Discounted optimal stopping for diffusions: free-boundary versus martingale approach

The free-boundary and the martingale approach are competitive methods of solving discounted optimal stopping problems for one-dimensional time-homogeneous regular diffusion processes on infinite time intervals. We provide a missing link showing the equivalence of these approaches for a problem, where the optimal stopping time is equal to the first exit time of the underlying process from a region restricted by two constant boundaries. We also consider several illustrating examples including the rational valuation of the perpetual American strangle option.