The problem comes from Karatzas's book 'Brownian motion and stochastic analysis'. Exercise 5.20.
Suppose $X$ is in the space of square integrable martingales with stationary, independent increments. Then $\langle X \rangle_t = t(EX_1^2), T \ge 0$.
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1What is $u_2$? What are your thoughts about the problem? – uniquesolution Mar 31 '17 at 06:41
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$\mathcal{u}_2$ should be a space of square integral martingales. Actually, I am not quite understand $t(EX_1^2)$. – Fly_back Mar 31 '17 at 06:46
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You have to see $E(X_{1}^2)$ like some kinda variance $\sigma^2$ for process. Standard brownian motion has variance 1, so its bracket process is 1*t.
That being said, depending on the tools you have there is two or three ways to solve this (One being really straightforward). Edit : Yes ! It is $t \cdot \sigma^2$ .
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You can use the fact that the quadratic variation is the unique a.s. continuous, monotone increasing and adapted process so that $⟨X⟩_0=0$ a.s. t that makes $X^2_t-⟨X⟩_t$ a martingale. This is precisely the definition of the quadratic variation and therefore if you can verify that this is a martingale you have shown that this is the quadratic variation.
El Duderino
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