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A note on stochastic dominance, uniform integrability, and lattice properties

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  • Nendel, Max

    (Center for Mathematical Economics, Bielefeld University)

Abstract

In this work, we discuss completeness for the lattice orders of first and second order stochastic dominance. The main results state that, both, first and second order stochastic dominance induce Dedekind super complete lattices, i.e. lattices in which every bounded nonempty subset has a countable subset with identical least upper bound and greatest lower bound. Moreover, we show that, if a suitably bounded set of probability measures is directed (e.g. a lattice), then the supremum and infimum w.r.t. first or second order stochastic dominance can be approximated by sequences in the weak topology or in the Wasserstein-1 topology, respectively. As a consequence, we are able to prove that a sublattice of probability measures is complete w.r.t. first order stochastic dominance or second order stochastic dominance and increasing convex order if and only if it is compact in the weak topology or in the Wasserstein-1 topology, respectively. This complements a set of characterizations of tightness and uniform integrability, which are discussed in a preliminary section.

Suggested Citation

  • Nendel, Max, 2025. "A note on stochastic dominance, uniform integrability, and lattice properties," Center for Mathematical Economics Working Papers 713, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:713
    as

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    File URL: https://pub.uni-bielefeld.de/download/3004913/3004914
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    References listed on IDEAS

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    1. Haim Levy, 2016. "Stochastic Dominance," Springer Books, Springer, edition 3, number 978-3-319-21708-6, December.
    2. Dianetti, Jodi & Ferrari, Giorgio & Fischer, Markus & Nendel, Max, 2019. "Submodular Mean Field Games. Existence and Approximation of Solutions," Center for Mathematical Economics Working Papers 621, Center for Mathematical Economics, Bielefeld University.
    3. Marco Scarsini & Alfred Muller, 2006. "Stochastic order relations and lattices of probability measures," Post-Print hal-00539119, HAL.
    4. Leskelä, Lasse & Vihola, Matti, 2013. "Stochastic order characterization of uniform integrability and tightness," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 382-389.
    Full references (including those not matched with items on IDEAS)

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