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 H\"{u}sler and Piterbarg (1999). We also point out the relation between certain results in Groeneboom (2010) and Janson, Louchard and Martin-L\"{o}f (2010).

Additional Metadata
Keywords Brownian motion, parabolic drift, Airy functions
MSC Brownian motion (msc 60J65)
Publisher Institute of Mathematical Statistics [etc.]
Journal Electronic Communications in Probability
Groeneboom, P, & Temme, N.M. (2011). The tail of the maximum of Brownian motion minus a parabola. Electronic Communications in Probability, 16, 458–466.