An explicit martingale version of the one-dimensional Brenier’s Theorem with full marginals constraint.

Summary We provide an extension of the martingale version of the Fréchet-Hoeffding coupling to the infinitely-many marginals constraints setting. In the two-marginal context, this extension was obtained by Beiglböck & Juillet [7], and further developed by Henry-Labordère & Touzi [40], see also [6]. Our main result applies to a special class of reward functions and requires some restrictions on the marginal distributions. We show that the optimal martingale transference plan is induced by a pure downward jump local Lévy model. In particular, this provides a new martingale peacock process (PCOC " Processus Croissant pour l'Ordre Convexe, " see Hirsch, Profeta, Roynette & Yor [43]), and a new remarkable example of discontinuous fake Brownian motions. Further, as in [40], we also provide a duality result together with the corresponding dual optimizer in explicit form. As an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.