Consider a standard Brownian motion \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)started at \(B_{0}=0\), and let

\[\tau:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=\alpha+\beta tight\}\]

denote the first hitting time of the straight line \(t \longmapsto \alpha+\beta t\) by \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\).

a) Compute the Laplace transform \(\mathbb{E}\left[\mathrm{e}^{-r \tau}ight]\) of \(\tau\) for all \(r>0\) and \(\alpha \geqslant 0\).

b) Compute the Laplace transform \(\mathbb{E}\left[\mathrm{e}^{-r \tau}ight]\) of \(\tau\) for all \(r>0\) and \(\alpha \leqslant 0\).

Use the stopping time theorem and the fact that \(\left(\mathrm{e}^{\sigma B_{t}-\sigma^{2} t / 2}ight)_{t \in \mathbb{R}_{+}}\)is a martingale for all \(\sigma \in \mathbb{R}\).