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Suppose that $X$ and $Y$ are normally distributed with mean zero and nonzero covariance. I want to know the distributions of $X | X - Y > c$ and $Y | X - Y > c$, which I believe should be jointly distributed as a multivariate normal distribution. I can compute the conditional expectations, using this answer and this answer, but I also want to know the variances and covariance.

That is, I want to compute $$\text E[X | X - Y > c], \quad \text E[Y | X - Y > c],$$ $$\text {Var}(X | X - Y > c), \quad \text {Var}(Y | X - Y > c),$$ and $$\text {Cov}(X, Y | X - Y > c).$$ How do I go about this?

quevivasbien
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If $X$ and $Y$ both have unit variance, then the formulas for this may be unilluminating but at least there are some formulas.

The trick is to express everything as integrals in terms of the variables $U=(X+Y)/\sqrt{2}$ and $V=(X-Y)/\sqrt{2}$.

This leads to a conditional covariance of $$K=\rho +\frac{(1-\rho)e^{-a^2}}{4\pi\, \Phi^2(-a)} -\frac{(1-\rho)e^{-a^2/2}a}{\sqrt{2\pi}\, \Phi(-a)}$$ where $a=c/\sqrt{2(1-\rho)}$, $\rho$ is the correlation between the original $X$ and $Y$, and $\Phi$ is the standard normal cdf.

The conditional variances are both equal to $1+\rho-K$.

Matt F.
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