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

a) Is the process \(t \longmapsto\left(2-B_{t}ight)^{+}\)a submartingale, a martingale or a supermartingale?

b) Is the process \(\left(\mathrm{e}^{B_{t}}ight)_{t \in \mathbb{R}_{+}}\)a submartingale, a martingale, or a supermartingale?

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

\[u:=\inf \left\{t \in \mathbb{R}_{+}: B_{t}=B_{2 t}ight\}\]

which represents the first intersection time of the curves \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)and \(\left(B_{2 t}ight)_{t \in \mathbb{R}_{+}}\).

Is \(u\) a stopping time?

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

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

which represents the first time geometric Brownian motion \(\mathrm{e}^{B_{t}-t / 2}\) crosses the straight line \(t \longmapsto \alpha+\beta t\). Is \(\tau\) a stopping time?

e) If \(\tau\) is a stopping time, compute \(\mathbb{E}[\tau]\) by the Doob Stopping Time Theorem 14.7 in each of the following two cases:

i) \(\alpha>1\) and \(\beta<0\), ii) \(\alpha<1\) and \(\beta>0\).