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


Author: Zhijian He
Journal: Math. Comp. 87 (2018), 2857-2870
MSC (2010): Primary 65D30, 65C05
DOI: https://doi.org/10.1090/mcom/3324
Published electronically: February 19, 2018
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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.


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Additional Information

Zhijian He
Affiliation: School of Mathematics, South China University of Technology, Guangzhou 510641, China
Email: hezhijian87@gmail.com

DOI: https://doi.org/10.1090/mcom/3324
Keywords: Quasi-Monte Carlo methods, singularities, discontinuities
Received by editor(s): February 10, 2017
Received by editor(s) in revised form: June 25, 2017
Published electronically: February 19, 2018
Additional Notes: This work was supported by the National Science Foundation of China under grant $71601189$.
Article copyright: © Copyright 2018 American Mathematical Society

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