Shadow couplings
Mathias Beiglboeck, Nicolas Juillet

TL;DR
This paper introduces a family of martingale couplings for probability measures in convex order, including the curtain martingale transport and a product-like coupling, with applications to optimal transport and Markov martingales.
Contribution
It constructs and characterizes a new family of martingale couplings with various optimality and geometric properties, expanding the understanding of solutions to the classical martingale coupling problem.
Findings
Introduces a family of martingale couplings with multiple characterizations.
Recovers the curtain martingale transport as a special case.
Identifies a coupling resembling the product coupling that optimizes a transport problem.
Abstract
A classical result of Strassen asserts that given probabilities on the real line which are in convex order, there exists a \emph{martingale coupling} with these marginals, i.e.\ a random vector such that and . 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…
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Stochastic processes and statistical mechanics · Random Matrices and Applications
