Compactness criterion for semimartingale laws and semimartingale optimal transport

• Main Authors: Liu, Chong, Neufeld, Ariel
• Other Authors: School of Physical and Mathematical Sciences
• Format: Article
• Language: English
• Published: 2021
• Subjects: Science::Mathematics; Limit Theorem and Weak Compactness; Semimartingale Optimal Transport
• Online Access: https://hdl.handle.net/10356/149289
• Institution: Nanyang Technological University

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. 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.