M. Beiglböck, N. Juillet:

"Shadow couplings";

arXiv.org e-Print archive,September 12 (submitted)(2016), S. 1 - 20.

A classical result of Strassen asserts that given probabilities μ,ν on the real line which are in convex order, there exists a \emph{martingale coupling} with these marginals, i.e.\ a random vector (X1,X2) such that X1∼μ,X2∼ν and E[X2|X1]=X1. Remarkably, it is a non trivial problem to construct particular solutions to this problem. In this article, 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 \cite{BeJu16, HeTo13, CaLaMa14, BeHeTo15} 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 \emph{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.

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