Stochastic methods for multiple integrals over unbounded regions

Recent work in the development of stochastic methods for multiple integrals over unbounded regions is reviewed and generalized. This includes randomization or deterministic rules, and new stochastic rules for integrals with multivariate Normal weight. Stochastic spherical–radial rules are also discussed. These rules use a spherical–radial transformation of the infinite integration region and combine stochastic rules for the infinite radial interval with stochastic rules for the spherical surface. Example problems taken from Bayesian statistical analysis and computational finance are used to illustrate the use of the different methods.