Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube
Provides theoretical error bounds for QMC in financial derivative pricing, where discontinuities and singularities are common.
This paper improves the error rate for randomized quasi-Monte Carlo integration of discontinuous integrands with boundary singularities from o(n^{-1/2}) to O(n^{-1/2-1/(4d-2)+ε}), with better rates when discontinuities are axis-aligned, and O(n^{-1+ε}) for QMC-friendly discontinuities.
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.