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Tail asymptotic for discounted aggregate claims with one-sided linear dependence and general investment return

  • Fenglong Guo
  • Dingcheng Wang
Articles
  • 3 Downloads

Abstract

In this study, we investigate the tail probability of the discounted aggregate claim sizes in a dependent risk model. In this model, the claim sizes are observed to follow a one-sided linear process with independent and identically distributed innovations. Investment return is described as a general stochastic process with cadlag paths. In the case of heavy-tailed innovation distributions, we are able to derive some asymptotic estimates for tail probability and to provide some asymptotic upper bounds to improve the applicability of our study.

Keywords

Poisson risk model tail probability one-sided linear process heavy-tailed distribution asymptotic upper bound investment return 

MSC(2010)

62E20 62P05 

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Notes

Acknowledgements

Guo's work was supported by National Natural Science Foundation of China (Grant No. 71501100) and the Open Project of Jiangsu Key Laboratory of Financial Engineering (Grant No. NSK2015-02). Wang's work was supported by National Natural Science Foundation of China (Grant No. 71271042). This article is one of the stage results of the Major Bidding Project of the Chinese National Social Science Foundation (Grant No. 17ZDA072).

References

  1. 1.
    Bankovsky D, Klüppelberg C, Maller R. On the ruin probability of the generalised Ornstein-Uhlenbeck process in the Cramér case. J Appl Probab, 2011, 48: 15–28CrossRefzbMATHGoogle Scholar
  2. 2.
    Bingham N H, Goldie C M, Teugels J L. Regular Variation. Cambridge: Cambridge University Press, 1987CrossRefzbMATHGoogle Scholar
  3. 3.
    Bowers N L, Gerber H U, Hickman J C, et al. Actuarial Mathematics, 2nd ed. Schaumburg: The Society of Actuaries, 1997Google Scholar
  4. 4.
    Breiman L. On some limit theorems similar to the arc-sin law. Theory Probab Appl, 1965, 10: 323–331MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cai J, Yang H. Ruin in the perturbed compound Poisson risk process under interest force. Adv in Appl Probab, 2005, 37: 819–835MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Collamore J F. Random recurrence equations and ruin in a Markov-dependent stochastic economic environment. Ann Appl Probab, 2009, 19: 1404–1458MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. New York: Springer, 1997CrossRefzbMATHGoogle Scholar
  8. 8.
    Gerber H U. Ruin theory in the linear model. Insurance Math Econom, 1982, 1: 213–217MathSciNetCrossRefGoogle Scholar
  9. 9.
    Guo F, Wang D. Finite- and infinite-time ruin probabilities with general stochastic investment return processes and bivariate upper tail independent and heavy-tailed claims. Adv Appl Probab, 2013, 45: 241–273MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guo F, Wang D. Uniform asymptotic estimates for ruin probabilities of renewal risk models with exponential Lévy process investment returns and dependent claims. Appl Stoch Models Bus Ind, 2013, 29: 295–313MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Guo F, Wang D. Uniform asymptotic estimates for ruin probabilities with exponential Lévy process investment returns and two-sided linear heavy-tailed claims. Comm Statist Theory Methods, 2015, 44: 4278–4306MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guo F, Wang D, Peng J. Tail asymptotic of discounted aggregate claims with compound dependence under risky investment. Comm Statist Theory Methods, 2018, 47: 1–21MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guo F, Wang D, Yang H. Asymptotic results for ruin probability in a two-dimensional risk model with stochastic investment returns. J Comput Appl Math, 2017, 325: 198–221MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heyde C C, Wang D. Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims. Adv Appl Probab, 2009, 41: 206–224MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hult H, Lindskog F. Ruin probabilities under general investments and heavy-tailed claims. Finance Stoch, 2011, 15: 243–265MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Klüppelberg C, Kostadinova R. Integrated insurance risk models with exponential Lévy investment. Insurance Math Econom, 2008, 42: 560–577MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mikosch T, Samorodnitsky G. The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann Appl Probab, 2000, 10: 1025–1064MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Paulsen J. On Cramér-like asymptotics for risk processes with stochastic return on investments. Ann Appl Probab, 2002, 12: 1247–1260MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Paulsen J, Gjessing H K. Ruin theory with stochastic return on investments. Adv Appl Probab, 1997, 29: 965–985MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Peng J, Huang J. Ruin probability in a one-sided linear model with constant interest rate. Statist Probab Lett, 2010, 80: 662–669MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999zbMATHGoogle Scholar
  22. 22.
    Tang Q, Tsitsiashvili G. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process Appl, 2003, 108: 299–325MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tang Q, Wang G, Yuen K C. Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model. Insurance Math Econom, 2010, 46: 362–370MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tang Q, Yuan Z. A hybrid estimate for the finite-time ruin probability in a bivariate autoregressive risk model with application to portfolio optimization. N Am Actuar J, 2012, 16: 378–397MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tang Q, Yuan Z. Randomly weighted sums of subexponential random variables with application to capital allocation. Extremes, 2014, 17: 467–493MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Wang D, Tang Q. Tail probabilities of randomly weighted sums of random variables with dominated variation. Stoch Models, 2006, 22: 253–272MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yang H, Zhang L. Martingale method for ruin probability in an autoregressive model with constant interest rate. Probab Engrg Inform Sci, 2003, 17: 183–198MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science in China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Jiangsu Key Laboratory of Financial Engineering and School of FinanceNanjing Audit UniversityNanjingChina
  2. 2.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduChina

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