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In this paper we demonstrate that the Riesz representation of excessivefunctions 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-dimensionalgeometric Brownian motion. After this, two classical optimal stopping problemsare discussed: 1) the optimal investment problem and 2) the valuation of theAmerican put option. It is seen in both of these problems that the boundary ofthe stopping region can be characterized as a unique solution of an integralequation arising immediately from the Riesz representation of the valuefunction. In Problem 2 the derived equation coincides with the standardwell-known equation found in the literature.

Riesz representation and optimal stopping with two case studies