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January  2016, 12(1): 31-43. doi: 10.3934/jimo.2016.12.31

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

Qingwu Gao 1, , Zhongquan Huang 1, , Houcai Shen 1, and  Juan Zheng 2,

1. 

International Center of Management Science and Engineering, School of Management and Engineering, Nanjing University, Nanjing, 210093, China, China, China
2. 

Department of Mathematics, Zaozhuang University, Zaozhuang, 277160, China

Received  December 2012 Revised  November 2014 Published  April 2015

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.
Keywords: subexponentiality, Asymptotics, quasi-asymptotic independence., random-time ruin probability, time-dependence.
Mathematics Subject Classification: Primary: 62E20; Secondary: 62P05, 91B3.
Citation: Qingwu Gao, Zhongquan Huang, Houcai Shen, Juan Zheng. Asymptotics for random-time ruin probability in a time-dependent renewal risk model with subexponential claims. Journal of Industrial & Management Optimization, 2016, 12 (1) : 31-43. doi: 10.3934/jimo.2016.12.31
References:
[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.  Google Scholar

[2]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. I. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extrems, 5 (2002), 303-336. doi: 10.1023/A:1025148622954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

S. I. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extrems, 5 (2002), 303-336. doi: 10.1023/A:1025148622954.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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