THE ULTIMATE RUIN PROBABILITY OF A DEPENDENT DELAYED-CLAIM RISK MODEL PERTURBED BY DIFFUSION WITH CONSTANT FORCE OF INTEREST

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 52, Issue 3, 2015, pp.895-906
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2015.52.3.895

Title & Authors

THE ULTIMATE RUIN PROBABILITY OF A DEPENDENT DELAYED-CLAIM RISK MODEL PERTURBED BY DIFFUSION WITH CONSTANT FORCE OF INTEREST

Gao, Qingwu; Zhang, Erli; Jin, Na;

Gao, Qingwu; Zhang, Erli; Jin, Na;

Abstract

Recently, Li [12] gave an asymptotic formula for the ultimate ruin probability in a delayed-claim risk model with constant force of interest and pairwise quasi-asymptotically independent and extended-regularly-varying-tailed claims. This paper extends Li's result to the case in which the risk model is perturbed by diffusion, the claims are consistently-varying-tailed and the main-claim interarrival times are widely lower orthant dependent.

Keywords

asymptotic ruin probability;delayed claim;diffusion;pairwise quasi-asymptotic independence;widely lower orthant dependence;

Language

English

References

1.

N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.

2.

Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models 25 (2009), no. 1, 76-89.

3.

D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Process. Appl. 49 (1994), no. 1, 75-98.

4.

P. Embrechts, C. Kluppelberg, and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997.

5.

Q. Gao and D. Bao, Asymptotic ruin probabilities in a generalized jump-diffusion risk model with constant force of interest, J. Korean Math. Soc. 51 (2014), no. 4, 735-749.

6.

Q. Gao and N. Jin, Randomly weighted sums of pairwise quasi upper-tail independent increments with application to risk theory, Comm. Statist. Theory Methods (2015), 10.1080/03610926.2013.851234.

7.

Q. Gao, N. Jin, and H. Shen, Asymptotic behavior of the finite-time ruin probability with pairwise quasi-asymptotically independent claims and constant interest force, Rocky Mountain J. Math. 44 (2014), no. 5, 1505-1528.

8.

Q. Gao and X. Liu, Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest, Statist. Probab. Lett. 83 (2013), no. 6, 1527-1538.

9.

Q. Gao and X. Yang, Asymptotic ruin probabilities in a generalized bidimensional risk model perturbed by diffusion with constant force of interest, J. Math. Anal. Appl. 419 (2014), no. 2, 1193-1213.

10.

Q. Gao and Y. Yang, Uniform asymptotics for the finite-time ruin probability in a general risk model with pairwise quasi-asymptotically independent claims and constant interest force, Bull. Korean Math. Soc. 50 (2013), no. 2, 611-626.

11.

J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, J. Theoret. Probab. 22 (2009), no. 4, 871-882.

12.

J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Statist. Probab. Lett. 83 (2013), no. 9, 2081-2087.

13.

J. Li and R. Wu, The Gerber-Shiu discounted penalty function for a compound binomial risk model with by-claims, Acta Math. Appl. Sin. Engl. Ser. 31 (2015), no. 1, 181-190.

14.

X. Liu, Q. Gao, and Y. Wang, A note on a dependent risk model with constant interest rate, Statist. Probab. Lett. 82 (2012), no. 4, 707-712.

15.

H. Meng and G. Wang, On the expected discounted penalty function in a delayed-claims risk model, Acta Math. Appl. Sin. Engl. Ser. 28 (2012), no. 2, 215-224.

16.

Q. Tang, Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation, Scand. Actuar. J. 2005 (2005), no. 1, 1-5.

17.

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.

18.

Q. Tang and G. Tsitsiashvili, Randomly weighted sums of subexponential random variables with application to ruin theory, Extremes 6 (2003), no. 3, 171-188.

19.

D. Wang, C. Su, and C. Zeng, Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory, Sci. China Ser. A 48 (2005), no. 10, 1379-1394.

20.

K. Wang, Y. Wang, and Q. Gao, Uniform asymptotics for the finite-time ruin probabil- ity of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab. 15 (2013), no. 1, 109-124.

21.

Y. Wang and D. Cheng, Basic renewal theorems for a random walk with widely de- pendent increments and their applications, J. Math. Anal. Appl. 384 (2011), no. 2, 597-606.

22.

H. R. Waters and A. Papatriandafylou, Ruin probabilities allowing for delay in claims settlement, Insurance Math. Econom. 4 (1985), no. 2, 113-122.

23.

Y. Xiao and J. Guo, The compound binomial risk model with time-correlated claims, Insurance Math. Econom. 41 (2007), no. 1, 124-133.

24.

H. Xie and W. Zou, Expected present value of total dividends in a delayed claims risk model under stochastic interest rates, Insurance Math. Econom. 46 (2010), no. 2, 415-422.

25.

H. Xie and W. Zou, On the expected discounted penalty function for the compound Poisson risk model with delayed claims, J. Comput. Appl. Math. 235 (2011), no. 8, 2392-2404.

26.

K. C. Yuen and J. Guo, Ruin probabilities for time-correlated claims in the compound binomial model, Insurance Math. Econom. 29 (2001), no. 1, 47-57.