Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 202 / 1 / 1 / 3 / Solution

For $\theta >0$, the Exponential martingale for Brownian motion and the Doob maximal inequality for a nonnegative submartingale giveTaking $\theta =a/s$ gives $e^{-a^2/(2s)}$. Apply the same argument to $-B$ and use the union bound: