Vadim Arkin, Alexander Slastnikov
The paper is devoted to the description of an approach to solving an optimal stopping problems for multidimensional diffusion processes. This approach is based on
connection between boundary problem for diffusion processes and Dirichlet problem for PDE of an elliptic type. The solution of a
Dirichlet problem is considered as a functional of the available continuation regions. The optimization of this functional will be carried out by variational methods. Unlike the heuristic ``smooth pasting" method the proposed approach allows to obtain, in principle, to find necessary and sufficient conditions for optimality of stopping time in a given class of continuation regions. The approach is applied to the solving an optimal stopping problem for a two-dimensional
geometric Brownian motion with objective functional, which is an expectation of a homogeneous function. We intend to discuss an
application of this optimal stopping problem to real option theory and optimal timing of investment.