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

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 $X$.

Choose stopping times $T_m\uparrow \infty$ such that $X^{T_m}$ is a bounded martingale. Part a constructs $[X^{T_m}]$. Uniqueness in the identityshows consistency on overlapping stopped intervals, so define $[X{]}_{t\wedge T_m}=[X^{T_m}{]}_t$. The stopped dyadic sums converge uniformly on every compact interval in probability, andis a local martingale. This localization constructs the quadratic variation of every continuous local martingale.

Letting $t\to \infty$ in the Burkholder-Davis-Gundy inequalities with exponent two gives absolute constants $c,C>0$ such thatSince $[X]$ is nondecreasing, $\mathbb{E}[X{]}_{\infty }={\sup }_t\mathbb{E}[X{]}_t$. Thus one of the two quantities in the question is finite exactly when the other is.