Timeline for Is the Itō integral $\int_0^TΦ_t\:{\rm d}B_t$ the mean-square limit of $\sum_{i=1}^nΦ_{t_{i-1}}(B_{t_i}-B_{t_{i-1}})$ as $\max_i(t_i-t_{i-1})\to 0$?

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Oct 21, 2016 at 23:09	comment	added	Elena Yudovina		Since the Ito martingale is an isometry, this is equivalent to asking whether $\\|\Phi^\varsigma-\Phi\\| \to 0$ as $\|\varsigma\| \to 0$, for mean-square-discontinuous $\Phi \in \overline{\mathcal E}$. It seems that the answer should be no, by letting $t$ be a point of mean-square-discontinuity of $\Phi$, and a sequence of partitions $\varsigma$ in which $t$ is always one of the points picked.
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