Consider the process (left(X_{t} ight)_{t in mathbb{R}_{+}})given by (X_{t}:=t Z, t in mathbb{R}_{+}), where (Z in{0,1}) is

Consider the process \(\left(X_{t}\right)_{t \in \mathbb{R}_{+}}\)given by \(X_{t}:=t Z, t \in \mathbb{R}_{+}\), where \(Z \in\{0,1\}\) is a Bernoulli random variable with \(\mathbb{P}(Z=1)=\mathbb{P}(Z=0)=1 / 2\). Given \(\epsilon \geqslant 0\), let the random time \(\tau_{\epsilon}\) be defined as

\[\tau_{\epsilon}:=\inf \left\{t>0: X_{t}>\epsilon\right\}\]
with \(\inf \emptyset=+\infty\), and let \(\left(\mathcal{F}_{t}\right)_{t \in \mathbb{R}_{+}}\)denote the filtration generated by \(\left(X_{t}\right)_{t \in \mathbb{R}_{+}}\).

a) Give the possible values of \(\tau_{\epsilon}\) in \([0, \infty]\) depending on the value of \(Z\).

b) Take \(\epsilon=0\). Is \(\tau_{0}:=\inf \left\{t>0: X_{t}>0\right\}\) an \(\left(\mathcal{F}_{t}\right)_{t \in \mathbb{R}_{+}-\text {-stopping time? Hint: Consider }}\) the event \(\left\{\tau_{0}>0\right\}\).

c) Take \(\epsilon>0\). Is \(\tau_{\epsilon}:=\inf \left\{t>0: X_{t}>\epsilon\right\}\) an \(\left(\mathcal{F}_{t}\right)_{t \in \mathbb{R}_{+}}\)-stopping time? Hint: Consider the event \(\left\{\tau_{\epsilon}>t\right\}\) for \(t \geqslant 0\).