Randomly Shifted Lattice Rules with Importance Sampling and Applications

In financial and statistical computations, calculating expectations often requires evaluating integrals with respect to a Gaussian measure. Monte Carlo methods are widely used for this purpose due to their dimension-independent convergence rate. Quasi-Monte Carlo is the deterministic analogue of Monte Carlo and has the potential to substantially enhance the convergence rate. Importance sampling is a widely used variance reduction technique. However, research into the specific impact of importance sampling on the integrand, as well as the conditions for convergence, is relatively scarce. In this study, we combine the randomly shifted lattice rule with importance sampling. We prove that, for unbounded functions, randomly shifted lattice rules combined with a suitably chosen importance density can achieve convergence as quickly as O(N-1+epsilon), given N samples for arbitrary epsilon values under certain conditions. We also prove that the conditions of convergence for Laplace importance sampling are stricter than those for optimal drift importance sampling. Furthermore, using a generalized linear mixed model and Randleman-Bartter model, we provide the conditions under which functions utilizing Laplace importance sampling achieve convergence rates of nearly O(N-1+epsilon) for arbitrary epsilon values.