I have X and Y following the dynamics
$dX(t)=X(t)(rdt+\sigma_{X}dW(t)$
$dY(t)=Y(t)(rdt+\sigma_{Y}dW(t)$
The components of the BM are independent.
What is the distribution of $ln(\frac{X(t)}{Y(t)})$?
By using the Ito, I got:
$N(ln\left(X(0)Y(0)\right) - 1/2 (\sigma_{X}^{2} +\sigma_{Y}^{2})t, (\sigma_{X}-\sigma_{Y})t)$
By using the rule of obtaining the distribution of a sum of two normal distributed variable, I got:
$N(ln\left(X(0)Y(0)\right) +1/2 \sigma_{Y}^{2} -1/2\sigma_{X}^{2})t, (\sigma_{X}-\sigma_{Y})t)$
Which mean is right? Thank you in advance!
Neither expression is correct, since interchanging $X$ with $Y$ corresponds to negating the expression $\log(X/Y)$, and hence negating the mean. In particular, the mean must be an antisymmetric function of $X$ and $Y$, which it is not in either of your two expressions.
With that being said, clearly the second expression is closer to being correct, where the mistake occurs in your constant term. By considering the degenerate case $\sigma_X=\sigma_Y=0$ in which case everything is deterministic and $X(t)=X(0)e^{rt}$ and similarly for $Y(t)$ should clarify what your constant term needs to be.