Stopping times. Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)be a standard Brownian motion started at 0 .

a) Consider the random time \(u\) defined by

\[u:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=B_{1}ight\}\]

which represents the first time Brownian motion \(B_{t}\) hits the level \(B_{1}\). Is \(u\) a stopping time?

b) Consider the random time \(\tau\) defined by

\[\tau:=\inf \left\{t \in \mathbb{R}_{+}: \mathrm{e}^{B_{t}}=\alpha \mathrm{e}^{-t / 2}ight\},\]

which represents the first time the exponential of Brownian motion \(B_{t}\) crosses the path of \(t \longmapsto \alpha \mathrm{e}^{-t / 2}\), where \(\alpha>1\).

Is \(\tau\) a stopping time? If \(\tau\) is a stopping time, compute \(\mathbb{E}\left[\mathrm{e}^{-\tau}ight]\) by applying the Stopping Time Theorem 14.7.

c) Consider the random time \(\tau\) defined by

\[\tau:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}^{2}=1+\alpha tight\}\]

which represents the first time the process \(\left(B_{t}^{2}ight)_{t \in \mathbb{R}_{+}}\)crosses the straight line \(t \longmapsto 1+\alpha t\), with \(\alpha<1\).

Is \(\tau\) a stopping time? If \(\tau\) is a stopping time, compute \(\mathbb{E}[\tau]\) by the Doob Stopping Time Theorem 14.7.