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.
Highlights
In this work, we discuss completeness for lattice orders arising from first and second order stochastic dominance, and their relation to tightness and uniform integrability, respectively
Given a lattice L, it is a well-known result, due to Birkhoff [3, Section X.12, Theorem X.20] and Frink [8], that L is complete, i.e. every nonempty subset of L has a least upper bound and a greatest lower bound, if and only if L is compact in the interval topology
Stochastic dominance or convex orders are present in many applications in microeconomics and decision theory
Summary
We discuss completeness for lattice orders arising from first and second order stochastic dominance, and their relation to tightness and uniform integrability, respectively. Two of the main results of this paper are a characterization of complete lattices w.r.t. first and second order stochastic dominance in terms of compactness in the weak topology and in the Wasserstein-1 topology, respectively (Theorem 3.5 and Theorem 3.12). Choosing S to be a singleton with π(S) > 0, we obtain the Dedekind super completeness of P(R) and P1(R), endowed with first order and second order stochastic dominance, as well as the approximation of suprema/infima in the weak topology and the Wasserstein-1 topology, respectively. We extend the results by Leskela and Vhiola [11] to R and combine them with the results from Section 2 and Section 3 in order to obtain a characterization of tightness and uniform integrability in terms of integrability conditions for a function ψ with certain properties and boundedness conditions w.r.t. first and second order stochastic dominance, respectively (Lemma 3.4 and Lemma 3.11).
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