= Localization and patching of quadratic variation
Stop a <continuous local martingale> at increasing level-and-time stopping times to obtain bounded <martingales>. Their limits of discrete <quadratic variation> sums agree before the earlier stopping time, because the sums commute exactly with stopping. These continuous limits patch into a continuous adapted nondecreasing process. On each compact interval, the chance that the stopping time occurs early tends to zero, giving <uniform convergence on compacts in probability> for the original sums.
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