Compactness criterion for semimartingale laws and semimartingale optimal transport
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- by Chong Liu and Ariel Neufeld PDF
- Trans. Amer. Math. Soc. 372 (2019), 187-231 Request permission
Abstract:
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.
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Additional Information
- Chong Liu
- Affiliation: Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
- Email: chong.liu@math.ethz.ch
- Ariel Neufeld
- Affiliation: Division of Mathematical Sciences, Nanyang Technological University, Singapore
- MR Author ID: 1028695
- Email: ariel.neufeld@ntu.edu.sg
- Received by editor(s): April 30, 2017
- Received by editor(s) in revised form: February 9, 2018
- Published electronically: March 19, 2019
- Additional Notes: Financial support by the NAP Grant and the Swiss National Foundation Grant SNF 200021$\_$153555 is gratefully acknowledged
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 187-231
- MSC (2010): Primary 60F05, 60G44, 93E20
- DOI: https://doi.org/10.1090/tran/7663
- MathSciNet review: 3968767