A note on pasting conditions for the American perpetual optimal stopping problem

The principles of smooth and continuous pasting play an important role in the study of optimal stopping problems with jump processes. These principles state that the optimal stopping boundary is selected so that the value function is smooth and continuous, respectively (depending on the behavior of the underlying process at the boundary). Extending the results of Alili & Kyprianou [Alili, L., Kyprianou, A.E., 2005. Some remarks on first passage of Lévy processes, the American put and pasting principles. Ann. Appl. Probab. 15, 2062–2080] we show that in the case of an American perpetual put under a Lévy process the optimal stopping point is in fact characterized as the only point that fulfills this smooth/continuous pasting condition.