Adaptive Multilevel Stochastic Approximation of the Value-at-Risk

Approximation stochastique adaptative à plusieurs niveaux de la valeur à risque

Stéphane Crépey (), Noufel Frikha (), Azar Louzi () and Jonathan Spence ()
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Stéphane Crépey: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité
Noufel Frikha: CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique
Azar Louzi: LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité
Jonathan Spence: Maxwell Institute for Mathematical Sciences, School of Mathematics - University of Edinburgh - The University of Edinburgh

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Abstract: 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.

Keywords: stochastic approximation; value-at-risk; nested Monte Carlo; multilevel Monte Carlo; adaptive Monte Carlo (search for similar items in EconPapers)
Date: 2024-08-13
Note: View the original document on HAL open archive server: https://hal.science/hal-04670735v1
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