Variational View to Optimal Stopping Problems for Diffusion Processes and Threshold Strategies

We describe a variational approach to solving optimal stopping problems for diffusion processes, as an alternative to the traditional approach based on the solution of the free-boundary problem. We study smooth pasting conditions from a variational point of view, and give some examples when the solution to free-boundary problem is not the solution to optimal stopping problem. A special attention is paid to threshold strategies which allow reduce optimal stopping problem to more simple one-parametric optimization. Necessary and sufficient conditions for threshold structure of optimal stopping time are derived. We apply these results to both investment timing and optimal abandon models.