Consider a Levy process $L$ in $\mathbb R^d$ written in the Levy-Kchinchine decomosition as $$L(t)=bt + W(t) + Z(t),$$ where $bt$ is the drift part, $W(t)$ is the Wiener part and $Z(t)$ is the jump part. Take a $d\times d$ matrix $A$ and consider solutions $X$ and $Y$ to the following equations
$$d X(t) = AX(t)\, dt + d L(t)$$
$$ dY(t) = AY(t)\, dt + dZ(t)+dbt$$
Put $U(t) = X(t) - Y(t)$. I was told that the process $U$ is idependent of $Y$ but I find it highly dubious. Where is the catch?
I think they are dependent but not sure how to prove it.