Adaptive importance sampling Monte Carlo simulation for general multivariate probability laws

We establish a parametric adaptive importance sampling variance reduction method for general multivariate probability laws. Employing the principle of bypass distributions makes it possible to develop adaptive algorithms without relying on particular properties of the target and proposal laws, both of which in the proposed framework are as general as the uniform law on the unit hypercube, without changing the sampling distribution at each iteration. We establish the asymptotic normality of the estimator of the desired mean and of the importance sampling parameter as the number of observations tends to infinity. Although implementation of the proposed methodology requires a small amount of initial work, it has the potential to yield substantial improvements in estimator efficiency in various general problem settings. To illustrate the applicability and effectiveness, we provide numerical results throughout, in which we apply exponential and normal bypass distributions, as well as demonstrate that well-known adaptive importance sampling formulations in the literature can be easily rewritten in the proposed framework.