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

\[\tau_{L}=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=Light\}\]

denote the first hitting time of the level \(L>0\) by \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\).

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

Use the Stopping Time Theorem 14.7 and the fact that \(\left(\mathrm{e}^{\sqrt{2 r} B_{t}-r t}ight)_{t \in \mathbb{R}_{+}}\)is a martingale when \(\underline{r>0}\).

b) Find the optimal level stopping strategy depending on the value of \(\underline{r>0}\) for the maximization problem

\[\operatorname{Sup}_{L>0} \mathbb{E}\left[\mathrm{e}^{-r \tau_{L}} B_{\tau_{L}}ight]\]