Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall

TL;DR

This paper establishes central limit theorems for the errors of multilevel stochastic approximation algorithms used to compute financial risk measures, supported by numerical validation.

Contribution

It provides the first theoretical analysis of the asymptotic distribution of errors for these algorithms, enhancing understanding of their statistical properties.

Findings

Central limit theorems are proven for the algorithms' estimation errors.

Numerical examples confirm the theoretical results.

The analysis applies to both original and averaged estimators.

Abstract

Cr\'epey, Frikha, and Louzi (2025) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions. Our findings are substantiated through a numerical example.