Some Remarks on First Passage of Lévy Processes, the American Put and Pasting

The purpose of this article is to provide, with the help of a fluctuation identity, a generic link between a number of known identities for the first passage time and overshoot above/below a fixed level of a Lévy process and the solution of Gerber and Shiu (1994), Boyarchenko and Levendorskǐi (1998, 1999, 2002), Chan (2000), Avram et al. (2002), Mordecki (2002), Asmussen et al. (2004) and Chesney and Jeanblanc (2004) to the American perpetual put optimal stopping problem. Furthermore, we make folklore precise and give necessary and sufficient conditions for smooth pasting to occur in the considered problem.