My question is regarding the usefulness of the law of the iterated logarithm, and its connection to stopping times. In many answers of this forum, I understand that some people often claim that some stopping times, such as
$$\tau_n = \inf\left\{t \geq 0 : B_t = n\right\}
$$
are finite almost surely because of the law of the iterated logarithm. I am familiar with the law of course, but I don't really see how we can argue that any stopping time is bounded if, say, in the example above $n$ is very large.
This takes me to ask the following two questions: does the law assure that $\mathscr{F}_{t}$-measurable stopping times are always bounded? Is there an example when the Law of the Iterated Logarithm doesn't help us to conclude that a stopping time is not bounded almost surely?
Best Answer
The law of the iterated logarithm states that almost surely: $$ \limsup_{t \to \infty} \frac{|B_t|}{\sqrt{2t\log(\log(t))}} = 1 $$
So asymptotically we have that $|B_t| \approx \sqrt{2t\log(\log(t))}$, and in particular $\limsup_{t \to \infty} |B_t| = \infty$. Since $P(B_t > 0, \forall t > 0) = 0$, the brownian motion must cross infinite times the zero line (use the Markov property), and since $\limsup_{t \to \infty} |B_t| = \infty$ we have that $\limsup B_t = \infty$ and $\liminf B_t = -\infty$. Finally by continuity of the paths you can conclude that any level $c$ must be attained.