In the process of thinking about this problem, I had a question regarding a related idea.
To summarize, I started by considering the expected number of times that an arithmetic brownian motion would cross a barrier $\beta$ (That resets after each hit) during a time interval. I thought I could start by considering the expected time until the first crossing, and then use the Markov property to get expected crossings. But (from Karatzas and Shreve), the expectation does not exist:
$$\mathbb{E}[t] = \int_0^\infty f(t, b) t dt = \infty$$
where $$f(t, b) = \frac{|b|}{\sqrt{2 \pi t^3}} e^{\frac{-b^2}{2t}}$$
I'm curious to see if there's still a way to get this expected crossings answer or at least some useful approximation of it. It seems to be a natural inverse: longer times means lower crossings, so in some sense we might expect that an undefined passage time expectation could still leave us with a finite expectation of crossing times.
Also, thanks to an answer on the post above, it could be useful to consider the average move that the brownian motion makes.Take $$\mathbb{E}[|S_t|] = 2 \int_0^\infty \frac{1}{\sqrt{2 \pi \sigma^2 t}} e^{\frac{-x^2}{2 \sigma ^2 t}}xdx = \frac{2 \sigma ^2 t}{\sqrt{2 \pi \sigma ^2 t}} \int_0^{-\infty} -e^u du = \sqrt{\frac{2}{\pi}} \sigma \sqrt{t} \rightarrow t = \frac{\pi}{2}\frac{\beta^2}{\sigma ^2}$$and so over a longer interval $T$ we would have $\approx \frac{2}{\pi}\frac{\sigma^2}{\beta^2}T$ crossings. According to my simulations, this formula is a close, but not exact fit for the actual number of crossing times.
So, are there other ways to get the crossings without considering passage times?
Or, even more generally (And the key point of this question), even when the moments aren't defined, is it sometimes possible to glean some useful information somehow about the expectation when it might be analytically undefined (Like only integrating over finite times etc)?
