The Annals of Applied Probability· 2026Q1
Adaptive multilevel stochastic approximation of the value-at-risk
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- 2026year
Short summary
A new adaptive multilevel stochastic approximation algorithm achieves O(ε−2|lnε|52) complexity for computing financial Value-at-Risk, improving upon the O(ε−52) of previous methods by adaptively selecting inner samples.
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Key points
- Introduces an adaptive multilevel stochastic approximation algorithm for Value-at-Risk (VaR) computation.
- Achieves a best-case complexity of O(ε−2|lnε|52), an improvement over the O(ε−52) of previous methods.
- The adaptation involves dynamically selecting the number of inner samples at each level.
- Addresses the challenge posed by the discontinuous Heaviside function in gradient estimation.
- Theoretical analysis is supported by numerical experiments.
AI-generated from the title and abstract; the full text is not read.
Abstract
Crépey, Frikha and Louzi (Finance Stoch. (2025) 1015–1074) 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−52), ε>0 being a prescribed accuracy, which is suboptimal compared to the canonical multilevel Monte Carlo performance. This suboptimality stems from the discontinuity of the 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|52). Our theoretical analysis is exemplified through numerical experiments.
The authors' abstract, as published at the source. The Annals of Applied Probability, 2026 · DOI ↗
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Field: Management Science and Operations Research
Management Science and Operations ResearchDecision Sciences