Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 201 / 4 / d / Solution

At the meeting time,The process $C$ is independent of $T$, so conditional on $T=u$ the meeting position is $N(0,u/2)$. Independently, $B_t-B_u$ is $N(0,t-u)$. ThereforeFor $0<s\le t$, integrate this conditional Gaussian distribution against the density from part c: