Stochastic Monge-Kantorovich Problem and its Duality

TL;DR

This paper establishes the existence of stochastic optimal transference plans, extends Kantorovich duality to stochastic settings, and discusses Wasserstein distances between probability kernels, advancing the theoretical framework of stochastic optimal transport.

Contribution

It introduces a stochastic version of the Monge-Kantorovich problem, proving existence, duality, and characterizations of optimal plans in a stochastic context.

Findings

Existence of stochastic optimal transference plan proven.

Stochastic Kantorovich duality established.

Wasserstein distance between probability kernels discussed.

Abstract

In this article we prove the existence of a stochastic optimal transference plan for a stochastic Monge-Kantorovich problem by measurable selection theorem. A stochastic version of Kantorovich duality and the characterization of stochastic optimal transference plan are also established. Moreover, Wasserstein distance between two probability kernels are discussed too.