Some remarks on first passage of Lévy processes, the American put and pasting principles

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 [Astin Bull. 24 (1994) 195.220], Boyarchenko and Levendorskii. [Working paper series EERS 98/02 (1998), Unpublished manuscript (1999), SIAM J. Control Optim. 40 (2002) 1663.1696], Chan [Original unpublished manuscript (2000)], Avram, Chan and Usabel [Stochastic Process. Appl. 100 (2002) 75.107], Mordecki [Finance Stoch. 6 (2002) 473.493], Asmussen, Avram and Pistorius [Stochastic Process. Appl. 109 (2004) 79.111] and Chesney and Jeanblanc [Appl. Math. Fin. 11 (2004) 207.225] 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.