  We consider the representation of the value of an optimal stopping problem of
a linear diffusion as an expected supremum of a known function. We establish an
explicit integral representation of this function by utilizing the explicitly
known 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.
