Shadow couplings
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- by Mathias Beiglböck and Nicolas Juillet PDF
- Trans. Amer. Math. Soc. 374 (2021), 4973-5002
Abstract:
A classical result of Strassen asserts that given probabilities $\mu , \nu$ on the real line which are in convex order, there exists a martingale coupling with these marginals, i.e. a random vector $(X_1,X_2)$ such that $X_1\sim \mu , X_2\sim \nu$ and $\mathbb {E}[X_2|X_1]=X_1$. Remarkably, it is a non-trivial problem to construct particular solutions to this problem. Based on the concept of shadow for measures in convex order, we introduce a family of such martingale couplings, each of which admits several characterizations in terms of optimality properties/geometry of the support set/representation through a Skorokhod embedding. As a particular element of this family we recover the (left-)curtain martingale transport, which has recently been studied (see Beiglböck, Henry-Labordère, and Touzi [Stochastic Process. Appl. 127 (2017), pp. 3005–3013]; Beiglböck and Juillet [Ann. Probab. 44 (2016), pp. 42–106]; Campi, Laachir, and Martini [Finance Stoch. 21 (2017), pp. 471–486; Henry-Labordère and Touzi [Finance Stoch. 20 (2016), pp. 635–668]) and which can be viewed as a martingale analogue of the classical monotone rearrangement. As another canonical element of this family we identify a martingale coupling that resembles the usual product coupling and appears as an optimizer in the general transport problem recently introduced by Gozlan et al. In addition, this coupling provides an explicit example of a Lipschitz kernel, shedding new light on Kellerer’s proof of the existence of Markov martingales with specified marginals.References
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Additional Information
- Mathias Beiglböck
- Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgensternplatz 1, 1090 Wien, Austria
- Nicolas Juillet
- Affiliation: Nicolas Juillet Institut de Recherche Mathématique Avancée, UMR 7501 Université de Strasbourg et CNRS 7 rue René-Descartes, 67000 Strasbourg, France
- MR Author ID: 841634
- ORCID: 0000-0002-1258-3034
- Received by editor(s): March 15, 2019
- Received by editor(s) in revised form: October 27, 2020
- Published electronically: April 27, 2021
- Additional Notes: The first author acknowledges support through FWF grant Y782. The second author was partially supported by the “Programme ANR JCJC GMT” (ANR 2011 JS01 011 01).
- © Copyright 2021 by the authors
- Journal: Trans. Amer. Math. Soc. 374 (2021), 4973-5002
- MSC (2020): Primary 60G42
- DOI: https://doi.org/10.1090/tran/8380
- MathSciNet review: 4273182