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We study the optimal stopping of an $\alpha$-dimensional Bessel bridge for the payoff $\phi(x)=x^n$, where $\alpha,n \in (0,\infty)$. 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.
A remarkable distribution found by Mardia and Dryden for the shape of a random triangle in ℝ2 whose vertices are displaced from their initial positions by independent identical symmetrical gaussian perturbations is re-derived via stochastic calculus.
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