Optimal Stopping of a Brownian Excursion and an $\alpha$-dimensional Bessel Bridge

We study the optimal stopping of an $\alpha$-dimensional Bessel bridge for the payoff $\phi(x)=x^n$, where $\alpha,n>0$. As a special case we consider the Brownian excursion with the identity function as the payoff ($\alpha=3,n=1$). For the Brownian excursion we can give an explicit solution but in the general case we provide a complete solution via a power series expansion.