Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals $1-\Phi(g(t))$.
Now define some $T$. I would like to know the probability that $B_t > g(t)$ at least once during the time $t\in [0;T]$.
Many thanks for your help.