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Differentiation of some functionals of risk processes

Author

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  • Stéphane Loisel

    (SAF - Laboratoire de Sciences Actuarielle et Financière - UCBL - Université Claude Bernard Lyon 1 - Université de Lyon)

Abstract

For general risk processes, the expected time-integrated negative part of the process on a fixed time interval is introduced and studied. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below zero studied by Dos Reis (1993) and the probability of ruin. Differentiation of other functionals of unidimensional and multidimensional risk processes with respect to the initial reserve level are carried out. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve which minimizes one of these risk measures, illustrate the variety of application fields and the benefits deriving from an efficient and effective use of such tools.

Suggested Citation

  • Stéphane Loisel, 2005. "Differentiation of some functionals of risk processes," Post-Print hal-00157739, HAL.
    • Handle: RePEc:hal:journl:hal-00157739
    DOI: 10.1239/jap/1118777177
    Note: View the original document on HAL open archive server: https://hal.science/hal-00157739v2
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    Citations

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    Cited by:

    1. Loisel, Stéphane & Trufin, Julien, 2014. "Properties of a risk measure derived from the expected area in red," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 191-199.
    2. Peggy Cénac & Stéphane Loisel & Véronique Maume-Deschamps & Clémentine Prieur, 2014. "Risk indicators with several lines of business: comparison, asymptotic behavior and applications to optimal reserve allocation," Post-Print hal-00816894, HAL.
    3. Romain Biard & Christophette Blanchet-Scalliet & Anne Eyraud-Loisel & Stéphane Loisel, 2013. "Impact of Climate Change on Heat Wave Risk," Risks, MDPI, vol. 1(3), pages 1-16, December.
    4. Lkabous, Mohamed Amine & Wang, Zijia, 2023. "On the area in the red of Lévy risk processes and related quantities," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 257-278.
    5. Florin Avram & Sooie-Hoe Loke, 2018. "On Central Branch/Reinsurance Risk Networks: Exact Results and Heuristics," Risks, MDPI, vol. 6(2), pages 1-18, April.
    6. Denuit, Michel & Robert, Christian Y., 2022. "Dynamic conditional mean risk sharing in the compound Poisson surplus model," LIDAM Discussion Papers ISBA 2022034, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    7. repec:hal:wpaper:hal-00870224 is not listed on IDEAS
    8. repec:hal:wpaper:hal-00816894 is not listed on IDEAS
    9. Delsing, G.A. & Mandjes, M.R.H. & Spreij, P.J.C. & Winands, E.M.M., 2019. "An optimization approach to adaptive multi-dimensional capital management," Insurance: Mathematics and Economics, Elsevier, vol. 84(C), pages 87-97.
    10. Mohamed Amine Lkabous & Jean-François Renaud, 2018. "A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model," Risks, MDPI, vol. 6(3), pages 1-11, August.
    11. Liu, Jingchen & Woo, Jae-Kyung, 2014. "Asymptotic analysis of risk quantities conditional on ruin for multidimensional heavy-tailed random walks," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 1-9.
    12. Esther Frostig & Adva Keren–Pinhasik, 2017. "Parisian ruin in the dual model with applications to the G/M/1 queue," Queueing Systems: Theory and Applications, Springer, vol. 86(3), pages 261-275, August.
    13. G. A. Delsing & M. R. H. Mandjes & P. J. C. Spreij & E. M. M. Winands, 2018. "An optimization approach to adaptive multi-dimensional capital management," Papers 1812.08435, arXiv.org.
    14. Macci, Claudio, 2008. "Large deviations for the time-integrated negative parts of some processes," Statistics & Probability Letters, Elsevier, vol. 78(1), pages 75-83, January.
    15. Cénac P. & Maume-Deschamps V. & Prieur C., 2012. "Some multivariate risk indicators: Minimization by using a Kiefer–Wolfowitz approach to the mirror stochastic algorithm," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 47-72, March.
    16. Romain Biard & Stéphane Loisel & Claudio Macci & Noel Veraverbeke, 2010. "Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation," Post-Print hal-00372525, HAL.
    17. Romain Biard, 2013. "Asymptotic multivariate finite-time ruin probabilities with heavy-tailed claim amounts: Impact of dependence and optimal reserve allocation," Post-Print hal-00538571, HAL.
    18. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
    19. Julien Callant & Julien Trufin & Pierre Zuyderhoff, 2022. "Some Expressions of a Generalized Version of the Expected Time in the Red and the Expected Area in Red," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 595-611, June.
    20. Cossette, Hélène & Marceau, Etienne & Trufin, Julien & Zuyderhoff, Pierre, 2020. "Ruin-based risk measures in discrete-time risk models," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 246-261.
    21. Zhu, Jinxia & Yang, Hailiang, 2009. "On differentiability of ruin functions under Markov-modulated models," Stochastic Processes and their Applications, Elsevier, vol. 119(5), pages 1673-1695, May.

    More about this item

    Keywords

    Ruin theory; Sample path properties; Optimal allocation; Multidimensional risk process; Risk measures;
    All these keywords.

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