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Reconciliation of probability measures

Part of: Foundations of probability theory

Published online by Cambridge University Press:  17 December 2018

JEAN BÉRARD  and
NICOLAS JUILLET
JEAN BÉRARD
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67 000 Strasbourg, France emails: [email protected]; [email protected]
NICOLAS JUILLET*
Affiliation:
Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67 000 Strasbourg, France emails: [email protected]; [email protected]
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Abstract

We discuss the reconciliation problem between probability measures: given n⩾2 probability spaces $(\Omega,{\mathcal{F}}_1,{\mathbb{P}}_1),\ldots,(\Omega,{\mathcal{F}}_n,{\mathbb{P}}_n)$ with a common sample space, does there exist an overall probability measure ${\mathbb{P}} \ \text{on} \ {\mathcal{F}} = \sigma({\mathcal{F}}_1,\ldots,{\mathcal{F}}_n)$ such that, for all i, the restriction of ${\mathbb{P}} \ \text{to} \ {\mathcal{F}}_i$ coincides with ${\mathbb{P}}_i$? General criteria for the existence of a reconciliation are stated, along with some counterexamples that highlight some delicate issues. Connections to earlier (recent and far less recent) work are discussed, and elementary self-contained proofs for the various results are given.

Keywords

Optimal transportjoint lawsigma-fieldsFarkas’ LemmaStrassen theorem

MSC classification

Primary: 60A05: Axioms; other general questions
Secondary: 90C08: Special problems of linear programming (transportation, multi-index, etc.)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1300 - 1310
Copyright
© Cambridge University Press 2018 

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References

Alon, S. & Lehrer, E. (2014) Subjective multi-prior probability: a representation of a partial likelihood relation. J. Econ. Theory 151, 476–492.CrossRefGoogle Scholar
Bertrand, J. & Puel, M. (2013) The optimal mass transport problem for relativistic costs. Calc. Var. Partial Differ. Equ. 46(1–2), 353–374.CrossRefGoogle Scholar
Billingsley, P. (1999) Convergence of Probability Measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York. A Wiley-Interscience Publication.CrossRefGoogle Scholar
Cambou, M. & Filipović, D. (2017) Model uncertainty and scenario aggregation. Math. Finance 27(2), 534–567.CrossRefGoogle Scholar
Dall’Aglio, G. (1961) Sulle distribuzioni doppie con margini assegnati sogette a delle limitazioni. G. Ist. Ital. Attuari 24, 94–108.Google Scholar
Dall’Aglio, G., Kotz, S. & Salinetti, G. (editors) (1991) Advances in Probability Distributions with Given Marginals, Mathematics and Its Applications, Vol. 67, Kluwer Academic Publishers Group, Dordrecht. Beyond the copulas, Papers from the Symposium on Distributions with Given Marginals held in Rome, April 1990.Google Scholar
Fréchet, M. (1957) Les tableaux de corrélation dont les marges et des bornes sont données. Ann. Univ. Lyon. Sect. A (3) 20, 13–31.Google Scholar
Jimenez, C. & Santambrogio, F. (2012) Optimal transportation for a quadratic cost with convex constraints and applications. J. Math. Pures Appl. (9) 98(1), 103–113.CrossRefGoogle Scholar
Kellerer, H. (1964) Maßtheoretische Marginalprobleme. Math. Ann. 153, 168–198.CrossRefGoogle Scholar
Kellerer, H. G. (1961) Funktionen auf Produkträumen mit vorgegebenen Marginal-Funktionen. Math. Ann. 144, 323–344.CrossRefGoogle Scholar
Korman, J., McCann, R. J. & Seis, C. (2015) An elementary approach to linear programming duality with application to capacity constrained transport. J. Convex Anal. 22(3), 797–808.Google Scholar
Korman, J., McCann, R. J. & Seis, C. (2015) Dual potentials for capacity constrained optimal transport. Calc. Var. Partial Differ. Equ. 54(1), 573–584.CrossRefGoogle Scholar
Levin, V. L. (1984) The problem of mass transfer in a topological space and probability measures with given marginal measures on the product of two spaces. Dokl. Akad. Nauk SSSR 276(5), 1059–1064.Google Scholar
Strassen, V. (1965) The existence of probability measures with given marginals. Ann. Math. Stat. 36, 423–439.CrossRefGoogle Scholar