The tail of the maximum of Brownian motion minus a parabola

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).