Since it is a martingale, it is easy to prove that $\mathrm{Var}[B]=\mathbb{E}([B,B]_t)$, where $[B,B]_t$ denotes the quadratic variation. But what is $[B,B]_t$ equal to, or equivalently what does \begin{equation} \lim_{\|\Pi\|\to 0}\sum_{i=0}^n(B_i-B_{i-1})^2 \end{equation} converge in probability to?
2026-02-23 10:22:29.1771842149
On the quadratic variation of the Brownian motion
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