Let \(\left(B_{t}ight)_{t \geqslant 0}\) be a one-dimensional Brownian motion, \(a, b>0\) and \(\tau:=\inf \left\{t \geqslant 0:\left|B_{t}ight|=b \sqrt{a+t}ight\}\). Show that

a) \(\mathbb{P}(\tau<\infty)=1\);

b) \(\mathbb{E} \tau=\infty \quad(b \geqslant 1)\);

c) \(\mathbb{E} \tau<\infty \quad(b<1)\).

Use in b) Wald's identities. For c) show that \(\mathbb{E}(\tau \wedge n) \leqslant a b^{2}\left(1-b^{2}ight)^{-1}\) for \(n \geqslant 1\).