SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B1-B29, January 2021.
We consider integration with respect to a $d$-dimensional spherical Gaussian measure arising from computational finance. Importance sampling (IS) is one of the most important variance reduction techniques in Monte Carlo (MC) methods. In this paper, two kinds of IS are studied in randomized quasi-MC (RQMC) setting, namely, the optimal drift IS (ODIS) and the Laplace IS (LapIS). Traditionally, the LapIS is obtained by mimicking the behavior of the optimal IS density with ODIS as its special case. We prove that the LapIS can also be obtained by an approximate optimization procedure based on the Laplace approximation. We study the promises and limitations of IS in RQMC methods and develop efficient RQMC-based IS procedures. We focus on how to properly combine IS with conditional MC (CMC) and dimension reduction methods in RQMC. In our procedures, the integrands are first smoothed by using CMC. Then the LapIS or the ODIS is performed, where several orthogonal matrices are required to be chosen to reduce the effective dimension. Intuitively, designing methods to determine all these optimal matrices seems infeasible. Fortunately, we prove that as long as the last orthogonal matrix is chosen elaborately, the choices of the other matrices can be arbitrary. This helps to significantly simplify the RQMC-based IS procedure. Due to the robustness and the superiority in efficiency of the gradient principal component analysis (GPCA) method, we use the GPCA method as an effective dimension reduction method in our RQMC-based IS procedures. Moreover, we prove the integrands obtained by the GPCA method are statistically equivalent. Numerical experiments illustrate the superiority of our proposed RQMC-based IS procedures.

中文翻译：

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拟蒙特卡洛方法中用于计算金融的有效重要性抽样

SIAM科学计算杂志，第43卷，第1期，B1-B29页，2021年1月。
我们考虑对由计算金融产生的$ d $维球形高斯测度进行积分。重要性抽样（IS）是蒙特卡洛（MC）方法中最重要的方差减少技术之一。本文研究了两种在随机准MC（RQMC）设置下的IS，即最佳漂移IS（ODIS）和Laplace IS（LapIS）。传统上，LopIS是通过以ODIS作为特例来模拟最佳IS密度的行为而获得的。我们证明了LapIS也可以通过基于Laplace近似的近似优化程序获得。我们研究RQMC方法中IS的前景和局限性，并开发有效的基于RQMC的IS程序。我们专注于如何在RQMC中将IS与条件MC（CMC）和降维方法正确结合。在我们的程序中 首先使用CMC平滑被积数。然后执行LapIS或ODIS，其中需要选择几个正交矩阵以减小有效尺寸。凭直觉，设计方法来确定所有这些最佳矩阵似乎是不可行的。幸运的是，我们证明只要精心选择了最后一个正交矩阵，其他矩阵的选择就可以是任意的。这有助于显着简化基于RQMC的IS过程。由于梯度主成分分析（GPCA）方法的鲁棒性和效率优势，我们在基于RQMC的IS程序中将GPCA方法用作有效的降维方法。此外，我们证明了通过GPCA方法获得的被积物在统计上是等效的。