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For two measures that are in convex-decreasing order, Nutz and Stebegg (Canonical supermartingale couplings, Ann. Probab. 46(6) 3351–3398, 2018) studied the optimal transport problem with supermartingale constraints and introduced two canonical couplings, namely the increasing and decreasing transport plans, that are optimal for a large class of cost functions. In the present paper we provide an explicit construction of the decreasing coupling by establishing a Brenier-type result: (a generalised version of) this coupling concentrates on the graphs of two functions. Our construction is based on the concept of the supermartingale shadow measure and requires a suitable extension of the results by Juillet (Stability of the shadow projection and the left-curtain coupling, Ann. Inst. H. Poincaré Probab. Statist. 52(4) 1823–1843, November 2016) and Beiglböck and Juillet (Shadow couplings, Trans. Amer. Math. Soc. 374 4973–5002, 2021) established in the martingale setting. In particular, we prove the stability of the supermartingale shadow measure with respect to initial and target measures introduce an infinite family of lifted supermartingale couplings that arise via shadow measure, and show how to explicitly determine the martingale points of each such coupling.
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A potential-based construction of the increasing supermartingale coupling
The increasing supermartingale coupling, introduced by Nutz and Stebegg (Ann. Probab. 46 (2018) 3351–3398) is an extreme point of the set of “supermartingale” couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively. Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (Electron. Commun. Probab. 27 (2022) 1–12) obtained in the martingale setting.
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- Award ID(s):
- 2106556
- PAR ID:
- 10535682
- Editor(s):
- Friz, Peter
- Publisher / Repository:
- IMS
- Date Published:
- Journal Name:
- The Annals of Applied Probability
- Volume:
- 33
- Issue:
- 5
- ISSN:
- 1050-5164
- Subject(s) / Keyword(s):
- Couplings, supermartingales, optimal transport.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1214/22-AAP1907
