Compactness criterion for semimartingale laws and semimartingale optimal transport Article Swipe
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· 2018
· Open Access
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· DOI: https://doi.org/10.1090/tran/7663
· OA: W2963147472
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 Lévy 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. \n \nAs 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.
