Consider bivariate copulas $C\_1$ and $C\_2$ with $\max\{C\_1(u,v), C\_2(u,v)\}< M\_2(u,v)$ for all $u,v \in(0,1)$, where $M\_2(u,v) := \min\{u,v\}$ is the Fréchet-Hoeffding upper bound.
Is there a copula $D$ with $\max\{C\_1(u,v), C\_2(u,v)\}\leq D(u,v) < M\_2(u,v)$ for all $u,v \in(0,1)$?


The problem here is that $\max\{C\_1(u,v), C\_2(u,v)\}$ is in general only a quasi-copula and I need a bound strictly below $M\_2$. I think such a bound should exist in full generality but I don't have a reference/proof for it.



Existence of copula bound pointwise strictly smaller than the Fréchet-Hoeffding upper bound
You can prove more. Let $F(u,v)$ be any $1$-Lipschitz function on $[0,1]^2$ such that $F(u,v)<\min(u,v)$ inside the square. Then there exists a copula $D(u,v)$ such that
$$
F(u,v)\le D(u,v)<\min(u,v)
$$
everywhere inside the square.


The second inequality will be immediate if we just construct the corresponding joint distribution measure $\mu$ on $[0,1]^2$ with full support. Indeed, then
$$
D(u,v)=\mu([0,u]\times [0,v])=\mu([0,u]\times [0,1])-\mu([0,u]\times(u,1])
\\
=u-\mu([0,u]\times(u,1])<u
$$
and similarly for $v$.


It remains to make the following two observations.


Observation 1: There is a copula $D\_0(u,v)>F(u,v)$ inside the unit square such that the associated $\mu$ has density $p\_0(u,v)\ge q\_0>0$ separated away from $0$ in some open neighborhood $\Omega$ of $\{(x,x):0<x<1\}$. To see it, just take a sufficiently fine countable partition of $(0,1)$ into disjoint intervals $I\_j$, consider the density $p=\sum\_j|I\_j|^{-1}\chi\_{I\_j\times I\_j}$ and mix two such distributions to take care of the corners. "Sufficiently fine" just means that the length of each $I\_j$ is much less than $u-F(u,u)$ for $u\in I\_j$. Now take any sequence $q\_0>q\_1>\dots \ge q\_0/2$ and


Observation 2: Suppose we have a copula $D\_n$ for which $D\_n(u,v)>F(u,v)$ inside the unit square and the density $p\_n$ of $\mu\_n$ is at least $q\_n$ in $\Omega$. Let $(u\_n,v\_n)\in(0,1)^2$ be any point outside the diagonal. Then, for every $\delta\_n\ge 0$, there exists a copula $D\_{n+1}$ with $D\_{n+1}>F$ inside the unit square, such that $p\_{n+1}\ge p\_n$ outside $\Omega$, $p\_{n+1}\ge q\_{n+1}$ in $\Omega$, and $(u\_n,v\_n)$ is in the support of $\mu\_{n+1}$. To do it, just choose two very short intervals $U,V$ of equal length containing $u\_n$ and $v\_n$ respectively and set
$$
p\_{n+1}=p\_n+t(-\chi\_{U\times U}-\chi\_{V\times V}+\chi\_{V\times U}+\chi\_{U\times V})
$$
with very small $t$. Notice that this changes $D\_n(u,v)$ only on a compact subset of the open unit square (say, $I\times I$ where $I$ is a closed interval such that $U\cup V\subset I\subset (0,1)$) and there the difference $D\_n-F$ is separated from $0$, so this surgery leaves it positive for small $t>0$.


Now just take any sequence $(u\_n,v\_n)\in (0,1)^2\setminus{\rm diag}$ dense in the square and run this recursion. Then take either the weak limit of $\mu\_n$, or the $L^1$-limit of $p\_n$, whichever you are more comfortable with, to get $D$.


