A mass transport approach to the optimization of adapted couplings of real valued random variables

TL;DR

This paper explores an optimization framework for adapted couplings of real-valued random variables, linking them to causal transport plans, establishing existence results, and providing illustrative examples within a mass transport perspective.

Contribution

It introduces a novel mass transport approach to optimize adapted couplings, connecting them to causal transport plans and characterizing solutions under mild conditions.

Findings

Existence of solutions under mild hypotheses.

Representation of adapted couplings via causal transport plans.

Multiple explicit examples demonstrating the framework.

Abstract

In this work, we investigate an optimization problem over adapted couplings between pairs of real valued random variables, possibly describing random times. We relate those couplings to a specific class of causal transport plans between probabilities on the set of real numbers endowed with a filtration, for which their provide a specific representation, which is motivated by a precise characterization of the corresponding deterministic transport plans. From this, under mild hypothesis, the existence of a solution to the problem investigated here is obtained. Furthermore, several examples are provided, within this explicit framework.