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Cluster Computing

, Volume 21, Issue 1, pp 997–1007 | Cite as

Large deviations for the stochastic present value of aggregate net claims with infinite variance in the renewal risk model and its application in risk management

  • Min Xiao
  • Sheng Cui
  • Ruixing Ming
  • Tao JiangEmail author
Article
  • 160 Downloads

Abstract

In insurance, if the insurer continuously invests her wealth in risk-free and risky assets, then the price process of the investment portfolio can be described as a geometric present Lévy process. People always are interested in estimating the tail distribution of the stochastic present value of aggregate claims. In this paper, the large deviation for the stochastic present value of aggregate net claims, when the net claim size distribution is of Pareto type with finite expectation are obtained. We conduct some simulations to check the accuracy of the result we obtained and consider a portfolio optimization problem that maximizes the expected terminal wealth of the insurer subject to a solvency constraint.

Keywords

Large deviations Regular variation Exponential Lévy process 

Notes

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 71671166) and the Talent start-up fund of China Three Georges University (Grant No. 1910103).

References

  1. 1.
    Heyde, C.C.: A contribution to the theory of large deviations for sums of independent random variables. Probab. Theory Relat. Fields 7(5), 303–308 (1967a)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Heyde, C.C.: On large deviation problems for sums of random variables which are not attracted to the normal law. Ann. Math. Stat. 38, 1575–1578 (1967b)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Heyde, C.C.: On large deviation probabilities in the case of attraction to a non-normal stable law. Sankyá 30, 253–258 (1968)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Nagaev, A.V.: Integral limit theorems taking into account large deviations when Cramér’s condition is not fullled I, II. Theory Probab. Appl. 14(51–64), 193–208 (1969a)CrossRefzbMATHGoogle Scholar
  5. 5.
    Nagaev, A.V.: Limit theorems for large deviations when Cramér’s conditions are violated. Fiz-Mat. Nauk. 7, 17–22 (1969b) (in Russian)Google Scholar
  6. 6.
    Nagaev, S.V.: Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cline, D.B.H., Hsing, T.: Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Texas A & M University (1991) (Preprint)Google Scholar
  8. 8.
    Tang, Q., Su, C., Jiang, T., Zhang, J.: Large deviations for heavy-tailed random sums in compound renewal model. Stat. Probab. Lett. 52(1), 91–100 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Liu, Y., Hu, Y.: Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails. Sci. China Math. 46(3), 383–395 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Tang, Q.: Insensitivity to negative dependence of the asymptotic behavior of precise large deviations. Electron. J. Probab. 11(4), 107–120 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liu, L.: Precise large deviations for dependent random variables with heavy tails. Stat. Probab. Lett. 79(9), 1290–1298 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sato, K.: Lévy Process and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  13. 13.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC, BocaRaton, FL (2004)zbMATHGoogle Scholar
  14. 14.
    Paulson, J.: Ruin models with investment income. Probab. Surv. 5, 416–434 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jiang, T., Cui, S., Ming, R.: Large deviations for stochastic present value of aggregate claims in the renewal risk model. Statist. Probab. Lett. 101, 83–91 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)CrossRefzbMATHGoogle Scholar
  17. 17.
    Embrechts, P., Kluppelberg, C., Mikosch, T.: Modeling Extremal Events for Insurance and Finance. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  18. 18.
    Konstantinides, D.G., Mikosch, T.: Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. Ann. Probab. 33, 1992–2035 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Borovkov, A.A., Borovkov, K.A.: Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions. Cambridge University Press, Cambridge (2008)CrossRefzbMATHGoogle Scholar
  20. 20.
    Zhou, Q., Luo, J.: Artificial neural network based grid computing of E-government scheduling for emergency management. Comput. Syst. Sci. Eng. 30(5), 327–335 (2015)Google Scholar
  21. 21.
    Zhou, Q., Luo, J.: The service quality evaluation of ecologic economy systems using simulation computing. Comput. Syst. Sci. Eng. 31(6), 453–460 (2016)MathSciNetGoogle Scholar
  22. 22.
    Merton, R.C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3(4), 373–413 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zhou, Q., Liu, R.: Strategy optimization of resource scheduling based on cluster rendering. Clust. Comput. 19(4), 2109–2117 (2016). doi: 10.1007/s10586-016-0655-9 CrossRefGoogle Scholar
  24. 24.
    Zhou, Q., Luo, J.: The study on evaluation method of urban network security in the big data era. Intell. Autom. Soft Comput. (2017). doi: 10.1080/10798587.2016.1267444

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of Statistic and MathematicsZhejiang Gongshang UniversityHangzhouChina
  2. 2.College of ScienceChina Three Gorges UniversityYichangChina

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