The tail of the maximum of Brownian motion minus a parabola

Abstract

We analyze the tail behavior of the maximum $N$ of $\{W(t)-t^2:t\ge0\}$, where $W$ is standard Brownian motion on $[0,\infty)$, and give an asymptotic expansion for ${\mathbb P}\{N\ge x\}$, as $x\to\infty$. This extends a first order result on the tail behavior, which can be deduced from Hüsler and Piterbarg (1999). We also point out the relation between certain results in Janson et al. (2010) and Groeneboom (2010).