  In many stochastic problems, the output of interest depends on an input
random vector mainly through a single random variable (or index) via an
appropriate univariate transformation of the input. We exploit this feature by
proposing an importance sampling method that makes rare events more likely by
changing the distribution of the chosen index. Further variance reduction is
guaranteed by combining this single-index importance sampling approach with
stratified sampling. The dimension-reduction effect of single-index importance
sampling also enhances the effectiveness of quasi-Monte Carlo methods. The
proposed method applies to a wide range of financial or risk management
problems. We demonstrate its efficiency for estimating large loss probabilities
of a credit portfolio under a normal and t-copula model and show that our
method outperforms the current standard for these problems.
