Optimal Skorokhod embedding given full marginals and Azéma -Yor peacocks
Källblad, Sigrid; Tan, Xiaolu; Touzi, Nizar (2017), Optimal Skorokhod embedding given full marginals and Azéma -Yor peacocks, The Annals of Applied Probability, 27, 2, p. 686-719. 10.1214/16-AAP1191
TypeArticle accepté pour publication ou publié
Journal nameThe Annals of Applied Probability
Institute of Mathematical Statistics
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Abstract (EN)We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval [0,1]. The problem is related to the study of extremal martingales associated with a peacock (“process increasing in convex order,” by Hirsch, Profeta, Roynette and Yor [Peacocks and Associated Martingales, with Explicit Constructions (2011), Springer, Milan]). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordère, Obłój, Spoida and Touzi [Ann. Appl. Probab. 26 (2016) 1–44]. Under technical conditions, we then characterize the optimal value and the solution to the dual problem. In particular, the optimal embedding corresponds to the Madan and Yor [Bernoulli 8 (2002) 509–536] peacock under their “increasing mean residual value” condition. We also discuss the associated martingale inequality.
Subjects / Keywordsmaximum of martingale given marginals; martingale transport problem; martingale inequality; Skorokhod embedding problem; peacocks
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An Explicit Martingale Version of the One-dimensional Brenier's Theorem with Full Marginals Constraint Henry-Labordère, Pierre; Tan, Xiaolu; Touzi, Nizar (2016) Article accepté pour publication ou publié