Compactness Criterion for Semimartingale Laws and Semimartingale Optimal Transport

We provide a compactness criterion for the set of laws $\mathfrak{P}^{ac}_{sem}(\Theta)$ 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 L\'evy triplets. Whereas boundedness of $\Theta$ implies tightness of $\mathfrak{P}^{ac}_{sem}(\Theta)$, 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 $\mathfrak{P}^{ac}_{sem}(\Theta)$ 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 $\mathfrak{P}^{ac}_{sem}(\Theta)$. We prove the existence of an optimal transport law $\widehat{\mathbb{P}}$ and obtain a duality result extending the classical Kantorovich duality to this setup.