TITLE: Proof of Ito quotient rule.
QUESTION [7 upvotes]: I've tried my best to search this problem but failed to find any on this site.
Please let me know if this problem is duplicated.
I would like to show the Ito's quotient rule as follows:
$$d(\frac{X}{Y})=\frac{X}{Y}(\frac{dX}{X} - \frac{dY}{Y} - \frac{dX}{X}\frac{dY}{Y}+(\frac{dY}{Y})^2)$$
My approach is to apply Ito's formula to
$$d(X\frac{1}{Y})=\frac{1}{Y}dX+Xd\frac{1}{Y}+dXd\frac{1}{Y}=\frac{X}{Y}(\frac{dX}{X}-\frac{dY}{Y}-\frac{dXdY}{XY})$$
Where I think $d(\frac{1}{Y})=-Y^{-2}dY$. However, I don't know where this term, $(\frac{dY}{Y})^2$, comes from? Any help will be appreciated! Thanks in advance!

REPLY [5 votes]: When you apply Ito's lemma,
$$
df(Y_t) = f'(Y_t) dY_t + \frac{1}{2}f''(Y_t) (dY_t)^2
$$
so
$$
d \left(Y^{-1}\right) = -Y^{-2} dY + 2 \times \frac{1}{2} \times Y^{-3}(dY)^2
$$