  We consider the representation of the value of a class of optimal stopping
problems of linear diffusions in a linearized form as an expected supremum of a
known function. We establish an explicit integral representation of this
representing function by utilizing the explicitly known marginals of the joint
probability distribution of the extremal processes. We also delineate
circumstances under which the value of a stopping problem induces directly this
representation and show how it is connected with the monotonicity of the
generator. We compare our findings with existing literature and show, for
example, how our representation is linked to the smooth fit principle and how
it coincides with the optimal stopping signal representation. The intricacies
of the developed integral representation are explicitly illustrated in various
examples arising in financial applications of optimal stopping.
