Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 29K / b / Solution

Use the time-inverted Brownian motion $B_u=u{W}_{1/u}$. For $s>0$, set $u=1/s$. ThenAs $0<s\le t$ corresponds exactly to $u\ge 1/t$, and the inequality at $s=0$ is automatic because $b>0$, this is the pathwise identitySince $B$ and $W$ have the same law by time inversion of Brownian motion,