Tail asymptotics for dependent subexponential differences

We study the asymptotic behavior of $\mathbb P(X-Y>u)$ as $u\to\infty$, where $X$ is subexponential, $Y$ is positive, and the random variables $X$ and $Y$ may be dependent. We give criteria under which the subtraction of $Y$ does not change the tail behavior of $X$. It is also studied under which conditions the comonotonic copula represents the worst-case scenario for the asymptotic behavior in the sense of minimizing the tail of $X-Y$. Some explicit construction of the worst-case copula is provided in other cases.