Stochastic order on metric spaces and the ordered Kantorovich monad
Tobias Fritz and Paolo Perrone
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Submission date: 29. Aug. 2018 (revised version: August 2018)
published in: Advances in mathematics, 366 (2020), art-no. 107081
DOI number (of the published article): 10.1016/j.aim.2020.107081
MSC-Numbers: 60E15, 28C15, 18C20
Keywords and phrases: stochastic order, stochastic dominance, Wasserstein distance, Kantorovich duality, Lawvere metric spaces, Eilenberg-Moore algebras
Link to arXiv: See the arXiv entry of this preprint.
In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces of probability measures with the usual stochastic order. It can be considered a metric analogue of the probabilistic powerdomain. Our proof of antisymmetry of the stochastic order on these spaces is more general than previously known results in this direction.
The spaces we consider, which we call L-ordered, are spaces where the order satisﬁes a mild compatibility condition with the metric itself, rather than merely with the underlying topology. As we show, this is related to the theory of Lawvere metric spaces, in which the partial order structure is induced by the zero distances.
We show that the algebras of the ordered Kantorovich monad are the closed convex subsets of Banach spaces equipped with a closed positive cone, with algebra morphisms given by the short and monotone aﬃne maps. Considering COMet as a locally posetal 2-category, the lax and oplax algebra morphisms are exactly the concave and convex short maps, respectively.
In the unordered case, we had identiﬁed the Wasserstein space as the colimit of the spaces of empirical distributions of ﬁnite sequences. We prove that this extends to the ordered setting as well by showing that the stochastic order arises by completing the order between the ﬁnite sequences, generalizing a recent result of Lawson. The proof holds on any metric space equipped with a closed partial order.