I have a conceptual question about Ito's lemma, in particular, the multiplication. Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. That I understand. However, it also states, that multiplying dx by itself yields dt. I understand that dx is proportional to the square root of time. Nevertheless, dx^2 is still a stochastic process. Nevertheless, in Ito's Lemma it is then treated as if it were part of the deterministic part of the formula. This I do not understand since the result is still a (albeit different) random process. Thanks very much for any enlightenment!
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$\begingroup$ Differentials are not processes. The relevant discrete time quantity is to look at the behavior of the square of a $N(0,h)$ random variable, divide it by $h$ (so as to compare it with $dt$) and send $h \to 0$. You find that the answer is just $1$ (in particular it is not random). Thus when you formally see $(dW)^2$ in an expansion, it is equivalent to $dt$. $\endgroup$– IanCommented Nov 28, 2017 at 1:39
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May I point your attention to my following paper, where I address this question in an intuitive manner on page 12:
von Jouanne-Diedrich, Holger, Ito, Stratonovich and Friends (May 18, 2017).
Available at SSRN: https://ssrn.com/abstract=2956257
The basic idea in the discrete case is to go through all possible branches and then square the differences. In each case you will see that it results in the same number - which renders the stochastic case, for all intents and purposes, deterministic!
Details can be found in the paper (and I would love to get your feedback on the paper - Thank you :-)
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