Let \(\left(X_{t}, \mathscr{F}_{t}ight)_{t \geqslant 0}\) be a \(d\)-dimensional stochastic process and \(A, A_{n}, C \in \mathscr{B}\left(\mathbb{R}^{d}ight), n \geqslant 1\). Then

a) \(A \subset C\) implies \(\tau_{A}^{\circ} \geqslant \tau_{C}^{\circ}\) and \(\tau_{A} \geqslant \tau_{C}\);

b) \(\tau_{A \cup C}^{\circ}=\min \left\{\tau_{A}^{\circ}, \tau_{C}^{\circ}ight\}\) and \(\tau_{A \cup C}=\min \left\{\tau_{A}, \tau_{C}ight\}\);

c) \(\tau_{A \cap C}^{\circ} \geqslant \max \left\{\tau_{A}^{\circ}, \tau_{C}^{\circ}ight\}\) and \(\tau_{A \cap C} \geqslant \max \left\{\tau_{A}, \tau_{C}ight\}\);

d) \(A=\bigcup_{n \geqslant 1} A_{n}\), then \(\tau_{A}^{\circ}=\inf _{n \geqslant 1} \tau_{A_{n}}^{\circ}\) and \(\tau_{A}=\inf _{n \geqslant 1} \tau_{A_{n}}\);

e) \(\tau_{A}=\inf _{n \geqslant 1}\left(\frac{1}{n}+\tau_{n}^{\circ}ight)\) where \(\tau_{n}^{\circ}=\inf \left\{s \geqslant 0: X_{S+1 / n} \in Aight\}\);

f) find an example of a set \(A\) such that \(\tau_{A}^{\circ}<\tau_{A}\).