Paradox with Ito Caculus

[quotes]

Hi everyone,

I have a peculiar problem here:

d(S/S) = d(1) =0

d(S/S) = dS /S+S d(1/S)
<product rule>

d(1/S) = -dS/ S^2 + (dS)^2/S^3

d(S/S) = dS/S - dS/S + (dS)^2/S^2 =(dS/S)^2

0 =? (dS/S)^2

 

[stochan]

\[d(XY)=YdX+XdY+dXdY\]

 

[quotes]

Thank you.(y)

How did you input formula BTW?

 

[quotes]

\[d\frac{X}{Y}=\frac{d X}{Y}-\frac{Xd Y}{Y^2}-\frac{d X d Y}{Y^2}+\frac{X d Y d Y}{Y^3}\]

 

[quotes]

\[d\frac{S}{S}=\frac{dS}{S}-\frac{SdS}{S^2}-\frac{dS^2}{S^2}+\frac{SdS^2}{S^3}=0\]