We have a stochastic process $ Y_t= \alpha t+ W_t$ where W is a standard brownian motion.
Is there a way to calculate the conditional probability with respect to $Y_1$ for this process to hit a barrier $B$ between $t=0$ and $t=1$ ?
$ \mathbb{P}(sup_{[0,1]} Y_t \geq B \mid Y_1)= ?$