## Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall

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Zhijian He and Xiaoqun Wang
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## Abstract:

Quantiles and expected shortfalls are usually used to measure risks of stochastic systems, which are often estimated by Monte Carlo methods. This paper focuses on the use of the quasi-Monte Carlo (QMC) method, whose convergence rate is asymptotically better than Monte Carlo in the numerical integration. We first prove the convergence of QMC-based quantile estimates under very mild conditions, and then establish a deterministic error bound of $O(N^{-1/d})$ for the quantile estimates, where $d$ is the dimension of the QMC point sets used in the simulation and $N$ is the sample size. Under certain conditions, we show that the mean squared error (MSE) of the randomized QMC estimate for expected shortfall is $o(N^{-1})$. Moreover, under stronger conditions the MSE can be improved to $O(N^{-1-1/(2d-1)+\epsilon })$ for arbitrarily small $\epsilon >0$.## References

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## Additional Information

**Zhijian He**- Affiliation: School of Mathematics, South China University of Technology, Guangzhou 510641, China
- MR Author ID: 1056493
- Email: hezhijian@scut.edu.cn
**Xiaoqun Wang**- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
- Email: wangxiaoqun@mail.tsinghua.edu.cn
- Received by editor(s): May 21, 2019
- Received by editor(s) in revised form: January 5, 2020, and April 18, 2020
- Published electronically: July 20, 2020
- Additional Notes: This work was supported by the National Science Foundation of China (No. 71601189), the Fundamental Research Funds for the Central Universities (No. 2019MS106), the Research Funds of South China University of Technology (No. D6191160), and the National Key R&D Program of China (No. 2016QY02D0301).
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 303-319 - MSC (2010): Primary 65D30, 65C05
- DOI: https://doi.org/10.1090/mcom/3555
- MathSciNet review: 4166462