Usually, American options can only be priced numerically. An important exception is the perpetual option, i.e. an American option with infinite maturity. Most mathematical finance textbooks treat this case. However, it is always assumed that the underlying is performing a geometric Brownian motion. I am looking for something simpler, namely the case when the underlying is performing a pure random walk. Does anyone know where I can find the corresponding formula? (I want it in the continuous, not discrete case).