I am currently working with the book An Introduction to Copulas by Roger Nelsen. I am having trouble solving one of the exercises in the first chapter. The exercise reads as follows:

Let $C$ be a copula, and let $(a,b)$ be a point in $[0,1]\times [0,1]$. For $(u,v)$ in $[0,1]\times [0,1]$, define $$K_{a,b}(u,v) = V_{C}([a(1-u), u+a(1-u)]\times [b(1-v), v+b(1-v]).$$ Show that $K_{a,b}$ is a copula.

For reference, the C-volume $V_{C}$ is defined as $V_{C}([u_{1},u_{2}]\times [v_{1},v_{2}]) = C(u_{2},v_{2}) - C(u_{2},v_{1}) - C(u_{1},v_{2}) + C(u_{1},v_{1})$.

The only thing I have not been able to show in this exercise is that $V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}]) \geq 0$ for any rectangle $[u_{1},u_{2}]\times [v_{1},v_{2}] \subset [0,1]\times [0,1]$ , which is the last defining property of copulas given in the book. Any help/hints for this problem would be greatly appreciated.

For reference, it has already been demonstrated in the book that copulas are continuous. The following inequalities has also been established for any copula $C(u,v)$: For every $(u,v)$ in the domain of $C$, we have $\text{max}(u+v-1,0)\leq C(u,v)\leq \text{min}(u,v)$. Further, it has been established that bivariate copulas are nondecreasing in both arguments.

I have tried writing out the whole expression for $V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}])$ and use the property that copulas are nondecreasing in both arguments to establish that $V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}]\geq 0$. I have also found out (if it is not wrong), that one can write

$V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}]) = V_{C}([a(1-u_{2}), u_{2} + a(1-u_{2})]\times [b(1-v_{2}),v_{2} + b(1-v_{2})]) - V_{C}([a(1-u_{2}), u_{2} + a(1-u_{2})]\times [b(1-v_{1}),v_{1} + b(1-v_{1})]) - V_{C}([a(1-u_{1}), u_{1} + a(1-u_{1})]\times [b(1-v_{2}),v_{2} + b(1-v_{2})]) + V_{C}([a(1-u_{1}), u_{1} + a(1-u_{1})]\times [b(1-v_{1}),v_{1} + b(1-v_{1})])$,

but I have not been able to use this to solve the exercise. However, the first and fourth term must be non-negative, due to $C$ being a copula. Therefore, I have tried to show that the sum of the first and fourth term must be greater than or equal to the sum of the second and third term, without success. I have also tried using the bounds above, but quickly got lost when splitting into cases based on the assumed relative sizes of the variables involved.

Thanks in advance to anyone that can provide any hints/help, and thanks to all that took the time to read this post.

Update: Using the fact that every $V_{C}$ in the above expression maps to $[0,1]$, one can see that $V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}]) \geq 0$ if $V_{C}([a(1-u_{2}), u_{2} + a(1-u_{2})]\times [b(1-v_{2}),v_{2} + b(1-v_{2})]) + V_{C}([a(1-u_{1}), u_{1} + a(1-u_{1})]\times [b(1-v_{1}),v_{1} + b(1-v_{1})]) \geq 2$

but this restriction is too strict to be useful. I am therefore currently looking for looser restrictions that might be useful. I have also tried to check whether or not the C-volume is monotonic in the size of the rectangle in $[0,1]\times [0,1]$ (if C is a copula), but have not been able to conclusively demonstrate this. In this pursuit, I also found this Is $H$-measure actually monotonic (at least on hyperrectangles)? , but I am not entirely convinced of the answer provided in the link.

Second update: I probably should have mentioned that not only has continuity been established, but something a bit stronger, namely that if $C$ is a copula, then for every $(u_{1},u_{2}), (v_{1},v_{2})$ in the domain of $C$, we have $|C(u_{2},v_{2}) - C(u_{1},v_{1})| \leq |u_{2}-u_{1}| + |v_{2}-v_{1}|$. I have been trying to use this to estimate relevant bounds on different terms of $V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}])$, but still without any luck.

Third update: I may have found a looser restriction than in the first update. If this is correct, it should be enough to show that $V_{C}([a(1-u_{2}), u_{2} + a(1-u_{2})]\times [b(1-v_{2}),v_{2} + b(1-v_{2})]) + V_{C}([a(1-u_{1}), u_{1} + a(1-u_{1})]\times [b(1-v_{1}),v_{1} + b(1-v_{1})]) + C(a(1-u_{2},v_{1}+b(1-v_{1})) - C(a(1-u_{2}), b(1-v_{1})) + C(a(1-u_{1}), v_{2}+b(1-v_{2})) - C(a(1-u_{1}), b(1-v_{2})) \geq v_{1} + v_{2}$.

This is because the first and second term has to be non-negative due to $C$ being a copula, and the difference between the third and fourth term and the difference between the fifht and the sixth term has to be non-negative due to $C$ being nondecreasing in both arguments. The remaining four terms of $V_{K_{a,b}}$ gives a net negative contribution to $V_{K_{a,b}}$, the upper bound of which I have estimated using the information in the second update.


It believe I have solved it now. If this is correct, it turns out that the answer was quite simple all along. Also, everything in the three updates above turned out to lead to a dead end as I was able to show using a special case that it was impossible in general to get a "lowest possible" bound on a C-volume that was also strictly positive. Hence, it should not be possible to verify the inequality I presented in the third update using only the information we have available. However, playing around with proving special cases led me to notice that if we write out $V_{K_{a,b}}([u_{1}, u_{2}]\times [v_{1}, v_{2}])$ in a particular manner, the solution presents itself quite nicely. Write all the four terms associated with $u_{i}, v_{j}$, where $i=1,2, j=1,2$, in one row for each pair $i,j$. Visually, this gives a $4\times 4$ matrix. One may now notice that each column of the matrix represents a C-volume. More specifically, one can write

\begin{align} V_{K_{a,b}}([u_{1},u_{2}]\times [v_{1},v_{2}]) &= V_{C}([u_{1}+a(1-u_{1}), u_{2}+a(1-u_{2})]\times [v_{1}+b(1-v_{1}), v_{2}+b(1-v_{2})]) \\ &+ V_{C}([u_{1}+a(1-u_{1}), u_{2}+a(1-u_{2})]\times [b(1-v_{2}), b(1-v_{1})]) \\ &+ V_{C}([a(1-u_{2}), a(1-u_{1})]\times [v_{1}+b(1-v_{1}), v_{2}+b(1-v_{2})]) \\ &+ V_{C}([a(1-u_{2}), a(1-u_{1})]\times [b(1-v_{2}), b(1-v_{1})]) \\ &\geq 0 \end{align}

because each C-volume is non-negative due to $C$ being a copula.


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