An Explicit Martingale Version of the One-Dimensional Brenier's Theorem with Full Marginals Constraint

Abstract

We extend the martingale version of the one-dimensional Brenier's theorem (Frechet-Hoeffding coupling), established in Henry-Labordere and Touzi to the infinitely-many marginals case. In short, their results give an explicit characterization of the optimal martingale transference plans as well as the optimal dual components of a two marginals discrete-time martingale transportation (MT) problem for a large class of reward functions. We consider here the limiting continuous-time case, which leads to an infinitely-many marginals MT problem. By approximation technique, we show that for a class of reward functions, the optimal martingale transference plan is provided by a pure downward jump local Levy model. In particular, it provides a new construction of the martingale peacock process (PCOC Processus Croissant pour l'Ordre Convexe, see Hirsch, Profeta, Roynette and Yor), and a new remarkable example of discontinuous fake Brownian motions. Further, as in, we also provide a duality result together with dual optimizer in explicit form. Finally, as an application to financial mathematics, our results give the model-independent optimal lower and upper bounds for variance swaps.