Coupling of probability distributions and an extremal problem for the divergence

Let $X$ and $Y$ be discrete random variables having probability distributions $P_X$ and $P_Y$, respectively. A necessary and sufficient condition is obtained for the existence of an $\alpha$-coupling of these random variables, i.e., for the existence of their joint distribution such that $\operatorname{Pr}\{X=Y\}=\alpha$, where $\alpha$, $0\le\alpha\le1$, is a given constant. This problem is closely related with the problem of determining the minima of the divergences $D(P_Z\,\|\,P_X)$ and $D(P_X\,\|\,P_Z)$ over all probability distributions $P_Z$ of a random variable $Z$ given $P_X$ and under the condition that $\operatorname{Pr}\{Z=X\}=\alpha$. An explicit solution for this problem is also obtained.