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Catastrophe Insurance Modeled by Shot-Noise Processes

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  • Thorsten Schmidt

    (Chemnitz University of Technology, Reichenhainer Str. 41, Chemnitz 09126, Germany)

Abstract

Shot-noise processes generalize compound Poisson processes in the following way: a jump (the shot) is followed by a decline (noise). This constitutes a useful model for insurance claims in many circumstances; claims due to natural disasters or self-exciting processes exhibit similar features. We give a general account of shot-noise processes with time-inhomogeneous drivers inspired by recent results in credit risk. Moreover, we derive a number of useful results for modeling and pricing with shot-noise processes. Besides this, we obtain some highly tractable examples and constitute a useful modeling tool for dynamic claims processes. The results can in particular be used for pricing Catastrophe Bonds (CAT bonds), a traded risk-linked security. Additionally, current results regarding the estimation of shot-noise processes are reviewed.

Suggested Citation

  • Thorsten Schmidt, 2014. "Catastrophe Insurance Modeled by Shot-Noise Processes," Risks, MDPI, vol. 2(1), pages 1-22, February.
    • Handle: RePEc:gam:jrisks:v:2:y:2014:i:1:p:3-24:d:33264
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    References listed on IDEAS

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    1. Timo Altmann & Thorsten Schmidt & Winfried Stute, 2008. "A Shot Noise Model For Financial Assets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(01), pages 87-106.
    2. Herbertsson, Alexander & Jang, Jiwook & Schmidt, Thorsten, 2011. "Pricing basket default swaps in a tractable shot noise model," Statistics & Probability Letters, Elsevier, vol. 81(8), pages 1196-1207, August.
    3. repec:fth:geneec:99.01 is not listed on IDEAS
    4. Esche, Felix & Schweizer, Martin, 2005. "Minimal entropy preserves the Lévy property: how and why," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 299-327, February.
    5. Moreno, Manuel & Serrano, Pedro & Stute, Winfried, 2011. "Statistical properties and economic implications of jump-diffusion processes with shot-noise effects," European Journal of Operational Research, Elsevier, vol. 214(3), pages 656-664, November.
    6. Schmidt, Thorsten & Stute, Winfried, 2007. "Shot-noise processes and the minimal martingale measure," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1332-1338, July.
    7. 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.
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    Cited by:

    1. Claudia Ceci & Katia Colaneri & Alessandra Cretarola, 2021. "Optimal Reinsurance and Investment under Common Shock Dependence Between Financial and Actuarial Markets," Papers 2105.07524, arXiv.org.
    2. Liu, Wenyue & Cadenillas, Abel, 2023. "Optimal insurance contracts for a shot-noise Cox claim process and persistent insured's actions," Insurance: Mathematics and Economics, Elsevier, vol. 109(C), pages 69-93.
    3. Zied Chaieb & Djibril Gueye, 2022. "Pricing zero-coupon CAT bonds using the enlargement of ltration theory: a general framework ," Post-Print hal-03745077, HAL.
    4. Sukono & Hafizan Juahir & Riza Andrian Ibrahim & Moch Panji Agung Saputra & Yuyun Hidayat & Igif Gimin Prihanto, 2022. "Application of Compound Poisson Process in Pricing Catastrophe Bonds: A Systematic Literature Review," Mathematics, MDPI, vol. 10(15), pages 1-19, July.
    5. Durga, N. & Muthukumar, P., 2019. "Existence and exponential behavior of multi-valued nonlinear fractional stochastic integro-differential equations with Poisson jumps of Clarke’s subdifferential type," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 155(C), pages 347-359.
    6. Sarah Bensalem & Nicolás Hernández-Santibáñez & Nabil Kazi-Tani, 2023. "A continuous-time model of self-protection," Finance and Stochastics, Springer, vol. 27(2), pages 503-537, April.
    7. Riza Andrian Ibrahim & Sukono & Herlina Napitupulu, 2022. "Multiple-Trigger Catastrophe Bond Pricing Model and Its Simulation Using Numerical Methods," Mathematics, MDPI, vol. 10(9), pages 1-17, April.
    8. Yiqing Chen, 2019. "A Renewal Shot Noise Process with Subexponential Shot Marks," Risks, MDPI, vol. 7(2), pages 1-8, June.
    9. Krzysztof Burnecki & Mario Nicoló Giuricich, 2017. "Stable Weak Approximation at Work in Index-Linked Catastrophe Bond Pricing," Risks, MDPI, vol. 5(4), pages 1-19, December.
    10. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2015. "A risk model with renewal shot-noise Cox process," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 55-65.
    11. Jang, Jiwook & Dassios, Angelos & Zhao, Hongbiao, 2018. "Moments of renewal shot-noise processes and their applications," LSE Research Online Documents on Economics 87428, London School of Economics and Political Science, LSE Library.
    12. Dassios, Angelos & Jang, Jiwook & Zhao, Hongbiao, 2015. "A risk model with renewal shot-noise Cox process," LSE Research Online Documents on Economics 64051, London School of Economics and Political Science, LSE Library.
    13. Masahiko Egami & Rusudan Kevkhishvili, 2016. "An Analysis of Simultaneous Company Defaults Using a Shot Noise Process," Discussion papers e-16-001, Graduate School of Economics , Kyoto University.
    14. Zied Chaieb & Djibril Gueye, 2022. "Pricing zero-coupon CAT bonds using the enlargement of ltration theory: a general framework," Papers 2208.02609, arXiv.org.
    15. Avanzi, Benjamin & Wong, Bernard & Yang, Xinda, 2016. "A micro-level claim count model with overdispersion and reporting delays," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 1-14.

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