Large deviations in rare events simulation: examples, counterexamples and alternatives

When simulating small probabilities, say of order 10-6 or less, by importance sampling, an established principle is to choose the importance sampling distribution as close to the conditional distribution given the rare event as possible. Implementing this often leads into large deviations calculations and exponential change of measure. We survey some of the standard examples where this approach works and supplement existing counterexamples with new ones. Difficulties often arise as consequence of reflecting barriers and we present an algorithm which at least in simple cases is able to deal with this problem. Also the case of heavy-tailed distributions is considere