Before revision:  In the present paper we consider the inverse problem, i.e. given prices of perpetual American options for different strikes  we show how to construct a time-homogeneous model for the stock price which reproduces the given option prices.
After revision:  In the present paper we consider the inverse problem, i.e. given prices of perpetual American options for different strikes we show how to construct a time-homogeneous stock price model which reproduces the given option prices.