computation of option prices as highlighted in the influential work [@GEK95]. In the context of utility maximization, [@RePEc:wsi:ijtafx:v:13:y:2010:i:03:p:459-477] connects the martingality of a local martingale with optimal terminal wealth. In all these cases, criteria for martingality of non-negative local martingales are substantial.

Recently also strict local martingales have attracted more attention and are used to model financial bubbles, c.f. [@Cox2005; @Jarrow2007; @MAFI:MAFI394; @kardaras2015; @Pal20101424; @Pro(2013)]. Examples of strict local martingales are given by [@Chybiryakov2007; @Delbaen1995; @ELY; @KellerRessel2015; @MU(2012); @protter2015strict]. Martingality of local martingales is also interesting from a purely probabilistic point of view, as it determines absolute continuity or singularity of laws. This observation was exploited in the seminal work [@doi:10.1137/1105027], c.f. also [@JM76; @KLS-LACOM1; @KLS-LACOM2] for a semimartingale perspective. Such absolute continuity results are valuable tools to study the statistics of stochastic processes, c.f. [@10.2307/4616544], and existence properties of SDEs and martingale problems, c.f. [@deprato; @KaraShre; @RY].

Let us shortly summarize the structure of the article. In Section \[First important Observations\] we present simple characterizations of martingality of a non-negative local martingale. These observations prove useful for establishing our main results. In Section  \[Hilbert-Vlaued Semimartingales and Semimartingale Problems on Stochastic Intervals\] we introduce our mathematical setting. The main results 