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-1

$dz_t^2$ 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 ...


4

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 ...


1

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 = ...


-4

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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