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$dz_t^2$ equals the quadratic variation (bracket process) $[dz_t]$. $[X_t]$ is defined as the process such that $X_t-[X_t]$ is a martingale. One can show by Ito Lemma, that the solution of the stochastic integral $(z\cdot z)_t=\int_0^t z dz$ leads to $[z]_t=t$, hence $d[z]_t=dz_t^2=dt$ (note that the general Ito formula is based on quadratic variations in ...


Beware, oversimplification ahead! (This means that the following is technically not correct, in fact it is false! But: It gives an intuition what is going on!) If you toss a coin and calculate heads as $-1$ and tails as $1$ you get a mean of $0$ with a variance of $1$. When you add up multiple coin tosses, i.e. create a random process $dz(t)$, the mean ...


Intuitively, because of the central limit theorem: wiener process is a limit of a random walk, and after n steps a random walk moves away from the origin by ~ $\sqrt{n}$ Edit: here is a complete answer. First the formula for the sum. The trick is the following simple observation: if $X_1,.. X_n$ are independent zero mean, then $E(\sum X_i)^2 = ...


Intuitively you are measuring the distance between two random variables which under limiting case turns out to be very small. So you can use one instead of another.

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