# Martingale optimal transport with stopping

## TL;DR

This paper addresses the martingale optimal transport problem involving stopping times, introducing a measure-valued approach, and characterizing solutions via viscosity solutions to HJB equations, with explicit solutions for certain costs.

## Contribution

It develops a measure-valued martingale framework for the problem and characterizes solutions through viscosity solutions of HJB equations, extending previous methods.

## Key findings

- Explicit solutions for a class of cost functions.
- The solution is the concave envelope of the cost function.
- Finite-dimensional approximations are characterized as viscosity solutions.

## Abstract

We solve the martingale optimal transport problem for cost functionals represented by optimal stopping problems. The measure-valued martingale approach developed in ArXiv: 1507.02651 allows us to obtain an equivalent infinite-dimensional controller-stopper problem. We use the stochastic Perron's method and characterize the finite dimensional approximation as a viscosity solution to the corresponding HJB equation. It turns out that this solution is the concave envelope of the cost function with respect to the atoms of the terminal law. We demonstrate the results by finding explicit solutions for a class of cost functions.