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

Abstract

Continuity of the value of the martingale optimal transport problem on the real line w.r.t. its marginals was recently established in [2] and [19]. We present a new perspective of this result using the theory of set-valued maps. In particular, using results from [4], 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.

A deep learning approach to data-driven model-free pricing and to martingale optimal transport

“…Being similar can for example mean to be close in Wasserstein distance or being from the same parametric family of probability distributions. (b) Since the proof of Theorem 3.1 relies on the continuity of the MOT problem, as stated in [4], [47], and [55], for which -at the moment -no extension to the case with more than two marginals or in multidimensional settings is known, we stick to the two marginal case in the one-dimensional setting. (c) The approach can be extended to the case with information on the variance as in [43], with Markovian assumptions as imposed in [25] and [53], or even more general constraints on the distribution (see e.g.…”