# Ito's Lemma: Multiplication Rule

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!

• 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 $dt$. – Ian Nov 28 '17 at 1:39