Consider a model of a financial market with a stock driven by a Lévy process and constant interest rate. A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formula for perpetual American put options...

In this paper we give the closed form solution of some optimal stopping problems for processes derived from a diffusion with jumps. Within the possible applications, the results can be interpreted as pricing perpetual American Options under diffusion-jump information.