  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.
