Let $X$ and $Y$ be two dependent normally distributed continuous random variables (their marginals are $\mathcal{N}(0, 1)$). I would like to find an upper bound on the probability that one is greater than the other by a given threshold, i.e. find a $\theta$ such that $P(Y > X + \delta) < \theta$, $\delta, \theta \in \mathbb{R}$
If $X$ and $Y$ were independent it would be easy as I could directly compute their joint density, but in the case were they are allowed to be dependent all I could do is find a numerical solution by discretizing the joint density (which looked far from trivial).