Multidimensional investment problem

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 to geometric Brownian motions. After this, a classical investment problem, also known as exchange-of-baskets-problem, is studied. It is seen that the boundary of the stopping region in this problem can be characterized as a unique solution of an integral equation arising immediately from the Riesz representation of the value function. The two-dimensional case is studied in more detail and a numerical algorithm is presented.