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?
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2Does this answer your question? Quadratic Variation of Brownian Motion – Kurt G. May 27 '23 at 19:02
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What I’m trying to ask is what it means to have a ‘finite’ variation. A p-variation of a process is another stochastic process $[X]^p_t$. In what sense is $[X]^p_t$ finite? The link you posted show that its expectation is finite, but it is not enough I guess. Does that mean $[X]^p_t<+\infty$ almost surely? But I’ve never seen it proven anywhere. – ric.san May 28 '23 at 07:58
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One important thing to note is that for the quadratic variation process, the partition is fixed before choosing your outcome $\omega \in \Omega$, where (as in the link) the 2-variation is a supremum over all partitions, which will be infinite (for most paths of Brownian motion) even on a finite interval. Which are you talking about? – George Jun 02 '23 at 15:37
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I'm referring to the quadratic variation, in the sense of your first definition @George – ric.san Jun 02 '23 at 18:22
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1Yes, a positive random variable with finite expectation is finite almost surely. – user6247850 Jun 02 '23 at 18:57
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@user6247850 according to what theroem? – ric.san Jun 03 '23 at 14:28
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That's a standard result in nearly every measure theory class. It follows readily from Markov's inequality. – user6247850 Jun 03 '23 at 15:10
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@ric.san it follows immediately from the definition of the integral for extended real valued functions – Andrew Jun 04 '23 at 23:41