# Martingale optimal transport duality

## TL;DR

This paper develops a dual representation for the Kantorovich functional in the context of martingale optimal transport, utilizing quotient sets, Choquet capacities, and a regularized topology on the Skorokhod space.

## Contribution

It introduces a novel dual formulation of the Kantorovich functional for martingale measures using quotient sets and Choquet capacities, with new regularization techniques on the Skorokhod space.

## Key findings

- Dual representation of Kantorovich functional established
- Choquet capacity generated by martingale measures constructed
- Regularized S-topology on Skorokhod space developed

## Abstract

We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given. Another important ingredient of our analysis is a regularized version of Jakubowski's $S$-topology on the Skorokhod space.