Abstract
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 $$\lim \sup_{\nu \rightarrow \infty} S^2(Q_\nu) = \infty.$$
Citation
Itrel Monroe. "Almost Sure Convergence of the Quadratic Variation of Martingales: A Counterexample." Ann. Probab. 4 (1) 133 - 138, February, 1976. https://doi.org/10.1214/aop/1176996192
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