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

## TL;DR

This paper analyzes the convergence properties of quasi-Monte Carlo methods for estimating quantiles and expected shortfalls, providing theoretical error bounds and demonstrating improved convergence rates over traditional Monte Carlo techniques.

## Contribution

It establishes convergence proofs and error bounds for QMC-based quantile and expected shortfall estimates, advancing the theoretical understanding of QMC in risk measurement.

## Key findings

- QMC quantile estimates converge under mild conditions.
- Deterministic error bound of $O(N^{-1/d})$ for QMC quantiles.
- MSE of randomized QMC for expected shortfall can be $o(N^{-1})$ or better under certain conditions.

## 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 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 $\epsilon>0$.