Compactness criterion for semimartingale laws and semimartingale optimal transport

We provide a compactness criterion for the set of laws P(Formula Persented) (Θ) on the Skorokhod space for which the canonical process X is a semimartingale having absolutely continuous characteristics with differential characteristics taking values in some given set Θ of Lévy triplets. Whereas boundedness of Θ implies tightness of P(Formula Persented) (Θ), closedness fails in general, even when choosing Θ to be additionally closed and convex, as a sequence of purely discontinuous martingales may converge to a diffusion. To that end, we provide a necessary and sufficient condition that prevents the purely discontinuous martingale part in the canonical representation of X to create a diffusion part in the limit. As a result, we obtain a sufficient criterion for P(Formula Persented)(Θ) to be compact, which turns out to be also a necessary one if the geometry of Θ is similar to a box on the product space. As an application, we consider a semimartingale optimal transport problem, where the transport plans are elements of P(Formula Persented) (Θ). We prove the existence of an optimal transport law P and obtain a duality result extending the classical Kantorovich duality to this setup.