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NEW RESULTS ON THE DISTRIBUTION OF DISCOUNTED COMPOUND POISSON SUMS

Published online by Cambridge University Press:  03 December 2018

Zhehao Zhang
Zhehao Zhang*
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, 111 Barry Street, Victoria 3010, Australia E-Mail: zhehaoz1@student.unimelb.edu.au
*
E-Mail: zhehaoz1@student.unimelb.edu.au
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Abstract

This paper focuses on the distribution of Poisson sums of discounted claims over a finite or infinite time period. It gives two new results when claim amounts follow Mittag-Leffler distributions and two new results when claim amounts follow gamma distributions. Further, as Mittag-Leffler distribution is of heavy-tailed nature and its moments only exist for order strictly smaller than one, this distribution can be used for modelling insurance whose claim amounts are extremely large, that is, catastrophe insurance.

Keywords

Poisson risk processdiscounted aggregate claimsMittag-Leffler distributionheavy tailedcatastrophe insurance

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 1 , January 2019 , pp. 169 - 187
Copyright
Copyright © Astin Bulletin 2018 

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References

Bateman, H. and Erdélyi, A. (1953) Higher Transcendental Functions, Vol. 3. New York: McGraw-Hill, 1953–1955.Google Scholar
Cahoy, D.O. (2013) Estimation of Mittag-Leffler parameters. Communications in Statistics-Simulation and Computation, 42(2), 303–315.CrossRefGoogle Scholar
Cruz, M.G., Peters, G.W. and Shevchenko, P.V. (2014) Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk. New Jersey: John Wiley & Sons.Google Scholar
Dufresne, D. (1990) The distribution of a perpetuity, with applications to risk theory and pension funding. Scandinavian Actuarial Journal, 1990(1), 39–79.CrossRefGoogle Scholar
Dufresne, D. (1996) On the stochastic equation l(x) = l[b(x + c)] and a property of gamma distributions. Bernoulli, 2(3), 287–291.Google Scholar
Dufresne, D. (1998) Algebraic properties of beta and gamma distributions, and applications. Advances in Applied Mathematics, 20(3), 285–299.CrossRefGoogle Scholar
Dufresne, D. and Zhang, Z. (2017) Discounted sums with renewal times. Manuscript submitted for publication.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (2013) Modelling Extremal Events: For Insurance and Finance, Vol. 33. New York: Springer Science & Business Media.Google Scholar
Garrido, J. and Léveillé, G. (2004) Impact of inflation and interest on aggregate claims. Wiley StatsRef: Statistics Reference Online.CrossRefGoogle Scholar
Gerber, H.U. (1979) An Introduction to Mathematical Risk Theory. S.S. Huebner Foundation for Insurance Education, Wharton School, University of Pennsylvania; Distributed by R.D. Irwin, Philadelphia; Homewood, Ill.Google Scholar
Gjessing, H.K. and Paulsen, J. (1997) Present value distributions with applications to ruin theory and stochastic equations. Stochastic Processes and Their Applications, 71(1), 123–144.CrossRefGoogle Scholar
Harrison, J.M. (1977) Ruin problems with compounding assets. Stochastic Processes and Their Applications, 5(1), 67–79.CrossRefGoogle Scholar
Jose, K.K. and Abraham, B. (2011) A count model based on Mittag-Leffler interarrival times. Statistica, 71(4), 501.Google Scholar
Jose, K.K., Uma, P., Lekshmi, V.S. and Haubold, H.J. (2010) Generalized Mittag-Leffler distributions and processes for applications in astrophysics and time series modeling. In Proceedings of the Third UN/ESA/NASA Workshop on the International Heliophysical Year 2007 and Basic Space Science, pp. 79–92. Berlin: Springer.CrossRefGoogle Scholar
Léveillé, G. and Garrido, J. (2001a) Moments of compound renewal sums with discounted claims. Insurance: Mathematics and Economics, 28(2), 217–231.Google Scholar
Léveillé, G. and Garrido, J. (2001b) Recursive moments of compound renewal sums with discounted claims. Scandinavian Actuarial Journal, 2001(2), 98–110.CrossRefGoogle Scholar
Léveillé, G., Garrido, J. and Wang, Y.F. (2010) Moment generating functions of compound renewal sums with discounted claims. Scandinavian Actuarial Journal, 2010(3), 165–184.CrossRefGoogle Scholar
Lin, G.D. (1998) On the Mittag–Leffler distributions. Journal of Statistical Planning and Inference, 74(1), 1–9.CrossRefGoogle Scholar
Lukacs, E. (1955) A characterization of the gamma distribution. The Annals of Mathematical Statistics, 26(2), 319–324.CrossRefGoogle Scholar
Nilsen, T. and Paulsen, J. (1996) On the distribution of a randomly discounted compound poisson process. Stochastic processes and Their applications, 61(2), 305–310.CrossRefGoogle Scholar
Paulsen, J. (1993) Risk theory in a stochastic economic environment. Stochastic processes and Their applications, 46(2), 327–361.CrossRefGoogle Scholar
Pillai, R. (1990) On Mittag-Leffler functions and related distributions. Annals of the Institute of statistical Mathematics, 42(1), 157–161.CrossRefGoogle Scholar
Sanders, D. (2005) The modelling of extreme events. British Actuarial Journal, 11(3), 519–557.CrossRefGoogle Scholar
Takács, L. (1954) On secondary processes generated by a poisson process and their applications in physics. Acta Mathematica Hungarica, 5(3–4), 203–236.Google Scholar
Tang, Q. (2005) The finite-time ruin probability of the compound poisson model with constant interest force. Journal of Applied Probability, 42(3), 608–619.CrossRefGoogle Scholar
Vervaat, W. (1979) On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Advances in Applied Probability, 11(4), 750–783.CrossRefGoogle Scholar
Wang, Y.F. (2010) The distribution of the discounted compound PH-renewal process. Ph.D. Thesis, Concordia University.Google Scholar
Wang, Y.F., Garrido, J. and Léveillé, G. (2018). The distribution of discounted compound PH-renewal processes. Methodology and Computing in Applied Probability, 20(1), 1–28.CrossRefGoogle Scholar