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

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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$.

Mathematics of Computation
Zhijian He
Zhijian He
Associate Professor

My research interests include Monte Carlo and quasi-Monte methods with applications in finance and statistics.