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 Lévy...
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| Format: | Journal Article |
| Language: | English |
| Published: |
2021
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| Online Access: | https://hdl.handle.net/10356/149289 |
| _version_ | 1811687404860342272 |
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| author | Liu, Chong Neufeld, Ariel |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Liu, Chong Neufeld, Ariel |
| author_sort | Liu, Chong |
| collection | NTU |
| description | 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. |
| first_indexed | 2024-10-01T05:15:47Z |
| format | Journal Article |
| id | ntu-10356/149289 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T05:15:47Z |
| publishDate | 2021 |
| record_format | dspace |
| spelling | ntu-10356/1492892021-05-24T01:47:34Z Compactness criterion for semimartingale laws and semimartingale optimal transport Liu, Chong Neufeld, Ariel School of Physical and Mathematical Sciences Division of Mathematical Sciences Science::Mathematics Limit Theorem and Weak Compactness 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 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. Accepted version Financial support by the NAP Grant Machine Learning based Algorithms in Finance and Insurance and the Swiss National Foundation Grant SNF 200021_153555 is gratefully acknowledged 2021-05-24T01:47:34Z 2021-05-24T01:47:34Z 2019 Journal Article Liu, C. & Neufeld, A. (2019). Compactness criterion for semimartingale laws and semimartingale optimal transport. Transactions of the American Mathematical Society, 372, 187-231. https://dx.doi.org/10.1090/tran/7663 0002-9947 https://hdl.handle.net/10356/149289 10.1090/tran/7663 372 187 231 en NAP Grant Machine Learning based Algorithms in Finance and Insurance Swiss National Foundation grant SNF 200021_153555 Transactions of the American Mathematical Society © 2019 American Mathematical Society (AMS). All rights reserved. This paper was published in Transactions of the American Mathematical Society and is made available with permission of American Mathematical Society (AMS). application/octet-stream |
| spellingShingle | Science::Mathematics Limit Theorem and Weak Compactness Semimartingale Optimal Transport Liu, Chong Neufeld, Ariel Compactness criterion for semimartingale laws and semimartingale optimal transport |
| title | Compactness criterion for semimartingale laws and semimartingale optimal transport |
| title_full | Compactness criterion for semimartingale laws and semimartingale optimal transport |
| title_fullStr | Compactness criterion for semimartingale laws and semimartingale optimal transport |
| title_full_unstemmed | Compactness criterion for semimartingale laws and semimartingale optimal transport |
| title_short | Compactness criterion for semimartingale laws and semimartingale optimal transport |
| title_sort | compactness criterion for semimartingale laws and semimartingale optimal transport |
| topic | Science::Mathematics Limit Theorem and Weak Compactness Semimartingale Optimal Transport |
| url | https://hdl.handle.net/10356/149289 |
| work_keys_str_mv | AT liuchong compactnesscriterionforsemimartingalelawsandsemimartingaleoptimaltransport AT neufeldariel compactnesscriterionforsemimartingalelawsandsemimartingaleoptimaltransport |