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The potential of the shadow measure
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Beiglböck, Mathias, Hobson, David (David G.) and Norgilas, Dominykas (2022) The potential of the shadow measure. Electronic Communications in Probability, 27 . pp. 1-12. doi:10.1214/22-ecp457 ISSN 1083-589X.
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Official URL: https://doi.org/10.1214/22-ecp457
Abstract
It is well known that given two probability measures μ and ν on
R
in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Statistics | ||||||
SWORD Depositor: | Library Publications Router | ||||||
Library of Congress Subject Headings (LCSH): | Probabilities, Convex functions, Martingales (Mathematics), Mathematical optimization, Partitions (Mathematics) | ||||||
Journal or Publication Title: | Electronic Communications in Probability | ||||||
Publisher: | Institute of Mathematical Statistics | ||||||
ISSN: | 1083-589X | ||||||
Official Date: | 2 March 2022 | ||||||
Dates: |
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Volume: | 27 | ||||||
Page Range: | pp. 1-12 | ||||||
DOI: | 10.1214/22-ecp457 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 28 November 2022 | ||||||
Date of first compliant Open Access: | 28 November 2022 |
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