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Moments of renewal shot-noise processes and their applications

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  • 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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    References listed on IDEAS

    as
    1. Willmot, Gordon E., 2007. "On the discounted penalty function in the renewal risk model with general interclaim times," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 17-31, July.
    2. Jang, Ji-Wook & Krvavych, Yuriy, 2004. "Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 97-111, August.
    3. Willmot, Gordon E. & Dickson, David C. M., 2003. "The Gerber-Shiu discounted penalty function in the stationary renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 403-411, July.
    4. Hans Gerber & Elias Shiu, 1998. "On the Time Value of Ruin," North American Actuarial Journal, Taylor & Francis Journals, vol. 2(1), pages 48-72.
    5. Hans Gerber & Elias Shiu, 2005. "The Time Value of Ruin in a Sparre Andersen Model," North American Actuarial Journal, Taylor & Francis Journals, vol. 9(2), pages 49-69.
    6. 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.
    7. Shang, Zhaoning & Goovaerts, Marc & Dhaene, Jan, 2011. "A recursive approach to mortality-linked derivative pricing," Insurance: Mathematics and Economics, Elsevier, vol. 49(2), pages 240-248, September.
    8. Dassios, Angelos & Zhao, Hongbiao, 2012. "Ruin by dynamic contagion claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 93-106.
    9. Joseph Abate & Ward Whitt, 2006. "A Unified Framework for Numerically Inverting Laplace Transforms," INFORMS Journal on Computing, INFORMS, vol. 18(4), pages 408-421, November.
    10. Zhu, Lingjiong, 2013. "Ruin probabilities for risk processes with non-stationary arrivals and subexponential claims," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 544-550.
    11. Vadim Linetsky, 2004. "The Spectral Decomposition Of The Option Value," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 7(03), pages 337-384.
    12. Ji‐Wook Jang, 2004. "Martingale Approach for Moments of Discounted Aggregate Claims," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 71(2), pages 201-211, June.
    13. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    14. Seal, Hilary L., 1983. "The poisson process: Its failure in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 2(4), pages 287-288, October.
    15. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    16. 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.
    17. Leveille, Ghislain & Garrido, Jose, 2001. "Moments of compound renewal sums with discounted claims," Insurance: Mathematics and Economics, Elsevier, vol. 28(2), pages 217-231, April.
    18. Woo, Jae-Kyung, 2010. "Some Remarks on Delayed Renewal Risk Models," ASTIN Bulletin, Cambridge University Press, vol. 40(1), pages 199-219, May.
    19. Thorsten Schmidt, 2014. "Catastrophe Insurance Modeled by Shot-Noise Processes," Risks, MDPI, vol. 2(1), pages 1-22, February.
    20. 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.
    21. Angelos Dassios & Hongbiao Zhao, 2014. "A Markov Chain Model for Contagion," Risks, MDPI, vol. 2(4), pages 1-22, November.
    22. Macci, Claudio & Torrisi, Giovanni Luca, 2011. "Risk processes with shot noise Cox claim number process and reserve dependent premium rate," Insurance: Mathematics and Economics, Elsevier, vol. 48(1), pages 134-145, January.
    23. Dassios, Angelos & Zhao, Hongbiao, 2014. "A Markov chain model for contagion," LSE Research Online Documents on Economics 60155, London School of Economics and Political Science, LSE Library.
    24. Torrisi, G. L., 2004. "Simulating the ruin probability of risk processes with delay in claim settlement," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 225-244, August.
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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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