Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 29 / 1 / b

Fix $T<\infty$. By uniform continuity of each path of the continuous local martingale on $[0,T+1]$, its modulustends to zero almost surely. For every $s\le T$, comparison of powers of each increment givesThe squared-increment sum converges in probability to $[X{]}_T$, by the dyadic definition of quadratic variation. Including the last full grid interval instead of the last partial interval changes that sum by a quantity bounded by $2{\eta }_n^2$, so the ceiling convention has the same limit. In particular these squared-increment sums are bounded in probability.

Since $p-2>0$, a quantity tending to zero almost surely multiplied by a family bounded in probability tends to zero in probability. HenceThis is uniform convergence on compacts in probability, and proves the superquadratic case of dyadic power variation of a continuous local martingale.