Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 25 / 3 / d

For every $t$, subtraction of the definitions givesThe difference of the two supplied continuous martingales is a square-integrable martingale on $[0,1]$; for each fixed mesh its finite sums are bounded. For this dyadic quadratic variation of a bounded continuous martingale, the Doob L2 maximal inequality therefore givesThus the dyadic approximations to quadratic variation are Cauchy for the expected squared uniform norm, as required. No monotonicity in time of the partially completed squared-increment sums is needed.