I'm calculating the probability that a standard Brownian motion path will cross a boundary.
I have $A$ and $B$ representing the PDFs for the Brownian motion going above a boundary function $a$ and for going below a boundary function $b$, respectively. Clearly, $a$ and $b$ are functions of time, and so are the PDFs, so I can't examine a single value, I have to take an average, so I did the easiest thing I could come up with, I used,
$$P_a={1 \over L} \cdot \int_0^L {A \over {A+B}} \ dt$$
For the probability the Brownian motion hits the boundary $a$ rather than $b$. If the PDFs become constant as I take $L$ to infinity, I can justify this. But, what about if $L$ is finite and the PDFs vary? For instance, take $a$ and $b$ to be half domes.
Basically, how do I probabilisticly interpret this ratio integral?