  We investigate the optimal stopping problems involving the supremum of a
diffusion. The starting point is the link between works of Peskir and
Meilijson, which we describe in a unified manner. The description developped
follows mainly the work of Peskir and relies on the maximality principle which
links the existence and form of the solution to an optimal stopping problem
with a special solution of a certain differential equaiton. We also discuss the
so-called optimal Skorokhod embedding problem and provide an explicit solution
for a large class of measures.
