Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Stéphane Crépey, Noufel Frikha, Azar Louzi, Gilles Pagès

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.

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

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.
Paper Structure (24 sections, 16 theorems, 298 equations, 1 figure, 4 tables)

This paper contains 24 sections, 16 theorems, 298 equations, 1 figure, 4 tables.

Table of Contents

  1. Nested Stochastic Approximation Algorithm
  2. Convergence Rate Analysis
  3. Complexity Analysis
  4. Averaged Nested Stochastic Approximation Algorithm
  5. Convergence Rate Analysis
  6. Complexity Analysis
  7. Multilevel Stochastic Approximation Algorithm
  8. Convergence Rate Analysis
  9. Complexity Analysis
  10. Averaged Multilevel Stochastic Approximation Algorithm
  11. Convergence Rate Analysis
  12. Complexity Analysis
  13. Financial Case Study
  14. Analytical and Simulation Formulas
  15. Numerical Results
  16. ...and 9 more sections

Key Result

theorem 1

Suppose that Assumptions asp:misc and asp:supE[sup] hold, and that $\mathbb{E}[|\varphi(Y, Z)|^{2+\delta}]<\infty$ for some $\delta>0$. If $\gamma_n=\gamma_1n^{-\beta}$, $n\geq1$, $\beta\in(\frac{1}{2},1]$, with $\lambda\gamma_1>1$ if $\beta=1$, then where

Figures (1)

  • Figure 1: Joint distributions of the renormalized VaR and ES estimation errors.

Theorems & Definitions (42)

  • remark 1
  • theorem 1
  • proof
  • corollary 1
  • proof
  • remark 2
  • remark 3
  • theorem 2
  • proof
  • remark 4
  • ...and 32 more