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Multivariate claim processes with rough intensities: Properties and estimation

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  • Hainaut, Donatien

Abstract

A Rough process shares most of features of fractional Brownian motion with a small Hurst index and its sample paths exhibit a high ruggedness compared to those of a Brownian motion. This article studies a multivariate claim process in which the instantaneous probability of claim occurrences has a rough dynamic. In this setting, the claim arrival intensities have an infinite quadratic variation and are not semi-martingales. Nevertheless, the joint moment generating function of claim processes and the integral of claim arrival intensities admits a representation in terms of solutions of fractional differential equations. A numerical procedure is next proposed to filter the most likely sample path of rough intensities from time-series of claims. To illustrate this work, we estimate one- and two-dimensional rough models to time-series of cyber-attacks targeting medical and other services in the US from 2014 to 2018.

Suggested Citation

  • Hainaut, Donatien, 2022. "Multivariate claim processes with rough intensities: Properties and estimation," Insurance: Mathematics and Economics, Elsevier, vol. 107(C), pages 269-287.
    • Handle: RePEc:eee:insuma:v:107:y:2022:i:c:p:269-287
    DOI: 10.1016/j.insmatheco.2022.08.010
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    1. Dupret, Jean-Loup & Hainaut, Donatien, 2021. "Portfolio insurance under rough volatility and Volterra processes," LIDAM Discussion Papers ISBA 2021026, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. Farkas, Sébastien & Lopez, Olivier & Thomas, Maud, 2021. "Cyber claim analysis using Generalized Pareto regression trees with applications to insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 92-105.
    3. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    4. Kumar, A. & Nane, Erkan & Vellaisamy, P., 2011. "Time-changed Poisson processes," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1899-1910.
    5. Cheung, Eric C.K. & Rabehasaina, Landy & Woo, Jae-Kyung & Xu, Ran, 2019. "Asymptotic correlation structure of discounted Incurred But Not Reported claims under fractional Poisson arrival process," European Journal of Operational Research, Elsevier, vol. 276(2), pages 582-601.
    6. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 406-424.
    7. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    8. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," LIDAM Reprints ISBA 2021014, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Eduardo Abi Jaber & Omar El Euch, 2019. "Multi-factor approximation of rough volatility models," Post-Print hal-01697117, HAL.
    10. Dupret, Jean-Loup & Hainaut, Donatien, 2021. "Portfolio insurance under rough volatility and Volterra processes," LIDAM Reprints ISBA 2021051, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    11. Corina D. Constantinescu & Jorge M. Ramirez & Wei R. Zhu, 2019. "An application of fractional differential equations to risk theory," Finance and Stochastics, Springer, vol. 23(4), pages 1001-1024, October.
    12. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 120-132.
    13. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," LIDAM Reprints ISBA 2021046, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    14. Jumarie, Guy, 2005. "Merton's model of optimal portfolio in a Black-Scholes Market driven by a fractional Brownian motion with short-range dependence," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 585-598, December.
    15. Omar El Euch & Mathieu Rosenbaum, 2019. "The characteristic function of rough Heston models," Mathematical Finance, Wiley Blackwell, vol. 29(1), pages 3-38, January.
    16. Jim Gatheral & Thibault Jaisson & Mathieu Rosenbaum, 2018. "Volatility is rough," Quantitative Finance, Taylor & Francis Journals, vol. 18(6), pages 933-949, June.
    17. Beghin, Luisa & Macci, Claudio, 2017. "Asymptotic results for a multivariate version of the alternative fractional Poisson process," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 260-268.
    18. Christian Bayer & Peter Friz & Jim Gatheral, 2016. "Pricing under rough volatility," Quantitative Finance, Taylor & Francis Journals, vol. 16(6), pages 887-904, June.
    19. Hainaut, Donatien, 2021. "A fractional multi-states model for insurance," LIDAM Discussion Papers ISBA 2021019, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    20. Dupret, Jean-Loup & Barbarin, Jérôme & Hainaut , Donatien, 2022. "Impact of rough stochastic volatility models on long-term life insurance pricing," LIDAM Reprints ISBA 2022022, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    21. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," LIDAM Discussion Papers ISBA 2021028, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
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    More about this item

    Keywords

    Fractional Brownian motion; Rough volatility; Cox process; Compound Poisson process; Cyber-risk;
    All these keywords.

    JEL classification:

    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies

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