A new family of one dimensional martingale couplings

TL;DR

This paper introduces a new family of martingale couplings for one-dimensional probability measures, generalizing existing couplings and providing bounds on expected absolute differences, with special cases for ordered measures.

Contribution

It presents a novel parametrized family of martingale couplings based on measure differences, including explicit inverse transform couplings and bounds on expected distances.

Findings

Contains the inverse transform martingale coupling explicitly.

Expected absolute difference is less than twice the Wasserstein-1 distance.

Generalizes to super and sub martingale couplings under convex order conditions.

Abstract

In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu$ and $\nu$ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $\mu$ and $\nu$. It contains the inverse transform martingale coupling which is explicit in terms of the associated cumulative distribution functions. The integral of $\vert x-y\vert$ with respect to each of these couplings is smaller than twice the $W_1$ distance between $\mu$ and $\nu$. When $\mu$ and $\nu$ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.