I'm studying Brownian motion and stopping time and I struggle when it comes to mix both. I know that $(B_t)$ is a Brownian motion and $(Y_t) = 3 + B_{2t}$.
I understand that $(Y_t)$ is not a martingale w.r.t. the natural filtration of $(B_{2t})$ because it is not measurable with w.r.t. the information available at time $T$, it is observed only at time $2t$.
I don't understand how I need to use the exit time of a brownian motion starting from $0$ out of a bond $(-a,b)$ including Doob's optional stopping to T^n to find $P(Y_T =7)$ after translation from 0 to 3. I only know that the answer is $1/3$