The potential of the shadow measure

TL;DR

This paper explores the shadow measure, a concept in optimal martingale transport, providing explicit construction and proving properties like associativity, thus advancing the understanding of martingale couplings under convex order constraints.

Contribution

It extends existing results by explicitly constructing the shadow measure and proving its properties, including existence, uniqueness, and associativity.

Findings

Explicit construction of the shadow measure.

Proof of the shadow measure's associativity.

Extension of existence and uniqueness results.

Abstract

It is well known that given two probability measures $\mu$ and $\nu$ on $\mathbb{R}$ in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglb\"{o}ck and Juillet (Ann. Probab. 44 (2016) 42-106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.