An Explicit Martingale Version of the One-dimensional Brenier's Theorem with Full Marginals Constraint
Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar (2016), An Explicit Martingale Version of the One-dimensional Brenier's Theorem with Full Marginals Constraint, Stochastic Processes and their Applications, 126, 9, p. 2800-2834. 10.1016/j.spa.2016.03.003
TypeArticle accepté pour publication ou publié
Journal nameStochastic Processes and their Applications
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Abstract (EN)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 , and further developed by Henry-Labordère & Touzi , see also . 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 ), and a new remarkable example of discontinuous fake Brownian motions. Further, as in , 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.
Subjects / KeywordsMartingale optimal transport; Brenier’s Theorem; PCOC; Fake Brownian motion
Showing items related by title and author.
Henry-Labordère, Pierre; Oudjane, Nadia; Tan, Xiaolu; Touzi, Nizar; Warin, Xavier (2017) Document de travail / Working paper