Let $\mathbf X$ be multivariate gaussian with some nontrivial convariance matrix. I'm looking for (preferably closed form, but small number of integrals to be tabulated is fine) formula for following expression:
$$ \mathbb{E}( \mathbb{I}(a_i < X_i < b_i) \exp( \mathbf{v}.\mathbf{X} )) $$
Where $\mathbf{a,b,v}$ some fixed real vectors.
This happens to have simple expression for $dim(\mathbf{X})\leq2$. I know that this relates to orthant probabilities but I'm unsure how to get there for general $\mathbf{v}$.