Only positive or only negative

Question

How can I calculate the following probability where $W$ denotes a standard Wiener process? $$\mathbb{P}\left(\left|W_{t}\right|>0:\forall t\in\left[1,2\right]\right)=?$$ So what is the probability, that a Wiener process stays only positive on time interval $\left[1,2\right]$ or only negative on time interval $\left[1,2\right]$?

Answer 1

I think I have found the solution.

Let me define the following random moments:$$\tau=\inf\left\{ t\geq1:W_{t}=0\right\} \;\;\;\text{and}\;\;\;\nu=\max\left\{ t\in\left[0,1\right]:W_{t}=0\right\}.$$ So $\tau$ is the first time, when the Wiener process visits the level $0$ after $t=1$, and $\nu$ is the last time on $0\leq t\leq1$ when the Wiener process visits the zero level. We have proven in class, that $\tau\overset{d}{=}\frac{1}{\nu}$, moreover it can be shown, that $\nu$ has arcsine distribution with the following CDF:$$F_{\nu}\left(t\right)=\frac{2}{\pi}\arcsin\sqrt{t},\;\;\;t\in\left[0,1\right].$$

We can redefine the main question: after $t=1$ what is the probability that the Wiener process crosses the level zero first time after $t=2$? More precisely, what is $$\mathbf{P}\left(\tau>2\right)=?$$

Knowing the statements above, $\mathbf{P}\left(\tau>2\right)$ is easy to calculate:$$\mathbf{P}\left(\tau>2\right)=\mathbf{P}\left(\frac{1}{\nu}>2\right)=\mathbf{P}\left(\nu<\frac{1}{2}\right)=\frac{2}{\pi}\arcsin\sqrt{\frac{1}{2}}=\frac{1}{2}.$$