Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 202 / 2 / a / Solution

The elementary discrete integration-by-parts identity isBy the supplied fact, $M^{(n)}\to M$ in $L^2$ of the uniform norm. DefineThenwhich proves i, whileis an $L^2$-bounded martingale, proving ii.

Choose a subsequence converging uniformly almost surely. For $s<t$, all complete dyadic increments between $s$ and $t$ contribute nonnegative squares; only the two boundary increments can affect monotonicity, and they vanish uniformly by continuity of $X$. Passing to the limit gives $[X{]}_s\le [X{]}_t$. Thus $[X]$ is nondecreasing and is the quadratic variation of $X$.