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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!

1 Answers 1

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

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    :I think you forgot $\frac12$ before $2Y^{-3}(dY)^2$ ?! am I right ? – 2017-01-17
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    thanks a lot! i think there isnt a 2 in front of dY^2 term. – 2017-01-17
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    @Pandaaaaaaa fixed thank you – 2017-01-17