Before revision:  Next we approximate the infinite dimensional dynamics by finite dimensional ones to which we associate suitable optimal stopping problems in R^n. Then, by taking the limit as n goes to infinity and by removing the smoothing on the payoff, we obtain an infinite dimensional variational inequality for the price of the American bond option. Moreover ,  the  first time at which the price of the American bond option equals the payoff turns out to be an optimalexercise time .
After revision:  Next we approximate the infinite dimensional dynamics by finite dimensional ones , the first time at which the price of the American bond option equals the payoff turns out to be an optimalexercise time .