A direct method for solving optimal stopping problems for L\'evy processes

We propose an alternative approach for solving a number of well-studied optimal stopping problems for L'evy processes. Instead of the usual method of guess-and-verify based on martingale properties of the value function, we suggest a more direct method by showing that the general theory of optimal stopping for strong Markov processes together with some elementary observations imply that the stopping set must be of a certain form for the optimal stopping problems we consider. The independence of increments and the strong Markov property of L'evy processes then allow us to use straightforward optimisation over a real-valued parameter to determine this stopping set. We illustrate this approach by applying it to the McKean optimal stopping problem (American put), the Novikov–Shiryaev optimal stopping problem and the Shepp–Shiryaev optimal stopping problem (Russian option).