Tail asymptotics for dependent subexponential differences

We study the asymptotic behavior of ℙ( X − Y > u ) as u → ∞, 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.