International Journal of Statistics and Actuarial Science

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Second-Order Asymptotic Expansion for the Ruin Probability of the Sparre Andersen Risk Process with Reinsurance and Stronger Semiexponential Claims

Received: Mar. 11, 2017    Accepted: Mar. 28, 2017    Published: Apr. 15, 2017
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Abstract

In this study the Sparre Andersen risk process with reinsurance is considered. The second-order asymptotic expansion for the ruin probability is obtained, when the claim sizes have the strongly semiexponential distribution. Moreover, numerical examples in cases proportional reinsurance and exsess stop loss reinsurance are provided.

DOI 10.11648/j.ijsas.20170102.12
Published in International Journal of Statistics and Actuarial Science ( Volume 1, Issue 2, May 2017 )
Page(s) 40-45
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Sparre Andersen Risk Process, Reinsurance, Ruin Probability, Second-Order Asymptotic Expansion, Semiexponential Distribution

References
[1] Albrecher H., Claramunt M. M., Mármol M. (2006). On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang (n) interclaim times. Insurance: Mathematics and Economics 37, pp. 324–334.
[2] Aleškeviciene A., Leipus R., Šiaulys J. (2009). Second-order asymptotics of ruin probabilities for semiexponential claims. Lithuanian Mathematical Journal, 49, 4, pp. 364–371.
[3] Aliyev R. T., Jafarova V. (2009). On the moments of the Sparre Andersen surplus process and it’s average value. 13th International Congress on Insurance: Mathematics and Economics, Istanbul, Turkey, p. 39.
[4] Asmussen S. (2000). Ruin Probabilities. World Scientific, 382 p.
[5] Baltrunas A. (1999). Second-order asymptotics for the ruin probability in the case of very large claims. Sib. Math. J., 40, pp. 1034–1043.
[6] Borovkov A. A. (2002). On subexponential distributions and asymptotics of the distribution of the maximum of sequential sums. Sib. Math. J., 43, pp. 995–1022.
[7] Dickson D. C., Waters H. R. (1996). Reinsurance and ruin. Insurance: Mathematics and Economics, 19, 1, pp. 61-80.
[8] Dickson D. C., Waters H. R. (1997). Relative reinsurance retention levels. ASTIN Bulletin, 27, 2, pp. 207–227. ASTIN BULLETIN, Vol. 27, No. 2. 1997, pp. 207-227.
[9] Dickson D. Proportional Reinsurance. Encyclopedia of Actuarial Science 2006.
[10] Dickson D. Insurance Risk and Ruin, Cambridge University Press, 2005, 229 p.
[11] Embrechts P., Veraverbeke N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Mathematics and Economics, 1, pp. 55–72.
[12] Foss S., Korshunov D., Zachary S. (2009). Convolutions of long-tailed and subexponential distributions, Preprint available at http://www.math.nsc.ru/LBRT/v1/foss
[13] Gerber H. U., Shiu E. W. (2005). The Time Value of Ruin in a Sparre Andersen Model. North American Actuarial Journal, 9 (2), pp. 49–84.
[14] Hald M., Schmidli H. (2004). On the maximisation of the adjustment coefficient under proportional reinsurance. ASTIN Bulletin, 34, pp. 75-83.
[15] Li S., Garrido J. (2004). On ruin for the Erlang (n) risk process, Insurance: Mathematics and Economics 34 (3), pp. 391–408.
[16] Li S., Dickson D. C. M. (2006). The maximum surplus before ruin in an Erlang (n) risk process and related problems, Insurance: Mathematics & Economics 38 (3), pp. 529–539.
[17] Luo S., Taksar M., Tsoi A. (2008). On reinsurance and investment for large insurance portfolios. Insurance: Mathematics and Economics, 42, 1, pp. 434–444.
[18] Schmidli H. (2002). On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 3, pp. 890-907.
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    Rovshan Aliyev. (2017). Second-Order Asymptotic Expansion for the Ruin Probability of the Sparre Andersen Risk Process with Reinsurance and Stronger Semiexponential Claims. International Journal of Statistics and Actuarial Science, 1(2), 40-45. https://doi.org/10.11648/j.ijsas.20170102.12

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    ACS Style

    Rovshan Aliyev. Second-Order Asymptotic Expansion for the Ruin Probability of the Sparre Andersen Risk Process with Reinsurance and Stronger Semiexponential Claims. Int. J. Stat. Actuarial Sci. 2017, 1(2), 40-45. doi: 10.11648/j.ijsas.20170102.12

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    AMA Style

    Rovshan Aliyev. Second-Order Asymptotic Expansion for the Ruin Probability of the Sparre Andersen Risk Process with Reinsurance and Stronger Semiexponential Claims. Int J Stat Actuarial Sci. 2017;1(2):40-45. doi: 10.11648/j.ijsas.20170102.12

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  • @article{10.11648/j.ijsas.20170102.12,
      author = {Rovshan Aliyev},
      title = {Second-Order Asymptotic Expansion for the Ruin Probability of the Sparre Andersen Risk Process with Reinsurance and Stronger Semiexponential Claims},
      journal = {International Journal of Statistics and Actuarial Science},
      volume = {1},
      number = {2},
      pages = {40-45},
      doi = {10.11648/j.ijsas.20170102.12},
      url = {https://doi.org/10.11648/j.ijsas.20170102.12},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ijsas.20170102.12},
      abstract = {In this study the Sparre Andersen risk process with reinsurance is considered. The second-order asymptotic expansion for the ruin probability is obtained, when the claim sizes have the strongly semiexponential distribution. Moreover, numerical examples in cases proportional reinsurance and exsess stop loss reinsurance are provided.},
     year = {2017}
    }
    

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Author Information
  • Department of Operation Research and Probability Theory, Applied Mathematics and Cybernetics, Baku State University, Institute of Control Systems of ANAS, Baku, Azerbaijan

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