A Variational Approach to Optimal Stopping Problems for Diffusion Processes

We describe a variational approach to the solution to optimal stopping problems for diffusion processes as an alternative to the traditional approach based on the solution of the Stefan (free-boundary) problem. The connection of this variational approach to smooth pasting conditions is established. We present an example where the solution to the Stefan problem is not the solution to an optimal stopping problem. On the basis of the proposed approach, we obtain the solution to an optimal stopping problem for a two-dimensional geometric Brownian motion with a homogeneous payoff function.