Monotonicity Preserving Transformations of MOT and SEP

TL;DR

This paper explores transformations that preserve monotonicity in martingale optimal transport and Skorokhod embedding problems, revealing symmetries and geometric insights into optimal solutions.

Contribution

It characterizes monotonicity preserving transformations linking MOT and SEP, providing new understanding of their symmetries and geometric structure of optimizers.

Findings

Transformations preserve monotonicity between MOT and SEP solutions.

Symmetries identified between different cost functions in MOT.

Geometric structure of optimizers clarified through SEP counterparts.

Abstract

Recently, \cite{BeJu16, BeNuTo16} established that optimizers to the martingale optimal transport problem (MOT) are concentrated on $c$-monotone sets. In this article we characterize monotonicity preserving transformations revealing certain symmetries between optimizers of MOT for different cost functions. Due to the intimate connection of MOT and the Skorokhod embedding problem (SEP) these transformations are also monotonicity preserving and disclose symmetries for certain solutions to the optimal SEP. Furthermore, the SEP picture allows to easily understand the geometry of these transformations once we have established the SEP counterparts to the known solutions of MOT based on the monotonicity principle for SEP which in turn allows to directly read off the structure of the MOT optimizers.