Let $A$ be the event that there is some $t$ such that $B_t=1+t^2$, where $B$ is a Brownian mation. Is there any way to compute the probability of $A$, or to approximate it well?
I ask, because we can calculate the probability that $B$ hits a line explicitly. First we change measure with Cameron-Martin, which allows us to reduce to the problem of a Brownian motion hitting a constant. Then we use the reflection principle, which gives the joint distribution of a Brownian motion and its maximum. This approach doesn't seem to generalize to the quadratic.
Thank you.