What I mean by a normalized Brownian motion is the following: $$ U(t) = \frac{W(t)}{\sqrt{t}}, $$ where W(t) is a standard Brownian motion (with $\mu = 0$ and $\sigma = 1$). Is there a name for such a stochastic process $U(t)$?
My question is that, for a given two-sided boundary $a > 0$, what is the distribution of the hitting time $T$? $$ T(a) = \inf_{t:t>0} \{|U(t)| > a\}$$ Or equivalently, for a given $t>0$, what is the distribution of its supremum before $T$? $$ M(t) = \sup_{s:s<t}\{U(s) \}$$
For my particular interest, it would be more helpful to consider all the above with respect to a Brownian motion in discrete time field $W(n), n = 1,2,\ldots$, but only considering $W(t)$ is good enough. Thank you!
By a variant of the Law of the Iterated Logarithm, we have $$ \limsup\limits_{t\rightarrow 0^{+}}{\frac{|W(t)|}{\sqrt{2t\log\log(1/t)}}} = 1$$ with probability one. It follows that $$ \limsup\limits_{t\rightarrow 0^{+}}{|U(t)|} = \limsup\limits_{t\rightarrow 0^{+}}{\frac{|W(t)|}{\sqrt{2t\log\log(1/t)}}\sqrt{2\log\log(1/t)}} = \infty$$ with probability one, since $\lim\limits_{t\rightarrow 0^{+}}{\sqrt{2\log\log(1/t)}} = \infty$. Hence $$ T(a) = \inf\{t>0: |U(t)|> a\} = 0$$ with probability one for all $a$.