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

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 and Juillet (2016), and further developed by Henry-Labordère and Touzi (in press), see also Beiglböck and Henry-Labordère (Preprint). 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 et al. (2011), and a new remarkable example of discontinuous fake Brownian motions. Further, as in Henry-Labordère and Touzi (in press), 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.