What is the expected value of $\tau$?

Question

$(B_t)_{t\geq0}$ is a SBM, $a>0, b>0$, and $\tau:=\inf\{t\geq0:B_t=b\sqrt{a+t}\}$, ($\inf\emptyset:=\infty$).

• Show that $\mathbb{E}(\tau)=\infty$, if $b\geq1$, and $\mathbb{E}(\tau)<\infty$ if $b\in(0,1)$...I tried using the Wald's identites, but unfortunately I got stuck.

Answer 1

Suppose $\mathbb E[\tau] < \infty$, then Wald's identity implies that $\mathbb E[B_\tau] = 0$.

However, since $B$ is continuous and $\tau < \infty$ almost surely, we have that $B_\tau = b\sqrt{a+\tau} \geq b\sqrt{a} > 0$, thus $$\mathbb E[B_\tau] > 0,$$ which is impossible.