Article Contents
Article Contents

# Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims

• This paper investigates the asymptotic behavior of the random-time ruin probability in a time-dependent renewal risk model with pairwise quasi-asymptotically independent and subexponential claims, where the time-dependence structure is constructed between a claim size and its inter-arrival time, and described by a conditional tail probability of the claim size given the inter-arrival time before the claim occurs. In particular, the results we obtained are also valid for the finite-time ruin probability.
Mathematics Subject Classification: Primary: 62E20; Secondary: 62P05, 91B30.

 Citation:

•  [1] A. Asimit and A. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model, Scand. Actuar. J., (2010), 93-104.doi: 10.1080/03461230802700897. [2] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.doi: 10.1017/CBO9780511721434. [3] Y. Chen and K. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stochastic Models, 25 (2009), 76-89.doi: 10.1080/15326340802641006. [4] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stoch. Proc. Appl., 49 (1994), 75-98.doi: 10.1016/0304-4149(94)90113-9. [5] P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997.doi: 10.1007/978-3-642-33483-2. [6] 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), 735-749.doi: 10.4134/JKMS.2014.51.4.735. [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), 1505-1528.doi: 10.1216/RMJ-2014-44-5-1505. [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, Stat. Probab. Lett., 83 (2013), 1527-1538.doi: 10.1016/j.spl.2013.02.018. [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), 1193-1213.doi: 10.1016/j.jmaa.2014.05.069. [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), 611-626.doi: 10.4134/BKMS.2013.50.2.611. [11] Q. Gao, E. Zhang and N. Jin, The ultimate ruin probability of a dependent delayed-claim risk model perturbed by diffusion with constant force of interest, to appear in Bull. Korean Math. Soc., (2014). [12] J. Kočetova, R. Leipus and J. Šiaulys, A property of the renewal counting process with application to the finite-time ruin probability, Lith. Math. J., 49 (2009), 55-61.doi: 10.1007/s10986-009-9032-1. [13] R. Leipus and J. Šiaulys, Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes, Insurance Math. Econom., 40 (2007), 498-508.doi: 10.1016/j.insmatheco.2006.07.006. [14] R. Leipus and J. Šiaulys, Asymptotic behaviour of the finite-time ruin probability in renewal risk model, Appl. Stoch. Models Bus. Ind., 25 (2009), 309-321.doi: 10.1002/asmb.747. [15] J. Li, Q. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv. Appl. Proba., 42 (2010), 1126-1146.doi: 10.1239/aap/1293113154. [16] S. I. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extrems, 5 (2002), 303-336.doi: 10.1023/A:1025148622954. [17] Q. Tang, Asymptotics for the finite time ruin probability in the renewal model with consistent variation, Stoch. Models, 20 (2004), 281-297.doi: 10.1081/STM-200025739. [18] K. Wang, Y. Wang and Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab., 15 (2013), 109-124.doi: 10.1007/s11009-011-9226-y. [19] Y. Wang, Z. Cui, K. Wang and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, J. Math. Anal. Appl., 390 (2012), 208-223.doi: 10.1016/j.jmaa.2012.01.025. [20] Y. Wang, Q. Gao, K. Wang and X. Liu, Random time ruin probability for the renewal risk model with heavy-tailed claims, J. Ind. Manag. Optim., 5 (2009), 719-736.doi: 10.3934/jimo.2009.5.719. [21] Y. Yang, R. Leipus, J. Šiaulys and Y. Cang, Uniform estimates for the finite-time ruin probability in the dependent renewal risk model, J. Math. Anal. Appl., 383 (2011), 215-225.doi: 10.1016/j.jmaa.2011.05.013.