Let $X, Y$ be continuous random variables. In short, I am interested in a property of the distribution of the difference $X-Y$ given some knowledge of their copula.
For simplicity, consider a probability transform $X=Q_1(U), Y= Q_0(V)$ where $U,V \sim \mathrm{Unif}(0,1)$ and $Q_0, Q_1$ are quantile functions (nondecreasing, left-continuous, you can assume some nice properties such as differentiability). We describe the dependence structure between $U, V$ (equivalently between $X, Y$) by a copula $\pi$ on $[0,1]^2$.
Fix $t\in\mathbb{R}$ and for a given copula $\pi$ bewteen $U, V$ define
$$ F_{\Delta,\pi}(t) := \mathbb P_\pi \big(Q_1(U) - Q_0(V) \le t\big), \quad Q_{\Delta,\pi}(t) := F^{-1}_{\Delta,\pi}(t)\text{ is its quantile function} $$ This describes the distribution of $X-Y$, knowing their marginals ($Q_0, Q_1$) and copula ($\pi$).
Fix $\tau \in (0.5,1)$ and $m \in [0,1-\tau].$ Consider the class of copulas
$$ \Pi_m(\tau)\ :=\ \Big\{\,\pi:\ \pi\ \text{is a copula on }[0,1]^2,\ \ \pi\big((\tau,1]\times(\tau,1]\big)\ \ge\ m\,\Big\}. $$
I am interested in solving
\begin{equation} \label{1} \sup_{\pi\in\Pi_m(\tau)}Q_{\Delta,\pi}(t);\quad\quad (1) \end{equation}
that is, the maximal possible quantile assuming $\pi\in\Pi_m(\tau)$.
- Without the tail constraint ($m=0$), it is easy to see that the countermonotone copula (the Fréchet–Hoeffding lower copula = antidiagonal) is the solution. EDIT: this is actually not true in general!
- With the additional tail rectangle constraint $\pi((\tau,1]^2) \ge m$, the countermonotone copula is no longer feasible.
Hence I am asking: which copula in $\Pi_m(\tau)$ solves (1)?