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

ArXiv ID: 2311.15333 “View on arXiv”

Authors: Unknown

Abstract

Crépey, Frikha, and Louzi (2023) 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.

Keywords: stochastic approximation, value-at-risk, expected shortfall, central limit theorem, multilevel acceleration, Financial Derivatives

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced stochastic approximation theory, multilevel Monte Carlo acceleration, and central limit theorems, while the empirical validation is limited to a single numerical example without code, backtests, or datasets.

  flowchart TD
    A["Research Goal<br>Asymptotic Error Analysis<br>of Multilevel SA for VaR & ES"] --> B["Key Methodology<br>Nested Stochastic Approximation<br>Algorithm & Multilevel Acceleration"]
    B --> C["Data/Input<br>Random Financial Loss<br>Distributions"]
    C --> D["Computational Process<br>Renormalized Estimation Errors<br>Central Limit Theorems"]
    D --> E["Key Findings<br>CLTs for Standard & Multilevel SA<br>Validated by Numerical Example"]