Convergence analysis of quasi-Monte Carlo sampling for quantile and expected shortfall
Provides theoretical guarantees for QMC-based risk measures, benefiting practitioners in finance and risk management who need efficient simulation methods.
This paper proves convergence of quasi-Monte Carlo (QMC) quantile estimates under mild conditions and establishes a deterministic error bound of O(N^{-1/d}) for quantile estimates. For expected shortfall, the mean squared error of randomized QMC is shown to be o(N^{-1}) and can be improved to O(N^{-1-1/(2d-1)+ε}) under stronger conditions.
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 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 $O(N^{-1-1/(2d-1)+ε})$ for arbitrarily small $ε>0$.