Adaptive Multilevel Stochastic Approximation of the Value-at-Risk

2024/08/12 by Stéphane Crépey, Noufel Frikha, Crépey, Stéphane +5 · 2 citations
Decision Sciences · #62G32 #62L20 #65C05 #91Gxx #Computational Finance (q-fin.CP) #FOS: Economics and business #FOS: Mathematics #Probability (math.PR) #Risk Management (q-fin.RM) #Risk and Portfolio Optimization

paper · pdf · doi:10.48550/arxiv.2408.06531

openalex publication_date 2024/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Crépey, Frikha, and Louzi (2025) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The best complexity of the scheme is in O(ε^-\frac52), ε>0 being a prescribed accuracy, which is suboptimal compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity ofthe Heaviside function involved in the biased stochastic gradient that is recursively evaluated to derive the value-at-risk. To mitigate this issue, this paper proposes and analyzes a multilevel stochastic approximation algorithm that adaptively selects the number of inner samples at each level, and proves that its best complexity is in O(ε^-2|lnε|^\frac52). Our theoretical analysis is exemplified through numerical experiments.

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• Optimized Multi-Level Monte Carlo Parametrization and Antithetic Sampling for Nested Simulations
• Mission-Aligned Learning-Informed Control of Autonomous Systems: Formulation and Foundations

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