**The question:**

$W$ denotes a Wiener process in filtration $mathcal{F}$, where $X$ is an $mathcal{F}_{0}$-measurable

*a,* Exponential

*b,* Cauchy

distributed random variable. Let us define $$tau=infleft{ tgeq0:left|W_{t}right|geq Xright} .$$

What is the probability, that $tau$ is finite, so $mathbf{P}left(tau<inftyright)=?$ And what is $mathbb{E}left(tauright)=?$ and $mathbb{E}left(W_{tau}cdotchi_{left{ tau<inftyright} }right)=?$, where $chi_{left{ tau<inftyright} }$ is the indicator variable of the $left{ tau<inftyright}$ event?

**Here are my thoughts so far…**

I know a Wiener process visits every deterministic level, but I don’t know if the same holds for random levels. Is there any plus boundary criterium? From the task I would say yes, there must be some criterium at least for the expected value for $X$, because why would the task give us two different random variables? I mean the exponential $X$ has finite expected value, but in the case of the Cauchy distribution we can’t say the same. For $mathbb{E}left(tauright)$ I would say it is $infty$, because a Wiener process visits every deterministic level with $1$ probability, but it costs $infty$ time in expectation, so $mathbb{E}left(tauright)=infty$. If the previous happens in the deterministic case, then it should holds for the random case as well, but I can’t say any compelling reason.

For $mathbb{E}left(W_{tau}cdotchi_{left{ tau<inftyright} }right)$ I wanted to use Wald’s identity, but there $mathbb{E}left(tauright)$ must be finite, so I couldn’t use it appropriately. I tried to use Doob’s Optional stopping theorem from this site: https://en.wikipedia.org/wiki/Optional_stopping_theorem. Unfortunatelly, I didn’t find the continuous-time version of the theorem so far, but as I know, the same holds for the continuous case. (Please, let me know, if I am wrong with this statement.) Using this theorem the answer should be $0$, but I am not so sure I can use this. Most of the times I have always used the words “bounded” and “finite” like they are synonims of each other. This example opened my eyes, that it is not necessary correct. I didn’t find it on wikipedia, but as I know Doob’s Optional stopping theorem is an “if and only if” statement. For example a Wiener process reaches every deterministic $aneq0$ level, so if $tau$ is the random level reaching time, then $mathbf{P}left(tau<inftyright)=1$, so we can say it is almost surely finite, but I can’t say any upper boundary for that, therefore we can’t use Doob’s Optional stopping theorem because $mathbb{E}left(W_{tau}right)=aneq W_{0}=0$. Is it a good train of thought? So if something is bounded or finite, then it doesn’t necessary mean the same.