Infinite horizon optimal stopping problems for a Lévy processes with a two-sided reward function are considered. A two-sided verification theorem is presented in terms of the overall supremum and the overall infimum of the process. A result to compute the angle of the value function at the optimal thresholds of the stopping region is given. To illustrate the results, the optimal stopping problem of a compound Poisson process with two-sided exponential jumps and a two-sided payoff function is solved. In this example, the smooth-pasting condition does not hold.
"Two-sided optimal stopping for Lévy processes." Electron. Commun. Probab. 26 1 - 12, 2021. https://doi.org/10.1214/21-ECP376