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A construction of the left-curtain coupling
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Hobson, David G. and Norgilas, Dominykas (2022) A construction of the left-curtain coupling. Electronic Journal of Probability, 27 . pp. 1-46. 147. doi:10.1214/22-ejp868 ISSN 1083-6489.
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Official URL: https://doi.org/10.1214/22-ejp868
Abstract
In a martingale optimal transport (MOT) problem mass distributed according to the law μ is transported to the law ν in such a way that the martingale property is respected. Beiglböck and Juillet (On a problem of optimal transport under marginal martingale constraints, Annals of Probability, 44(1):42-106, 2016) introduced a solution to the MOT problem which they baptised the left-curtain coupling. The left-curtain coupling has been widely studied and shown to have many applications, including to martingale inequalities and the model-independent pricing of American options. Beiglböck and Juillet proved existence and uniqueness, proved optimality for a family of cost functions, and proved that when μ is a continuous distribution, mass at x is mapped to one of at most two points, giving lower and upper functions. Henry-Labordère and Touzi (An explicit martingale version of Brenier’s theorem, Finance and Stochastics, 20:635-668, 2016) showed that the left-curtain coupling is optimal for an extended family of cost functions and gave a construction of the upper and lower functions under an assumption that μ and ν are continuous, together with further simplifying assumptions of a technical nature.
In this article we construct these upper and lower functions in the general case of arbitrary centred measures in convex order, and thereby give a complete construction of the left-curtain coupling. In the case where μ has atoms these upper and lower functions are to be interpreted in the sense of a lifted martingale.
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): | Martingales (Mathematics) | ||||||
Journal or Publication Title: | Electronic Journal of Probability | ||||||
Publisher: | Institute of Mathematical Statistics | ||||||
ISSN: | 1083-6489 | ||||||
Official Date: | 17 November 2022 | ||||||
Dates: |
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Volume: | 27 | ||||||
Page Range: | pp. 1-46 | ||||||
Article Number: | 147 | ||||||
DOI: | 10.1214/22-ejp868 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Date of first compliant deposit: | 28 November 2022 | ||||||
Date of first compliant Open Access: | 28 November 2022 |
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