On the stability of the martingale optimal transport problem: A set-valued map approach

TL;DR

This paper introduces a set-valued map approach to analyze the stability of the martingale optimal transport problem, demonstrating continuity and compactness properties of the set of optimal measures with respect to marginals.

Contribution

It provides a novel perspective using set-valued maps to establish continuity and compactness results in martingale optimal transport, extending previous findings.

Findings

Set of martingale measures is continuous w.r.t. marginals.

Compactness of the set of optimizers is established.

Upper hemicontinuity of optimizers w.r.t. marginals is shown.

Abstract

Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in Backhoff-Veraguas and Pammer [2] and Wiesel [21]. We present a new perspective of this result using the theory of set-valued maps. In particular, using results from Beiglb\"ock, Jourdain, Margheriti, and Pammer [5], we show that the set of martingale measures with fixed marginals is continuous, i.e., lower- and upper hemicontinuous, w.r.t. its marginals. Moreover, we establish compactness of the set of optimizers as well as upper hemicontinuity of the optimizers w.r.t. the marginals.