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Pré-Publication, Document De Travail Année : 2024

Adaptive Multilevel Stochastic Approximation of the Value-at-Risk

Approximation stochastique adaptative à plusieurs niveaux de la valeur à risque

Résumé

Crépey, Frikha, and Louzi (2023) introduced a multilevel stochastic approximation scheme to compute the value-at-risk of a financial loss that is only simulatable by Monte Carlo. The optimal complexity of the scheme is in $O(\varepsilon^{-5/2})$, $\varepsilon>0$ being a prescribed accuracy, which is suboptimal when 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 optimal complexity is in $O(\varepsilon^{-2}|\ln{\varepsilon}|^{5/2})$. Our theoretical analysis is exemplified through numerical experiments.
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Dates et versions

hal-04670735 , version 1 (13-08-2024)

Identifiants

  • HAL Id : hal-04670735 , version 1

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Stéphane Crépey, Noufel Frikha, Azar Louzi, Jonathan Spence. Adaptive Multilevel Stochastic Approximation of the Value-at-Risk. 2024. ⟨hal-04670735⟩
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