Compactness criterion for semimartingale laws and semimartingale optimal transport - Liu, Chong (2021)
We provide a compactness criterion for the set of laws $ \mathfrak{P}^{ac}_{sem}(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 $ \mathfrak{P}^{ac}_{sem}(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 $ \mathfrak{P}^{ac}_{sem}(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.