Quasi-Monte Carlo for discontinuous integrands with singularities along the boundary of the unit cube

Abstract:

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 $O(n^{-1/2-1/(4d-2)+\epsilon })$ for arbitrarily small $\epsilon >0$. 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 $O(n^{-1+\epsilon })$ 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.