I'm attempting to solve the following question but I'm really unsure of what my approach should be. I've some progress but I need a lot of help, as I'm not sure if I'm anywhere near the right track:
For part (a) intuitively, $$\textbf{(1)} \quad \mu > \sigma^2/2 \implies \lim S_t = \infty \\ \textbf{(2)} \quad \mu < \sigma^2/2 \implies \lim S_t = 0$$ almost surely.
Note that $t W_{1/t}$ is also Brownian motion so that it suffices to show that $$\lim_{t \rightarrow \infty} S_0 \exp(t \left(W_{1/t} + \mu - \frac{\sigma^2}{2}\right) = \infty \text{ or } 0$$
This holds if and only if $$\bigcup_{k = N}^\infty \bigcap_{0<s <1/k \\ s \in \mathbb{Q}} \left \{ S_0 \exp \left(\frac{W_s + \mu - \sigma^2/2}{s} \right) \ge M \right \} \text{ or } \bigcup_{k = N}^\infty \bigcap_{0<s <1/k \\ s \in \mathbb{Q}} \left \{ S_0 \exp \left(\frac{W_s + \mu - \sigma^2/2}{s} \right) \leq 1/M \right \}$$ for each $M \in \mathbb{N}$ for the two cases, respectively.
Both of these events are $\mathcal{F}_{1/N}$ measurable, where $\mathcal{F}_{1/N} = \sigma(W_t : t \leq 1/N)$ for every $N$ and thus $\mathcal{F}_0^+ = \bigcap_{\epsilon > 0} \mathcal{F}_\epsilon$ measurable. By Blumenthal's 0-1 law, their probabilities are thus 0 or 1 so we simply need to show that they are positive, but I have no idea how to do this, or if there is an easier way. Please help if you can!
First, forget about the exponent and $S_0$. Try to show the following:
If $a \neq 0$ then $W_t + a t$ tends to $-\infty$ or $\infty$ almost surely.
The statement you want follows immediately from this above fact.
To show this fact, there are loads of ways. Here's an easy way to do it: try to show that there is a random $T < \infty$ a.s. so that for all $t \geq T$ we have $|W_t| \leq t^{2/3}$ (loads of different bounds here would work, by the way).
Why is this the case? Again, there are loads of ways to do this, but recall that the reflection principle says $$P(\max_{s \leq t} |W_s| \geq M) \leq 2 P(|W_t| \geq M)$$
and so $$\sum_{n \geq 1} P( \max_{t \in [n,n+1]} |W_t| \geq n^{2/3}) < \infty\,.$$
This shows that we eventually have $|W_t| \leq t^{2/3}$ by the Borel Cantelli lemma.