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][1]).
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**Let me first review the result** For simplicity, let $n=2$.
Let $\mathcal{S}\equiv \mathcal{S}_1\times \mathcal{S}_2 \subseteq [0,1]^2$, where each $\mathcal{S}_i$ is such that $\{0,1\}\in \mathcal{S}_i$ for each $i=1,2$.
A $2$-D subcopula is a function $\bar{C}:\mathcal{S}\rightarrow [0,1]$ such that:
**1.** $\bar{C}$ is non decreasing. That is, the volume under $\bar{C}$ of each $2$-D box whose vertices are elements of $\mathcal{S}$ is $\geq 0$.
- For example, take $(a_1,b_1)\in \mathcal{S}$ and $(a_2,b_2)\in \mathcal{S}$ with $a_1\leq a_2$ and $b_1\leq b_2$. These 2 elements of $\mathcal{S}$ form the box with 4 vertices $$(a_1,b_1)\in \mathcal{S}, (a_2,b_1)\in \mathcal{S}, (a_1,b_2)\in \mathcal{S}, (a_2,b_2)\in \mathcal{S}$$ Condition 1 requires that $$\bar{C}(a_1,b_1)+\bar{C}(a_2,b_2)-\bar{C}(a_1,b_2)-\bar{C}(a_2,b_1)\geq 0$$
- As another example, take $(0,0)\in \mathcal{S}$ and $(a,b)\in \mathcal{S}$. These 2 elements of $\mathcal{S}$ form the box with 4 vertices $$(0,0)\in \mathcal{S}, (a,0)\in \mathcal{S}, (0,b)\in \mathcal{S}, (a,b)\in \mathcal{S}$$ Condition 1 requires that $$\bar{C}(a,b)+\bar{C}(0,0)-\bar{C}(a,0)-\bar{C}(0,b)\underbrace{=}_{\text{See condition 2 below}} C(a,b)\geq 0$$
**2.** $\bar{C}(u) = 0$ for any $u \in \mathcal{S}$ that has at least one component equal to 0.
**3.** $\bar{C}(u) = u_i$ for any $u \in \mathcal{S}$ that has all components, except the $i$-th, equal to 1.
A $2$-D copula is a $2$-D subcopula for which $\mathcal{S}=[0,1]^2$.
"Extension Lemma": Let $\bar{C}:\mathcal{S}\rightarrow [0,1]$ be a $2$-D subcopula with domain $\mathcal{S}$. Then, there exists a proper $2$-D copula $C$ such that $C(u) = \bar{C}(u)$ for all $u\in \mathcal{S}$.
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**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 $\bar{C}: \underbrace{\mathcal{S}_1\times \mathcal{S}_2}_{\equiv \mathcal{S}}\rightarrow[0,1]$.
Let us consider the simple setting where $\mathcal{S}_1$ and $\mathcal{S}_2$ are **finite**.
Consider the box with vertices $(a,b),(c,b), (a,d), (c,d)$, which we call $B$.
Assume that **at least one** of the following four conditions holds: $a\notin \mathcal{S}_1$; $b\notin \mathcal{S}_2$; $c\notin \mathcal{S}_1$; $d\notin \mathcal{S}_2$. That is, the vertices of $B$ are not all elements of $\mathcal{S}$.
Further, assume that if $B$ contains any box $D$ whose 4 vertices are all elements of $\mathcal{S}$, then the volume of $D$ under $\bar{C}$ is equal to 0.
Can we construct a copula $C$ that extends $\bar{C}$ and such that
$$
C(a,b)+C(c,d)-C(a,d)-C(c,b)=0 \quad ?
$$
I've tried to tweak the traditional proof of the above "Extension Lemma" to achieve my result (see p.16-17 [here][1] with $n=2$), but I haven't been successful. The proof of the lemma is based on a bilinear interpolation procedure which just ensures that $C(a,b)+C(c,d)-C(a,d)-C(c,b)\geq 0$ and not $C(a,b)+C(c,d)-C(a,d)-C(c,b)= 0$. Do you have suggestions on how to proceed?
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**I'm reporting here some questions on the answer below.** In this picture I consider 6 possible cases. Each quadrant is $[0,1]^2$. The pink dots are the elements of $\mathcal{S}\equiv \mathcal{S}_1\times \mathcal{S}_2$ (recall that in my simplified setting $\mathcal{S}$ is finite).
[![enter image description here][2]][2]
Consider case 1, where there is only one box, $B$, whose volume we want to be zero (small black quadrant in the middle). Associated with box $B$, there is only one atomic box, $D$, from the sub-copula. Your proposal consists of uniformly redistributing mass from $B$ to $D\setminus B$ (yellow area), without modifying the value of the copula at the pink dots (because otherwise we would modify the sub-copula $\bar{C}$ which is instead fixed).
Consider case 2, where again there is only one box, $B$, whose volume we want to be zero (black rectangle in the middle). Associated with box $B$, there are 6 atomic boxes, $\{D_i\}_{i=1}^6$, from the sub-copula. For each $i=1,...,6$, let $B_i\equiv B\cap D_i$. Your proposal consists of uniformly redistributing mass from $B_i$ to $D\setminus B_i$ for each $i=1,...,6$ (yellow areas), without modifying the value of the copula at the pink dots.
Case 4 seems similar to case 1.
I have one question at this stage: **(Q1)** Is the "uniform" shift of the mass necessary here? If yes, why?
Let us now move to the pathological cases which, if I have understood correctly your answer, are cases 3,5, and 6. Consider, for instance, case 3. Here there are two boxes, $B_1,B_2$, whose volume we want to be zero. Associated with box $B_1$, there are 4 atomic boxes, $\{D_{i,1}\}_{i=1}^4$, from the sub-copula. Associated with box $B_2$, there are also 4 atomic boxes, $\{D_{i,2}\}_{i=1}^4$, from the sub-copula. $B_1$ and $B_2$ share one atomic box. This means that the blue region should accommodate part of the mass coming from both $B_1$ and $B_2$. Your discussion suggests that this creates issues in the "mass-redistribution" procedure. My second question is: **(Q2)** Why is this an issue?
**To sum up: I'm not sure about when/why the "mass-redistribution" procedure fails in the case where two or more boxes are required to have zero volume.**
[1]: https://rady.ucsd.edu/faculty/directory/valkanov/pub/classes/mfe/docs/copula_ch2.pdf
[2]: https://i.sstatic.net/Xr6U0.jpg