Context: Need to compute the probability that a 2D Gaussian random walk falls within distance $ d $ of some point $ p $ on the next step. (Assume the covariance $ \Sigma $ is the identity matrix $ I $.)
Considering that there's no analytic expression for the CDF of a multivariate normal (according to Wikipedia), I expect that the same is true here... If that's the case, then how can I go about approximating such an integral?
The best thing I can think of is to approximate the circular area with a portion of an annulus, which would allow me to use the CDF of the univariate standard normal distribution ($F$). So if the RW is at $ S_t $ at time $ t $, the probability that it falls within the annulus which touches the circular region at the next time step is
$$ 2 [F(|S_t-p|+d) - F(|S_t-p|-d)] $$
if $ |S_t-p| > d $.