Unifying constructions of martingales associated with processes increasing in the convex order, via Lévy and Sato sheets

In this paper, we present a unified framework for our previous constructions of martingales with the same one-dimensional marginals as particular cases of processes increasing in the convex order. This framework encompasses our former uses of Lévy sheets, Sato sheets and self-decomposable laws. New examples of processes increasing in the convex order are also exhibited, but we do not know how to associate to them martingales with the same one-dimensional marginals.