Shadow couplings

Abstract

A classical result of Strassen asserts that given probabilities $\mu, \nu$ on the real line which are in convex order, there exists a \emph{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 $E[X_2|X_1]=X_1$. 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.