Before revision:   In particular, our construction reproduces the stopping boundaries obtained in Madan and Yor , 2002 and Brown et al. , 2001a . We demonstrate with a counterexample that our technical assumption is necessary and propose an extended embedding for n=3.
After revision:  In particular, our construction reproduces the stopping boundaries obtained in Madan and Yor , 2002 and Brown et al. , 2001a . We demonstrate with a counterexample that our technical assumption is necessary for the explicit embedding, as demonstrated with a counterexample. We discuss extensions to the general case giving details when n=3.