  We present a methodology for obtaining explicit solutions to infinite time
horizon optimal stopping problems involving general, one-dimensional, It\^o
diffusions, payoff functions that need not be smooth and state-dependent
discounting. This is done within a framework based on dynamic programming
techniques employing variational inequalities and links to the probabilistic
approaches employing $r$-excessive functions and martingale theory. The aim of
this paper is to facilitate the the solution of a wide variety of problems,
particularly in finance or economics.
