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

Author

Listed:
  • 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)

  • Gilles Pagès

    (LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation - SU - Sorbonne Université - CNRS - Centre National de la Recherche Scientifique - UPCité - Université Paris Cité)

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.

Suggested Citation

  • Stéphane Crépey & Noufel Frikha & Azar Louzi & Gilles Pagès, 2023. "Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall," Post-Print hal-04304985, HAL.
    • Handle: RePEc:hal:journl:hal-04304985
    DOI: 10.1214/24-EJP1246
    Note: View the original document on HAL open archive server: https://hal.science/hal-04304985v3
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    References listed on IDEAS

    as
    1. Aharon Ben‐Tal & Marc Teboulle, 2007. "An Old‐New Concept Of Convex Risk Measures: The Optimized Certainty Equivalent," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 449-476, July.
    2. Stéphane Crépey & Noufel Frikha & Azar Louzi, 2024. "A Multilevel Stochastic Approximation Algorithm for Value-at-Risk and Expected Shortfall Estimation," Working Papers hal-04037328, HAL.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    value-at-risk; expected shortfall; stochastic approximation; multilevel Monte Carlo; Polyak-Ruppert averaging; central limit theorem;
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