Let $X_s$ be a continuous martingale and $Q\nu$ be an increasing sequence of partitions of [0, 1]. Let $$S^2(Q_\nu) = \sum_{t_i\in Q_\nu} (X_{t_i} - X_{t_{i - 1}})^2.$$ An example is given in which $$\lim \sup_{\nu \rightarrow \infty} S^2(Q_\nu) = \infty.$$
@article{1176996192,
author = {Monroe, Itrel},
title = {Almost Sure Convergence of the Quadratic Variation of Martingales: A Counterexample},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 133-138},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996192}
}
Monroe, Itrel. Almost Sure Convergence of the Quadratic Variation of Martingales: A Counterexample. Ann. Probab., Tome 4 (1976) no. 6, pp. 133-138. http://gdmltest.u-ga.fr/item/1176996192/