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Notes on discrete compound Poisson model with applications to risk theory

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  • Zhang, Huiming
  • Liu, Yunxiao
  • Li, Bo

Abstract

Probability generating function (p.g.f.) is a powerful tool to study discrete compound Poisson (DCP) distribution. By applying inverse Fourier transform of p.g.f., it is convenient to numerically calculate probability density and do parameter estimation. As an application to finance and insurance, we firstly show that in the generalized CreditRisk+ model, the default loss of each debtor and the total default of all debtors are both approximately equal to a DCP distribution, and we give Le Cam’s error bound between the total default and a DCP distribution. Next, we consider geometric Brownian motion with DCP jumps and derive its rth moment. We establish the surplus process of the difference of two DCP distributions, and numerically compute the tail probability. Furthermore, we define the discrete pseudo compound Poisson (DPCP) distribution and give the characterizations and examples of DPCP distribution, including the strictly decreasing discrete distribution and the zero-inflated discrete distribution with P(X=0)>0.5.

Suggested Citation

  • Zhang, Huiming & Liu, Yunxiao & Li, Bo, 2014. "Notes on discrete compound Poisson model with applications to risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 325-336.
    • Handle: RePEc:eee:insuma:v:59:y:2014:i:c:p:325-336
    DOI: 10.1016/j.insmatheco.2014.09.012
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    1. Rulliere, Didier & Loisel, Stephane, 2004. "Another look at the Picard-Lefevre formula for finite-time ruin probabilities," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 187-203, October.
    2. H. P. Galliher & Philip M. Morse & M. Simond, 1959. "Dynamics of Two Classes of Continuous-Review Inventory Systems," Operations Research, INFORMS, vol. 7(3), pages 362-384, June.
    3. Ole E. Barndorff-Nielsen & David G. Pollard & Neil Shephard, 2012. "Integer-valued L�vy processes and low latency financial econometrics," Quantitative Finance, Taylor & Francis Journals, vol. 12(4), pages 587-605, January.
    4. Dufresne, François & Gerber, Hans U. & Shiu, Elias S. W., 1991. "Risk Theory with the Gamma Process," ASTIN Bulletin, Cambridge University Press, vol. 21(2), pages 177-192, November.
    5. Gómez-Déniz, Emilio & Sarabia, José María & Calderín-Ojeda, Enrique, 2011. "A new discrete distribution with actuarial applications," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 406-412, May.
    6. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    7. Hossack,I. B. & Pollard,J. H. & Zehnwirth,B., 1999. "Introductory Statistics with Applications in General Insurance," Cambridge Books, Cambridge University Press, number 9780521652346.
    8. Chan, Beda, 1982. "Recursive formulas for discrete distributions," Insurance: Mathematics and Economics, Elsevier, vol. 1(4), pages 241-243, October.
    9. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, September.
    10. Hossack,I. B. & Pollard,J. H. & Zehnwirth,B., 1999. "Introductory Statistics with Applications in General Insurance," Cambridge Books, Cambridge University Press, number 9780521655347.
    11. Gerber, Hans U., 1984. "Error bounds for the compound poisson approximation," Insurance: Mathematics and Economics, Elsevier, vol. 3(3), pages 191-194, July.
    12. Chuancun Yin, 2013. "Optimal dividend problem for a generalized compound Poisson risk model," Papers 1305.1747, arXiv.org, revised Feb 2014.
    13. Dhaene, Jan & Pril, Nelson De, 1994. "On a class of approximative computation methods in the individual risk model," Insurance: Mathematics and Economics, Elsevier, vol. 14(2), pages 181-196, May.
    14. Ross,Sheldon M., 2011. "An Elementary Introduction to Mathematical Finance," Cambridge Books, Cambridge University Press, number 9780521192538.
    15. Panjer, Harry H., 1981. "Recursive Evaluation of a Family of Compound Distributions," ASTIN Bulletin, Cambridge University Press, vol. 12(1), pages 22-26, June.
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