## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 686-719 |

Number of pages | 34 |

Journal | Annals of Applied Probability |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2017 |

## Keywords

- Martingale inequality
- Martingale transport problem
- Maximum of martingale given marginals
- Peacocks
- Skorokhod embedding problem

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty