Copula of $C_{s(X),t(Y)}$ A question on copula: If $s: \Bbb R\to \Bbb R$ is an increasing function, and $t: \Bbb R\to \Bbb R$ is a decreasing function, find the copula $C_{s(X),t(Y)}$ of $(s(X), t(Y))$ in terms of $C_{X,Y}$. (Assume $X$ and $Y$ to be continuous random variables)
 A: Let's introduce some notation, we have the cumulative distribution of $(X,Y)$, $X$ and $Y$ :
$$F_{(X,Y)}(x,y) = \mathbb{P}(X\leq x, Y \leq y) \; , \;\; F_X(x)=\mathbb{P}(X\leq x) \; \mbox{ and } \; F_Y(y)=\mathbb{P}(Y\leq y) \; .$$
Likewise for $(s(X),t(Y))$, $s(X)$ and $t(Y)$ :
$$F_{(s(X),t(Y))}(u,v) = \mathbb{P}(s(X)\leq u, t(Y) \leq v) \; , \;\; F_{s(X)}(u)=\mathbb{P}(s(X)\leq u) \; \mbox{ and } \\ \; F_{t(Y)}(v)=\mathbb{P}(t(Y)\leq v) \; .$$
Now, we first establish the relationship between $F_{(X,Y)}$ and $F_{(s(X),t(Y))}$ :
$$F_{(X,Y)}(x,y) = \mathbb{P}(X\leq x, Y \leq y) = \mathbb{P}(s(X)\leq s(x), t(Y) \geq t(y))$$
The last step is obtained by applying the functions $s$ and $t$ since $s$ preserves order and $t$ reverses it. This can be further transformed into
$$F_{(X,Y)}(x,y) = \mathbb{P}(s(X)\leq s(x)) - \mathbb{P}(s(X)\leq s(x), t(Y) \leq t(y)) = F_{s(X)}(s(x)) - F_{(s(X),t(Y))}(s(x),t(y))$$
Since our random variables are continuous, we assume that the difference between $t(Y) \leq t(y)$ and $t(Y)<t(y))$ is just a set of zero measure.
Now, to transform this into a statement about copulas, note that
$$C_{(X,Y)}(a,b) = F_{(X,Y)}(F_X^{-1}(a),F_Y^{-1}(b))$$
Thus, plugging $x=F_X^{-1}(a)$ and $y=F_Y^{-1}(b)$ into our previous formula, we get
$$F_{(X,Y)}(F_X^{-1}(a),F_Y^{-1}(b)) = F_{s(X)}(s(F_X^{-1}(a))) - F_{(s(X),t(Y))}(s(F_X^{-1}(a)),t(F_Y^{-1}(b)))$$
The left hand side is the copula $C_{(X,Y)}$, the right hand side still needs some work. Note that
$$F_{s(X)}(s(F_X^{-1}(a))) = \mathbb{P}(s(X)\leq s(F_X^{-1}(a))) = \mathbb{P}(X\leq F_X^{-1}(a)) = F_X(F_X^{-1}(a)) = a$$
and likewise
$$F_{t(Y)}(s(F_Y^{-1}(b))) = \mathbb{P}(t(Y)\leq t(F_Y^{-1}(b))) = \mathbb{P}(Y\geq F_Y^{-1}(b)) = 1-F_Y(F_Y^{-1}(b)) = 1-b$$
Combining all results we obtain for the relationship between the copulas
$$C_{(X,Y)}(a,b) = a-C_{(s(X),t(Y))}(a,1-b) \; .$$
