Before revision: We address the problem of optimally exercising American options based on the assumption that the underlying stock's price follows a Brownian bridge whose final value coincides with the strike price. In order to do so, we solve the discounted optimal stopping problem endowed with the gain function G(x) = (S - x)^+ and a Brownian bridge whose final value equals S.
After revision: We address the problem of optimally exercising American options based on the assumption that the knowledge of the holder is restricted to the price history of the underlying asset, we allow for the disclosure of future information about the terminal price of the asset by modeling it as a Brownian bridge whose final value equals S.