Monte Carlo and Quasi-Monte Carlo Methods 2010 pp 685-694 | Cite as

# High-Discrepancy Sequences for High-Dimensional Numerical Integration

## Abstract

In this paper, we consider a sequence of points in [0, 1]^{ d }, which are distributed only on the diagonal line between \((0,\ldots ,0)\) and \((1,\ldots ,1)\). The sequence is constructed based on a one-dimensional low-discrepancy sequence. We apply such sequences to *d*-dimensional numerical integration for two classes of integrals. The first class includes isotropic integrals. Under a certain condition, we prove that the integration error for this class is \(O(\sqrt{\log N}/N)\), where *N* is the number of points. The second class is called as Kolmogorov superposition integrals for which, under a certain condition, we prove that the integration error for this class is \(O((\log N)/N)\).

## Keywords

Option Price Diagonal Line Software Implementation Normal Distribution Function Risk Free Rate## Notes

### Acknowledgements

The author thanks the anonymous referees for their valuable comments. This research was supported by KAKENHI(22540141).

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