Optimal Stopping of a Brownian Excursion and an $α$-dimensional Bessel Bridge
David Hobson, Jingfei Liu
TL;DR
The paper addresses optimal stopping for $\alpha$-dimensional Bessel bridges with power payoff $x^n$, unifying the Brownian excursion ($\alpha=3,n=1$) as a key special case. It develops a free-boundary framework with self-similar structures, yielding an explicit solution for the Brownian excursion and a power-series construction for the general $\alpha,n$ case, along with a verification scheme. For the Brownian excursion, the optimal boundary is determined by $C=2e^{-C^{2}/2}\int_{0}^{C}e^{u^{2}/2}\,du$, and the value function is given piecewise by a closed form in the continuation region and by $x$ in the stopping region. In the general setting, the problem reduces to a squared-Bessel representation with a unique root $Z_{\alpha,n}$ of $F_{\alpha,n}(Z)=0$ that defines the boundary, together with a power-series solution for the associated value function; explicit solvable cases (e.g., $n=\alpha-2$ or $\alpha=n$) illustrate the approach. The results extend the literature on optimal stopping for Brownian and Bessel bridges and offer insights applicable to diffusion control and pinning-type financial problems.
Abstract
We study the optimal stopping of an $α$-dimensional Bessel bridge for the payoff $φ(x)=x^n$, where $α,n>0$. As a special case we consider the Brownian excursion with the identity function as the payoff ($α=3,n=1$). 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.