Let $(W_t)$ be a Wiener process and for $a>0$ define stopping time: $$\tau = \inf \left\{t>0: W_t + at = 5\right\}$$
a) show $\tau < \infty$ a.s;
b) compute $\mathbb{E}\tau$.
I have done the first part and in the second part I'm trying to show that $\mathbb{E}\tau < \infty$, because then $\mathbb{E}W_t = 0 \implies \mathbb{E}\tau = \frac{5}{a}$. I think I should use $W_t^2 - t$ matringale, but I cannot use it in the right way.
Edit. Ok, I've managed to do it.