COMPACTNESS CRITERION FOR SEMIMARTINGALE LAWS AND SEMIMARTINGALE OPTIMAL TRANSPORT

We provide a compactness criterion for the set of laws beta(ac)(sem) (Theta) 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 (Theta) of Levy triplets. Whereas boundedness of Theta implies tightness of beta(ac)(sem) (Theta) closedness fails in general, even when choosing Theta 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 beta(ac)(sem)(Theta) to be compact, which turns out to be also a necessary one if the geometry of Theta 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 beta(ac)(sem)(Theta). We prove the existence of an optimal transport law (P) over cap and obtain a duality result extending the classical Kantorovich duality to this setup.