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
Volume 1, Issue 2, May 2017, Pages: 40-45
Received: Mar. 11, 2017; Accepted: Mar. 28, 2017; Published: Apr. 15, 2017
Views 1381      Downloads 58
Author
Rovshan Aliyev, Department of Operation Research and Probability Theory, Applied Mathematics and Cybernetics, Baku State University, Institute of Control Systems of ANAS, Baku, Azerbaijan
Article Tools
Follow on us
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.
Keywords
Sparre Andersen Risk Process, Reinsurance, Ruin Probability, Second-Order Asymptotic Expansion, Semiexponential Distribution
To cite this article
Rovshan Aliyev, 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. Vol. 1, No. 2, 2017, pp. 40-45. doi: 10.11648/j.ijsas.20170102.12
Copyright
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186