Stability of martingale optimal transport and weak optimal transport

Abstract

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans π 1 , π 2 , . . . converges weakly to a transport plan π, then π is also optimal (between its marginals).Alfonsi, Corbetta and Jourdain [3] asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since optimal transport plans π are not characterized by a 'monotonicity'-property of supp π.In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet [32]. An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan et al.

Stability of the Weak Martingale Optimal Transport Problem

“…However this has previously not found much use as this cost functions does not fall in the realm of the typical regularity assumptions. Specifically, stability of WOT and MOT is established in two separate results in [7]. In the present article, we unify and extend these results, specifically we will prove the following: Theorem 1.1 (Stability).…”

“…As in the classical case, the WOT problem then admits a minimiser and a natural duality relation holds, see [3]. As in the case of classical optimal transport, stability is a more delicate question; in [7,Theorem 1.3] is established for continuous cost functions that are sufficiently bounded.…”

“…We also refer to [31,18,22] for the multidimensional case and to [8,13] for connections to Skorokhod problem. Stability of MOT, that is the continuous dependence of value and optimiser on the marginals, is a crucial property in light of its numerical resolution and the imperfect data coming available in financial applications and was recently established in [27,7,34].…”

“…Recently the notion of martingale C-monotonicity in [7] was introduced for WMOT to show stability of MOT.…”

The Wasserstein space of stochastic processes

“…The case N = 2 of Theorem 1.7 corresponds to [10, Lemma 2.6] which has recently seen applications to the Schrödinger problem (see e.g. the survey of Leonard [55]), martingale optimal transport, the weak transport problem (see [36,6] among many others), the theory of linear transfers ( [25]), and mathematical finance / martingale optimal transport, we refer to [10,15,2,12].…”