Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

PRECISE LARGE DEVIATIONS FOR AGGREGATE LOSS PROCESS IN A MULTI-RISK MODEL

  • Tang, Fengqin (School of Mathematics Sciences Huaibei Normal University) ;
  • Bai, Jianming (School of Management Lanzhou University)
  • Received : 2014.01.17
  • Published : 2015.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider a multi-risk model based on the policy entrance process with n independent policies. For each policy, the entrance process of the customer is a non-homogeneous Poisson process, and the claim process is a renewal process. The loss process of the single-risk model is a random sum of stochastic processes, and the actual individual claim sizes are described as extended upper negatively dependent (EUND) structure with heavy tails. We derive precise large deviations for the loss process of the multi-risk model after giving the precise large deviations of the single-risk model. Our results extend and improve the existing results in significant ways.

Keywords

References

  1. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.
  2. Y. Chen, K. C. Yuen, and K.W. Ng, Precise large deviations of random sums in presence of negative dependence and consistent variation, Methodol. Comput. Appl. Probab. 13 (2011), no. 4, 821-833. https://doi.org/10.1007/s11009-010-9194-7
  3. Y. Chen and W. P. Zhang, Large deviations for random sums of negatively dependent random variables with consistently varying tails, Statist. Probab. Lett. 77 (2007), no. 5, 530-538. https://doi.org/10.1016/j.spl.2006.08.021
  4. N. Ebrahimi and M. Ghosh, Multivariate negative dependence, Commun. Stat. Theor. M. 10 (1981), no. 4, 307-337. https://doi.org/10.1080/03610928108828041
  5. P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extremal Events, Springer, Berlin, 1997.
  6. C. Kluppelberg and T. Mikosch, Large deviations of heavy-tailed random sums with applications in insurance and finance, J. Appl. Probab. 34 (1997), no. 2, 293-308. https://doi.org/10.2307/3215371
  7. Z. H. Li and X. B. Kong, A new risk model based on policy entrance process and its weak convergence properties, Appl. Stoch. Models Bus. Ind. 23 (2007), no. 3, 235-246. https://doi.org/10.1002/asmb.669
  8. L. Liu, Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett. 79 (2009), no. 9, 1290-1298. https://doi.org/10.1016/j.spl.2009.02.001
  9. T. Mikosch and A. V. Nagaev, Large deviations of heavy-tailed sums with applications in insurance, Extremes 1 (1998), no. 1, 81-110. https://doi.org/10.1023/A:1009913901219
  10. K. W. Ng, Q. Tang, J. A. Yan, and H. Yang, Precise large deviations for the prospective-loss process, J. Appl. Probab. 40 (2003), no. 2, 391-400. https://doi.org/10.1239/jap/1053003551
  11. K. W. Ng, Q. Tang, J. A. Yan, and H. Yang, Precise large deviations for sums of random variables with consistently varying tails, J. Appl. Probab. 41 (2004), no. 1, 93-107. https://doi.org/10.1239/jap/1077134670
  12. X. Shen, Z. Lin, and Y. Zhang, Precise large deviations for the actual aggregate loss process, Stoch. Anal. Appl. 27 (2009), no. 5, 1000-1013. https://doi.org/10.1080/07362990903136512
  13. Q. Tang, Insensitivity to negative dependence of the asymptotic behavior of precise large deviations, Electron. J. Probab. 11 (2006), no. 4, 107-120. https://doi.org/10.1214/EJP.v11-304
  14. Q. Tang, C. Su, T. Jiang, and J. Zhang, Large deviations for heavy-tailed random sums in compound renewal model, Statist. Probab. Lett. 52 (2001), no. 1, 91-100. https://doi.org/10.1016/S0167-7152(00)00231-5
  15. Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl. 108 (2003), no. 2, 299-325. https://doi.org/10.1016/j.spa.2003.07.001
  16. Y. Wang, K. Wang, and D. Cheng, Precise large deviations for sums of negatively associated random variables with common dominatedly varying tails, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 6, 1725-1734. https://doi.org/10.1007/s10114-005-0745-8
  17. K. Y. Wang, Y. Yang, and J. G. Lin, Precise large deviations for widely orthant dependent random variables with dominatedly varying tails, Front. Math. China. 7 (2012), no. 5, 919-932. https://doi.org/10.1007/s11464-012-0227-0
  18. Y. Yang, K. Wang, R. Leipus, and J. Siaulys, Precise large deviations for actual aggregate loss process in a dependent compound customer-arrival-based insurance risk model, Lith. Math. J. 53 (2013), no. 4, 448-470. https://doi.org/10.1007/s10986-013-9221-9