  The study further explores randomized QMC (RQMC), which maintains the QMC
convergence rate and facilitates computational efficiency analysis. Emphasis is
laid on integrating randomly shifted lattice rules, a distinct RQMC quadrature,
with IS,a classic variance reduction technique. The study underscores the
intricacies of establishing a theoretical convergence rate for IS in QMC
compared to MC, given the influence of problem dimensions and smoothness on
QMC. The research also touches on the significance of IS density selection and
its potential implications. The study culminates in examining the error bound
of IS with a randomly shifted lattice rule, drawing inspiration from the
reproducing kernel Hilbert space (RKHS). In the realm of finance and
statistics, many problems boil down to computing expectations, predominantly
integrals concerning a Gaussian measure. This study considers optimal drift
importance sampling (ODIS) and Laplace importance sampling (LapIS) as common
importance densities. Conclusively, the paper establishes that under certain
conditions, the IS-randomly shifted lattice rule can achieve a near $O(N^{-1})$
error bound.
