Open Access
2022 The potential of the shadow measure
Mathias Beiglböck, David Hobson, Dominykas Norgilas
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Electron. Commun. Probab. 27: 1-12 (2022). DOI: 10.1214/22-ECP457


It is well known that given two probability measures μ and ν on 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. Beiglböck 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.


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Mathias Beiglböck. David Hobson. Dominykas Norgilas. "The potential of the shadow measure." Electron. Commun. Probab. 27 1 - 12, 2022.


Received: 20 April 2021; Accepted: 22 February 2022; Published: 2022
First available in Project Euclid: 2 March 2022

MathSciNet: MR4389158
Digital Object Identifier: 10.1214/22-ECP457

Primary: 60G42

Keywords: Convex order , Couplings , Martingales , Optimal transport , peacocks

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