Irreducible convex paving for decomposition of multi-dimensional\n martingale transport plans

2017/02/27 by Hadrien De March, De March, Hadrien, Nizar Touzi +1 · 5 citations
Mathematics · Computer Science · #Point processes and geometric inequalities #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1702.08298

Abstract

Martingale transport plans on the line are known from Beiglbock & Juillet to\nhave an irreducible decomposition on a (at most) countable union of intervals.\nWe provide an extension of this decomposition for martingale transport plans in\nR^d, d larger than one. Our decomposition is a partition of R^d consisting of a\npossibly uncountable family of relatively open convex components, with the\nrequired measurability so that the disintegration is well-defined. We justify\nthe relevance of our decomposition by proving the existence of a martingale\ntransport plan filling these components. We also deduce from this decomposition\na characterization of the structure of polar sets with respect to all\nmartingale transport plans.\n

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