单位立方体边界的不连续积分函数的准蒙特卡洛方法
Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube
AI总结：
本文研究了处理单位立方体边界处奇异性的随机准蒙特卡洛方法，发现其误差率可达到O(n^{-1/2-1/(4d-2)+ε})，并指出若不连续边界与坐标轴平行则可进一步提高精度。
摘要：
本文研究了随机准蒙特卡洛（QMC）采样在处理单位立方体边界处具有奇异性的不连续积分函数时的应用。已知随机QMC的均方误差仅为o(n^{-1/2})。本文发现，在某些温和条件下，随机QMC的期望误差为O(n^{-1/2-1/(4d-2)+ε})，其中ε>0任意小。此外，若不连续边界与某些坐标轴平行，则可获得更好的收敛速率。作为副产品，我们发现当不连续性边界为QMC友好（即所有不连续边界与坐标轴平行）时，期望误差率可达O(n^{-1+ε})。这些结果可用于评估金融工程中某些典型问题的QMC精度。
英文摘要：
This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $[0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only $o(n^{-1/2})$ for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of $O(n^{-1/2-1/(4d-2)+ε})$ for arbitrarily small $ε>0$. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains $O(n^{-1+ε})$ if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering.
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