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An expansion formula for Hawkes processes and application to cyber-insurance derivatives

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  • Hillairet, Caroline
  • Réveillac, Anthony
  • Rosenbaum, Mathieu

Abstract

In this paper we provide an expansion formula for Hawkes processes which involves the addition of jumps at deterministic times to the Hawkes process in the spirit of the well-known integration by parts formula (or more precisely the Mecke formula) for Poisson functional. Our approach allows us to provide an expansion of the premium of a class of cyber insurance derivatives (such as reinsurance contracts including generalized Stop-Loss contracts) or risk management instruments (like Expected Shortfall) in terms of so-called shifted Hawkes processes. From the actuarial point of view, these processes can be seen as “stressed” scenarios. Our expansion formula for Hawkes processes enables us to provide lower and upper bounds on the premium (or the risk evaluation) of such cyber contracts and to quantify the surplus of premium compared to the standard modeling with a homogeneous Poisson process.

Suggested Citation

  • Hillairet, Caroline & Réveillac, Anthony & Rosenbaum, Mathieu, 2023. "An expansion formula for Hawkes processes and application to cyber-insurance derivatives," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 89-119.
    • Handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:89-119
    DOI: 10.1016/j.spa.2023.02.012
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    References listed on IDEAS

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    1. Gerber, Hans U., 1982. "On the numerical evaluation of the distribution of aggregate claims and its stop-loss premiums," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 13-18, January.
    2. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    3. Adrian Baldwin & Iffat Gheyas & Christos Ioannidis & David Pym & Julian Williams, 2017. "Contagion in cyber security attacks," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(7), pages 780-791, July.
    4. Dassios, Angelos & Zhao, Hongbiao, 2012. "Ruin by dynamic contagion claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 93-106.
    5. Chen Peng & Maochao Xu & Shouhuai Xu & Taizhong Hu, 2017. "Modeling and predicting extreme cyber attack rates via marked point processes," Journal of Applied Statistics, Taylor & Francis Journals, vol. 44(14), pages 2534-2563, October.
    6. Gabriele Stabile & Giovanni Luca Torrisi, 2010. "Risk Processes with Non-stationary Hawkes Claims Arrivals," Methodology and Computing in Applied Probability, Springer, vol. 12(3), pages 415-429, September.
    7. Reijnen, Rajko & Albers, Willem & Kallenberg, Wilbert C.M., 2005. "Approximations for stop-loss reinsurance premiums," Insurance: Mathematics and Economics, Elsevier, vol. 36(3), pages 237-250, June.
    8. Caroline Hillairet & Ying Jiao & Anthony Réveillac, 2018. "Pricing formulae for derivatives in insurance using the Malliavin calculus ," Post-Print hal-01561987, HAL.
    9. Jang, Jiwook & Dassios, Angelos, 2013. "A bivariate shot noise self-exciting process for insurance," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 524-532.
    10. Privault, Nicolas, 2021. "Recursive computation of the Hawkes cumulants," Statistics & Probability Letters, Elsevier, vol. 177(C).
    11. Albers, Willem, 1999. "Stop-loss premiums under dependence," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 173-185, May.
    12. Dassios, Angelos & Jang, Jiwook, 2003. "Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity," LSE Research Online Documents on Economics 2849, London School of Economics and Political Science, LSE Library.
    13. Dassios, Angelos & Zhao, Hongbiao, 2017. "A generalised contagion process with an application to credit risk," LSE Research Online Documents on Economics 68558, London School of Economics and Political Science, LSE Library.
    14. Angelos Dassios & Hongbiao Zhao, 2017. "A Generalized Contagion Process With An Application To Credit Risk," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-33, February.
    15. Xuefeng Gao & Xiang Zhou & Lingjiong Zhu, 2018. "Transform analysis for Hawkes processes with applications in dark pool trading," Quantitative Finance, Taylor & Francis Journals, vol. 18(2), pages 265-282, February.
    16. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
    17. de Lourdes Centeno, Maria, 2005. "Dependent risks and excess of loss reinsurance," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 229-238, October.
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