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Medientyp:
E-Artikel
Titel:
Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall
Beteiligte:
He, Zhijian;
Wang, Xiaoqun
Erschienen:
American Mathematical Society (AMS), 2020
Erschienen in:
Mathematics of Computation, 90 (2020) 327, Seite 303-319
Sprache:
Englisch
DOI:
10.1090/mcom/3555
ISSN:
0025-5718;
1088-6842
Entstehung:
Anmerkungen:
Beschreibung:
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 ) O(N^{-1/d}) for the quantile estimates, where d d is the dimension of the QMC point sets used in the simulation and N 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 ) o(N^{-1}) . Moreover, under stronger conditions the MSE can be improved to O ( N − 1 − 1 / ( 2 d − 1 ) + ϵ ) O(N^{-1-1/(2d-1)+\epsilon }) for arbitrarily small ϵ > 0 \epsilon >0 .