSDE] starts in a possibly explosive diffusion setting and gives a condition for martingality, which is in the spirit of absolute continuity conditions for laws of semimartingales as given by [@JS]. The proof of his condition is based on the Dambis-Dubins-Schwarz theorem and hence restricted to a continuous path setting.

A condition for martingality in a jump-diffusion setting can be deduced from the main result of [@CFY]. [@KMK; @Mayerhofer2011568] provide conditions for affine processes. Based on an extension of probability measures [@kardaras2015] provide techniques to construct strict local martingales. [@BR16] study martingality from a perspective of weak convergence.

In this article we raise the question how martingality of non-negative local martingales driven by Hilbert-space valued semimartingales on stochastic intervals relates to path properties of their driving processes. Building our framework on semimartingale problems (SMPs) in the sense of [@J79] allows us to connect martingality with several interesting properties of semimartingales. To briefly describe the concept let us be given a filtered space, a process and a triplet, which serves as a candidate for semimartingale characteristics. A probability measure under which the process is a semimartingale with the candidate 