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Stochastic order relations and lattices of probability measures

Author

Listed:
  • Marco Scarsini

    (GREGH - Groupement de Recherche et d'Etudes en Gestion à HEC - HEC Paris - Ecole des Hautes Etudes Commerciales - CNRS - Centre National de la Recherche Scientifique, Dipartimento di Scienze Economiche e Aziendali - LUISS - Libera Università Internazionale degli Studi Sociali Guido Carli [Roma])

  • Alfred Muller

Abstract

We study various partially ordered spaces of probability measures and we determine which of them are lattices. This has important consequences for optimization problems with stochastic dominance constraints. In particular we show that the space of probability measures on $\mathbb{R}$ is a lattice under most of the known partial orders, whereas the space of probability measures on $\mathbb{R}^d$ typically is not. Nevertheless, some subsets of this space, defined by imposing strong conditions on the dependence structure of the measures, are lattices.

Suggested Citation

  • Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
    • Handle: RePEc:hal:journl:hal-00539119
    DOI: 10.1137/040611021
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    Citations

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    Cited by:

    1. Jae Youn Ahn & Sebastian Fuchs, 2020. "On Minimal Copulas under the Concordance Order," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 762-780, March.
    2. Axel Anderson & Lones Smith, 2024. "The Comparative Statics of Sorting," American Economic Review, American Economic Association, vol. 114(3), pages 709-751, March.
    3. Dianetti, Jodi, 2023. "Strong Solutions to Submodular Mean Field Games with Common Noise and Related McKean-Vlasov FBSDES," Center for Mathematical Economics Working Papers 674, Center for Mathematical Economics, Bielefeld University.
    4. Piotr Więcek, 2017. "Total Reward Semi-Markov Mean-Field Games with Complementarity Properties," Dynamic Games and Applications, Springer, vol. 7(3), pages 507-529, September.
    5. Martin Kaae Jensen, 2018. "Distributional Comparative Statics," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 85(1), pages 581-610.
    6. Kocourek, Pavel & Steiner, Jakub & Stewart, Colin, 0. "Boundedly rational demand," Theoretical Economics, Econometric Society.
    7. Gerhold, Stefan & Gülüm, I. Cetin, 2019. "Peacocks nearby: Approximating sequences of measures," Stochastic Processes and their Applications, Elsevier, vol. 129(7), pages 2406-2436.
    8. Leskelä, Lasse & Vihola, Matti, 2013. "Stochastic order characterization of uniform integrability and tightness," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 382-389.
    9. Benjamin Brooks & Alexander Frankel & Emir Kamenica, 2022. "Information Hierarchies," Econometrica, Econometric Society, vol. 90(5), pages 2187-2214, September.
    10. Miguel Carrión & Uwe Gotzes & Rüdiger Schultz, 2009. "Risk aversion for an electricity retailer with second-order stochastic dominance constraints," Computational Management Science, Springer, vol. 6(2), pages 233-250, May.
    11. Ghosh, Gagan & Liu, Heng, 2021. "Sequential auctions with ambiguity," Journal of Economic Theory, Elsevier, vol. 197(C).
    12. Gentzkow, Matthew & Kamenica, Emir, 2017. "Bayesian persuasion with multiple senders and rich signal spaces," Games and Economic Behavior, Elsevier, vol. 104(C), pages 411-429.
    13. Bar Light, 2021. "Stochastic Comparative Statics in Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 797-810, May.
    14. Christopher P. Chambers & Siming Ye, 2023. "Haves and Have-Nots: A Theory of Economic Sufficientarianism," Papers 2301.08666, arXiv.org, revised Sep 2023.
    15. Shaked, Moshe, 2007. "Stochastic comparisons of multivariate random sums in the Laplace transform order, with applications," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1339-1344, July.
    16. Aurélien Alfonsi & Jacopo Corbetta & Benjamin Jourdain, 2019. "Sampling Of One-Dimensional Probability Measures In The Convex Order And Computation Of Robust Option Price Bounds," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-41, May.
    17. Christopher Chambers & Alan Miller & Ruodu Wang & Qinyu Wu, 2024. "Max-stability under first-order stochastic dominance," Papers 2403.13138, arXiv.org.

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