Stopping Time / First Passage Time

First passage times have a brutal punchline: Brownian motion hits every level almost surely, yet the expected time to do so is infinite. Interviewers love this one precisely because it breaks your intuition.

Let ${\tau }_a=\inf \{t\ge 0:W_t=a\}$ be the first time a standard Brownian motion $W_t$ hits level $a>0$.

Part A

Using the reflection principle, derive $\mathbb{P}({\tau }_a\le t)$ in terms of the standard Gaussian CDF $\Phi (\cdot )$, and show that $\mathbb{P}({\tau }_a<\infty )=1$.

Part B

Show that $\mathbb{E}[{\tau }_a]=\infty$, even though ${\tau }_a$ is finite almost surely.

Part C (optional, for context)

The Laplace transform of ${\tau }_a$ is $\mathbb{E}[e^{-\lambda {\tau }_a}]=e^{-a\sqrt{2\lambda }}$ for $\lambda \ge 0$. Explain how this immediately signals that $\mathbb{E}[{\tau }_a]$ diverges.