Bounds on $P(Y, X)$ with $P(Y)$ and $P(X)$ known, as well as $X \geq Y$ Suppose you know the marginal distribution of two random variables, $P(Y)$ and $P(X)$. There are well-known bounds on the joint distribution $P(X, Y)$ that use this information. 
However, suppose you also know that $P(X \geq Y) = 1$. How does this tighten the bounds on $P(X, Y)$? I am especially interested in bounding $E[XY]$, where both variables are continuous. 
 A: Update: A sharp lower bound is given in Corollary 2.4 in Nutz, Marcel, and Ruodu Wang. "The Directional Optimal Transport." arXiv preprint arXiv:2002.08717 (2020).
A bound exploiting the inequality constraint is given by 

Smith, Woollcott. "Inequalities for bivariate distribution with x ≤ y
  and marginals given." Communications in Statistics-Theory and Methods
  12.12 (1983): 1371-1379.

It proceeds by bounding the CDF $P(X \leq x, Y \leq y) = H(x, y)$. The uppper bound is given by 
$$min(P(X \leq x), P(Y \leq y)).$$ 
This is the standard Frechet-Hoeffding bound. This bound is sharp under the available information and the restriction $P(X \geq Y) = 1$.
A lower bound is
$$P(X \leq x) - max(0, min(P(Y \leq y) - P(Y \leq x), P(X \leq y) - P(X \leq x))).$$
According to the paper, this bound may "define a probability density function with negative densities". 
I do not know how to get at "nice" representations of the resulting bounds on $E[XY]$. Without the inequality constraint, sharp bounds on this expectations, using the quantile functions $Q_X$ and $Q_Y$ of the variables, are
$$E[XY]_U = \int_0^1 Q_X(u)Q_Y(u)du$$
and 
$$E[XY]_L = \int_0^1 Q_X(u)Q_Y(1 - u)du,$$
see Aronow, Peter M., Donald P. Green, and Donald KK Lee. "Sharp bounds on the variance in randomized experiments." The Annals of Statistics 42.3 (2014): 850-871. 
The upper bound $E[XY]_U$ is sharp if $P(X \geq Y) = 1$, but the lower bound could be improved upon. This lower bound is givne in the Nutz/Wang paper.
