On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the \(d\)-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function associated with the problem. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem. These equations allow to formulate a discretization scheme to obtain an approximate solution. The approach is illustrated through the optimal stopping problem of a \(d\)-dimensional Wiener process with a positive definite quadratic form reward function.