Optimal transport and its applications

The series of working papers contains recent advances on some aspects of optimal transport theory and its applications. Optimal transport is a classic topic in mathematics, developed by G. Monge and L. Kantorovich, concerning coupling between two or more measures by minimizing some cost function. Multi-marginal optimal transport problems are also addressed in the working paper series on Joint mixability and negative dependence and on Robust risk aggregation in the disguise of dependence modeling and risk management. This working paper series contains some new directions of optimal transport which are not necessarily connected.

A brief description of each working paper is provided below to explain its main results and the logical structure across papers.

We study the framework of distorted optimal transport by minimizing a distorted expected cost instead of the expected cost, and obtain several results in this framework.

This paper proposes a tractable framework of composite sorting for workers and jobs in the labour market. The theory is built based on optimal transport on the real line with concave cost functions. This framework allows for the study of within-job wage dispersion within occupations in the United States.

This paper establishes in Theorem 2.1 that, for any two probability measures on the real line increasing in convex order with the second one atomless, there exists a martingale transport that is backward Monge, or forward injective. Such martingale transport is also dense in a natural sense (Theorem 2.3), and a refined version of Strassen's theorem (Theorem 3.1) is obtained. It is conjectured that a version of Theorem 2.1 should also hold true on general spaces but we could not find a proof.

This paper connects joint mix dependence (JM) and some classic negative dependence concepts (ND). In contrast to JM which solves a multi-marginal optimal transport problem, a combination of JM and ND solves a corresponding problem under uncertainty. This paper is cross-listed as WP14 of Joint mixability and negative dependence. The journal version is to appear in Mathematics of Operations Research (2024).

This paper contains the general framework of simultaneous mass transport, that is, mass transport between vector-valued measures. The new framework is motivated by the need to transport resources of different types simultaneously, i.e., in single trips, from specified origins to destinations. The mathematical structure of simultaneous transport is very different from the classic setting of optimal transport, leading to many new challenges and results.

We introduce the directional optimal transport problem where origins on the real line can only be transported to destinations to the right of the origins. We obtain an optimal coupling for supermodular costs, which admits manifold characterizations: geometric, order-theoretic, as optimal transport, through the cdf, and via the transport kernel. The journal version is published in Annals of Applied Probability (2022).