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Moments of renewal shot-noise processes and their applications

Author

Listed:
  • Jang, Jiwook
  • Dassios, Angelos
  • Zhao, Hongbiao
Abstract
In this paper, we study the family of renewal shot-noise processes. The Feynmann–Kac formula is obtained based on the piecewise deterministic Markov process theory and the martingale methodology. We then derive the Laplace transforms of the conditional moments and asymptotic moments of the processes. In general, by inverting the Laplace transforms, the asymptotic moments and the first conditional moments can be derived explicitly; however, other conditional moments may need to be estimated numerically. As an example, we develop a very efficient and general algorithm of Monte Carlo exact simulation for estimating the second conditional moments. The results can be then easily transformed to the counterparts of discounted aggregate claims for insurance applications, and we apply the first two conditional moments for the actuarial net premium calculation. Similarly, they can also be applied to credit risk and reliability modelling. Numerical examples with four distribution choices for interarrival times are provided to illustrate how the models can be implemented.

Suggested Citation

  • 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.
    • Handle: RePEc:ehl:lserod:87428
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    File URL: http://eprints.lse.ac.uk/87428/
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    References listed on IDEAS

    as
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    Cited by:

    1. Lautier, Jackson P. & Pozdnyakov, Vladimir & Yan, Jun, 2023. "Pricing time-to-event contingent cash flows: A discrete-time survival analysis approach," Insurance: Mathematics and Economics, Elsevier, vol. 110(C), pages 53-71.
    2. Fouad Marri & Franck Adékambi & Khouzeima Moutanabbir, 2018. "Moments of Compound Renewal Sums with Dependent Risks Using Mixing Exponential Models," Risks, MDPI, vol. 6(3), pages 1-17, August.
    3. Hainaut, Donatien, 2021. "Moment generating function of non-Markov self-excited claims processes," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 406-424.
    4. 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).
    5. Jang, Jiwook & Qu, Yan & Zhao, Hongbiao & Dassios, Angelos, 2023. "A Cox model for gradually disappearing events," LSE Research Online Documents on Economics 112754, London School of Economics and Political Science, LSE Library.

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    More about this item

    Keywords

    renewal shot-noise processes; discounted aggregate claims; actuarial net premium; piecewise-deterministic Markov processes; martingale method; Monte Carlo exact simulation; credit risk; reliability;
    All these keywords.

    JEL classification:

    • G32 - Financial Economics - - Corporate Finance and Governance - - - Financing Policy; Financial Risk and Risk Management; Capital and Ownership Structure; Value of Firms; Goodwill
    • F3 - International Economics - - International Finance
    • G3 - Financial Economics - - Corporate Finance and Governance

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