A construction of processes with one-dimensional martingale marginals, associated with a Lévy process, via its Lévy sheet

We give some adequate extension, in the framework of a general L´evy process, of our previous construction of processes with one-dimensional martingale marginals, done originally in the set-up of Brownian motion. The L´evy process framework allows us to streamline our previous arguments, as well as to reach a larger class of such processes, even in the Brownian case. We give some illustrations of our construction when the L´evy process is either a Gamma process, or a Poisson process. We also work in the fractional Brownian and stable frameworks.