Assume that $X=(X_1,X_2)^T$ is a random vector that is multivariate normal with mean vector $\mu$ and variance-covariance matrix $\Sigma$. I.e. we have $$ X \sim N_2(\mu, \Sigma).$$
I am interested in the truncated version of this normal distribution and assume that $X_1$ and $X_2$ are smaller than some threshold $c$. Hence I am searching for $X^*=(X_1^*,X_2^*)^T$ that is jointly trucnated normal
$$ X^*~\sim TN_2(\mu, \Sigma,c)$$ While there are formulas for the expectation and the variance of the univariate truncated normal I do not know how to compute the covariance of $X_1^*$ and $X_2^*$: $\operatorname{Cov}(X_1^*,X_2^*)= \operatorname{Cov}(X_1,X_2\mid X_1<c,X_2<c)$.