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

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• 作者：Pierre Henry-Labordè ; re^a ; ^{pierre.henry-labordere@sgcib.com" class="auth_mail" title="E-mail the corresponding author} ; Xiaolu Tan^b ; ^{tan@ceremade.dauphine.fr" class="auth_mail" title="E-mail the corresponding author} ; Nizar Touzi^c ; ^{nizar.touzi@polytechnique.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Martingale optimal transport ; Brenier&rsquo ; s Theorem ; PCOC ; Fake Brownian motion
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：126
• 期：9
• 页码：2800-2834
• 全文大小：526 K

文摘

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.