Optimal martingale transport between radially symmetric marginals in general dimensions

TL;DR

This paper characterizes the optimal couplings for the martingale transport problem between radially symmetric distributions in any dimension, establishing their structure and uniqueness for minimizing a specific cost functional.

Contribution

It provides a complete description and proof of uniqueness for the optimal martingale couplings between radially symmetric marginals in arbitrary dimensions.

Findings

Optimal couplings are explicitly characterized for radially symmetric marginals.

Uniqueness of the optimal solution is established.

The results hold for the cost functional with exponent 0<p≤1 in any dimension.

Abstract

We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $\mu, \nu$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimension $d$ is arbitrary.