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.

翻译：本研究进一步探讨了随机化QMC（RQMC），该方法保留了QMC的收敛速率，并便于计算效率分析。重点在于将随机移位格点规则（一种独特的RQMC求积方法）与经典方差缩减技术重要性采样（IS）相结合。研究强调了在QMC中建立IS理论收敛速率的复杂性——与MC相比，QMC受问题维度和光滑性的影响。本研究还探讨了IS密度选择的重要性及其潜在影响。最终，在再生核希尔伯特空间（RKHS）的启发下，研究了结合随机移位格点规则的IS误差界。在金融与统计学领域，许多问题可归结为计算期望值，主要是关于高斯测度的积分。本研究将最优漂移重要性采样（ODIS）和拉普拉斯重要性采样（LapIS）作为常见的重要性密度。最后，本文证明：在特定条件下，IS-随机移位格点规则可达到接近$O(N^{-1})$的误差界。