Suppose we wave a Brownian Motion $B_t$ on the interval [0,1] and I would like to compute the probability that in intersects a given curve of the following type:
$f(-1)=-\infty $, $f(1)=0$ and f is strictly increasing.
E.g $f$ could be $\displaystyle{-\frac{1}{x} +1}$.
Is there a way I could to this type or problems without Monte-Carlo methods i.e maybe arrive at a PDE or an integral ?
Clarification: A Brownian motion in the interval [0,1] is a random function $B_t$: $[0,1] -> \mathbb{R} $. Such that $B_0=0$ and the increments are independent normal distributed such that $B_{t_1}-B_{t_2}$ $=N(0,t_1-t_2) $ for $t_1>t_2$.