  In this paper we demonstrate that the Riesz representation of excessive
functions is a useful and enlightening tool to study optimal stopping problems.
After a short general discussion of the Riesz representation we concretize,
firstly, on a d-dimensional and, secondly, a space-time one-dimensional
geometric Brownian motion. After this, two classical optimal stopping problems
are discussed: 1) the optimal investment problem and 2) the valuation of the
American put option. It is seen in both of these problems that the boundary of
the stopping region can be characterized as a unique solution of an integral
equation arising immediately from the Riesz representation of the value
function. In Problem 2 the derived equation coincides with the standard
well-known equation found in the literature.
