The Annals of Probability

Irreducible convex paving for decomposition of multidimensional martingale transport plans

Hadrien De March and Nizar Touzi

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Abstract

Martingale transport plans on the line are known from Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in $\mathbb{R}^{d}$, $d\ge1$. Our decomposition is a partition of $\mathbb{R}^{d}$ consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.

Article information

Source
Ann. Probab., Volume 47, Number 3 (2019), 1726-1774.

Dates
Received: March 2017
Revised: June 2018
First available in Project Euclid: 2 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aop/1556784031

Digital Object Identifier
doi:10.1214/18-AOP1295

Mathematical Reviews number (MathSciNet)
MR3945758

Zentralblatt MATH identifier
07067281

Subjects
Primary: 60G42: Martingales with discrete parameter
Secondary: 49N05: Linear optimal control problems [See also 93C05]

Keywords
Martingale optimal transport irreducible decomposition polar sets

Citation

De March, Hadrien; Touzi, Nizar. Irreducible convex paving for decomposition of multidimensional martingale transport plans. Ann. Probab. 47 (2019), no. 3, 1726--1774. doi:10.1214/18-AOP1295. https://projecteuclid.org/euclid.aop/1556784031


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