Sure, I understand - it's a very nice correction, thanks. I mean that I need to find change of measure which makes the same change of distributions, i.e. $Q_X\to Q_Y$ such that $$ dX_t = \sigma dw_t\to dY_t = \sqrt{a}\sigma dw_t. $$ I ask this question as a new question mathoverflow.net/questions/51103/… so I will be happy if can forward our discussion to the new question.

Yes, it is extremely strange result that $$ \frac{dQ_Y}{dQ_X}=0 $$ if $|a|\neq 1$. You can mention that if we take 1a instead of a we should have that $$ \frac{dQ_Y}{dQ_X} =1/\frac{dQ_X}{dQ_Y} = \infty $$ - under the condition that your result is true of course.

I think, my question is not correct, I will reask it.

springerlink.com/content/0383j321m62n1u3t/fulltext.pdf Proposition 1

Haha, have you mentioned that the definition of $R(T,A)$ does not coincide with the definition of $I_A(X_\tau)$. Of course you can say it's obvious that if two triangles have the same sides, they are equal - but maybe you remember that this simple fact also need to be proved.

It's not a proof )