Optimal Skorokhod embedding given full marginals and Azema-Yor peacocks

TL;DR

This paper addresses the optimal Skorokhod embedding problem with full marginals, establishing duality, explicit solutions for maximum-dependent rewards, and connections to martingale inequalities, advancing the understanding of extremal martingales and peacocks.

Contribution

It introduces a duality framework for the full-marginal SEP, derives explicit solutions for maximum-based rewards, and links the problem to martingale inequalities, extending prior work on peacocks and martingale transport.

Findings

Established a general duality result for the full-marginal SEP.

Derived explicit characteristics of solutions for maximum-dependent reward functions.

Connected the SEP solutions to associated martingale inequalities.

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). 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-Labordere, Obloj, Spoida and Touzi. Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality.