Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 30K / e / Solution

Choose $N$ with $T\le N$. By part (d),Each $B_k{1}_{\{T\ge k\}}$ is ${\mathcal{F}}_{k-1}$-measurable. Distinct summands are orthogonal: if $j<k$, conditioning on ${\mathcal{F}}_{k-1}$ makes every factor except ${\xi }_k$ known and $\mathbb{E}[{\xi }_k\mid {\mathcal{F}}_{k-1}]=0$. The cross terms with $M_0$ vanish in the same way, while ${\xi }_k^2=1$. Expanding the square therefore givesThis is the stopped martingale isometry in a Rademacher filtration.