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Step in Ito's Lemma Derivation
Can anyone please clarify how (dXt)2 = dt?
Re: Step in Ito's Lemma Derivation
It's not true -- there's a missing [$]\sigma^2(t,X_t)[$].
As for [$]dW_t^2 = dt[$], first [$]E[W_t^2] = t[$]. The latter probably suffices for Ito's lemma.
As for [$]dW_t^2 = dt[$], first [$]E[W_t^2] = t[$]. The latter probably suffices for Ito's lemma.
Re: Step in Ito's Lemma Derivation
Alan, skafetaur was referring just to the highlighted text, which I think gives the substitution to be made into the text you were referring to.
skafetaur, the highlighted text is actually wrong because both sides belong in a closed integral for the equation to be correct.
Ignoring that, though, every dXt is a normal variable with mean 0 and variance dt. So dXt/sqrt(dt) is a standard normal variable and a sum of n independent (dXt)2/dt results is a chi-squared variable with n degrees of freedom. This will have mean n and variance 2n.
So a sum of n (dXt)2 results will be dt times chi-squared variable with n degrees of freedom, and will have mean ndt and variance 2ndt2.
So as n ==> ∞ and dt ==> 0 — as in a closed integral — the mean is preserved but the variance disappears.
skafetaur, the highlighted text is actually wrong because both sides belong in a closed integral for the equation to be correct.
Ignoring that, though, every dXt is a normal variable with mean 0 and variance dt. So dXt/sqrt(dt) is a standard normal variable and a sum of n independent (dXt)2/dt results is a chi-squared variable with n degrees of freedom. This will have mean n and variance 2n.
So a sum of n (dXt)2 results will be dt times chi-squared variable with n degrees of freedom, and will have mean ndt and variance 2ndt2.
So as n ==> ∞ and dt ==> 0 — as in a closed integral — the mean is preserved but the variance disappears.
Re: Step in Ito's Lemma Derivation
My first remark is still correct. But your argument for why [$]dW_t^2 = dt[$] is much better than mine.
Re: Step in Ito's Lemma Derivation
You are correct; I read the dXt2 as the dWt2 that it was probably meant to be.
Re: Step in Ito's Lemma Derivation
I think I was the first, twenty or so years ago, to use chi-squared to show Ito convergence, and more importantly rate of convergence. I might still be the only person who makes note of this.
The rest of you are happy to assume continuity.
The rest of you are happy to assume continuity.
Re: Step in Ito's Lemma Derivation
I’ve moved beyond the point of getting excited about this stuff, but I’d recommend following Shreve’s approach and first getting the economics sorted in a binomial setting and then taking a limit. Or, if I were to be radical, consider Rob Anderson’s approach to construct a nonstandard Brownian motion as a symmetric random walk on infinitesimals. No limit taking is required, although there is a requirement to deal with liftings and the taking of standard parts.