A note on stochastic dominance, uniform integrability and lattice properties

Nendel M. A note on stochastic dominance, uniform integrability and lattice properties. Bulletin of the London Mathematical Society. 2020;52(5):907-923.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, that is, 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 (for example, a lattice), then the supremum and infimum with respect to first-order 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 with respect to 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