This question is about the extension of subcopulas to copulas, shown in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look backward and forward." Institute of Mathematical Statistics Lecture Notes-Monograph Series, 28, 1–14. [106, 133]. A proof for the 2-D case is in Nelsen (2006) "An introduction to copulas" (see p.16-17 here)
Let me first review the result. Let $\mathcal{I}\equiv \mathcal{I}_1\times \mathcal{I}_2 \times ... \times \mathcal{I}_n\subseteq [0,1]^n$, where each $\mathcal{I}_i$ is such that $\{0,1\}\in \mathcal{I}_i$ for each $i=1,...,n$.
An $n$-D subcopula is a function $\bar{C}:\mathcal{I}\rightarrow [0,1]$ such that:
- $\bar{C}$ is non decreasing. That is, the volume under $\bar{C}$ of each $n$-D box whose vertices are elements of $\mathcal{I}$ is $\geq 0$.
For example, when $n=2$, take $(x_1,y_1)\in \mathcal{I}$ and $(x_2,y_2)\in \mathcal{I}$ with $x_1\leq y_1$ and $x_2\leq y_2$. These 2 elements of $\mathcal{I}$ form the box with 4 vertices $$(x_1,y_1)\in \mathcal{I}, (x_2,y_1)\in \mathcal{I}, (x_1,y_2)\in \mathcal{I}, (x_2,y_2)\in \mathcal{I}$$ Condition 1 requires that $$\bar{C}(x_1,y_1)+\bar{C}(x_2,y_2)-\bar{C}(x_1,y_2)-\bar{C}(x_2,y_1)\geq 0$$
$\bar{C}(u) = 0$ for any $u \in \mathcal{I}$ that has at least one component equal to 0.
$\bar{C}(u) = u_i$ for any $u \in \mathcal{I}$ that has all components, except the $i$-th, equal to 1.
An $n$-D copula is an $n$-D subcopula for which $\mathcal{I}=[0,1]^n$.
"Extension Lemma": Let $\bar{C}:\mathcal{I}\rightarrow [0,1]$ be an $n$-D subcopula with domain $\mathcal{I}$. Then, there exists a proper $n$-D copula $C$ such that $C(u) = \bar{C}(u)$ for all $u\in \mathcal{I}$.
My question is about whether some specific constraints can be enforced while constructing a copula $C$ extending a subcopula $\bar{C}$.
More precisely, let us consider a subcopula $\bar{C}$. Let $\mathcal{D}$ be the collection of $n$-D boxes whose vertices are elements of $\mathcal{I}$. Suppose that some of these boxes have been constrained to have volume zero under $\bar{C}$. That is, Condition 1 above is enforced as equality ($=0$ instead of $\geq 0$). Let $\mathcal{D}^0\subset \mathcal{D}$ be the collection of boxes with volume zero under $\bar{C}$.
Let $C$ be a copula extending $\bar{C}$. Let $\mathcal{A}$ be the collection of $n$-D boxes whose vertices are elements of $[0,1]^n$. Observe that $\mathcal{D}\subset \mathcal{A}$. Further, observe that each box in $\mathcal{D}^0$ has also volume zero under $C$ because $C$ is an extension of $\bar{C}$.
Suppose that, while constructing the copula $C$, I want to enforce some box(es) $B\in \mathcal{A}\setminus \mathcal{D}$ to have volume zero under $C$. $B$ is not "any" box in $ \mathcal{A}\setminus \mathcal{D}$. In particular, $B$ should satisfy the requirement that, every box $\tilde{B}\in \mathcal{D}$ contained in $B$ (i.e., every "sub-box" $\tilde{B}$ of $B$ such that $\tilde{B}\in \mathcal{D}$) has volume zero under $\bar{C}$ (i.e., $\tilde{B}\in \mathcal{D}^0$).
Am I free to do that? Do I need to "reprove" anything?
I've tried to tweak the traditional proof of the above "Extension Lemma" to achieve my result (see p.16-17 here with $n=2$), but I haven't been successful. The proof of the lemma is based on a bilinear interpolation procedure which makes it hard to introduce variations. Do you have suggestions on how to proceed?