triplet as characteristics is called a solution to the SMP. We particularly allow for infinite-dimensions and stochastic lifetimes.

Let $Z$ be a non-negative local martingale (usually) defined on a filtered probability space equipped with a solution to a given SMP. In the diffusion setting this boils down to the assumption that $Z$ is driven by  . We now summarize our main contributions.

Firstly, we prove under mild topological assumptions that if $Z$ is a martingale then a modified SMP, comparable to the modified SDE , has a solution, c.f. Theorem \[main theorem new 2\]. This is not only an existence result for solutions to SMPs, but also provides a condition for strict local martingality via contradiction.

Secondly, we replace the topological assumption by a uniqueness condition on the modified SMP and assume existence of a solution. Our uniqueness condition is related to the definition of local uniqueness used by [@JS] in their study of absolute continuity of laws. In Markovian settings it is implied by well-posedness of the SMP. If the SMP corresponds to a not necessarily Markovian SDE the uniqueness is implied by pathwise-uniqueness. In this setting we formulate abstract convergence conditions in terms of a sequence of stopping 