American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Discount-offering and demand-rejection decisions for substitutable products with different profit levels
  • JIMO Home
  • This Issue
  • Next Article
    Global stabilization for ball-and-beam systems via state and partial state feedback
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. doi: 10.1080/03461230802700897. Google Scholar

[2]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation,, Cambridge University Press, (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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1080/03461230802700897. Google Scholar

[2]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation,, Cambridge University Press, (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. 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. 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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jmaa.2011.05.013. Google Scholar

[1]

Yuebao Wang, Qingwu Gao, Kaiyong Wang, Xijun Liu. Random time ruin probability for the renewal risk model with heavy-tailed claims. Journal of Industrial & Management Optimization, 2009, 5 (4) : 719-736. doi: 10.3934/jimo.2009.5.719
[2]

Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012
[3]

Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial & Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058
[4]

Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
[5]

Xiaoqing Liang, Lihua Bai. Minimizing expected time to reach a given capital level before ruin. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1771-1791. doi: 10.3934/jimo.2017018
[6]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007
[7]

Yayun Zheng, Xu Sun. Governing equations for Probability densities of stochastic differential equations with discrete time delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3615-3628. doi: 10.3934/dcdsb.2017182
[8]

P.E. Kloeden, Pedro Marín-Rubio. Equi-Attraction and the continuous dependence of attractors on time delays. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 581-593. doi: 10.3934/dcdsb.2008.9.581
[9]

Weizhong Huang, Xianyi Wu. Credibility models with dependence structure over risks and time horizon. Journal of Industrial & Management Optimization, 2015, 11 (2) : 365-380. doi: 10.3934/jimo.2015.11.365
[10]

Yinghui Dong, Guojing Wang. Ruin probability for renewal risk model with negative risk sums. Journal of Industrial & Management Optimization, 2006, 2 (2) : 229-236. doi: 10.3934/jimo.2006.2.229
[11]

Jerry L. Bona, Laihan Luo. Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 15-50. doi: 10.3934/dcdss.2011.4.15
[12]

Bernard Ducomet. Asymptotics for 1D flows with time-dependent external fields. Conference Publications, 2007, 2007 (Special) : 323-333. doi: 10.3934/proc.2007.2007.323
[13]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79
[14]

Barry Simon. Zeros of OPUC and long time asymptotics of Schur and related flows. Inverse Problems & Imaging, 2007, 1 (1) : 189-215. doi: 10.3934/ipi.2007.1.189
[15]

David Lipshutz. Exit time asymptotics for small noise stochastic delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3099-3138. doi: 10.3934/dcds.2018135
[16]

Vladimir Varlamov. Eigenfunction expansion method and the long-time asymptotics for the damped Boussinesq equation. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 675-702. doi: 10.3934/dcds.2001.7.675
[17]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701
[18]

Marie Doumic, Miguel Escobedo. Time asymptotics for a critical case in fragmentation and growth-fragmentation equations. Kinetic & Related Models, 2016, 9 (2) : 251-297. doi: 10.3934/krm.2016.9.251
[19]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Time asymptotics of structured populations with diffusion and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4087-4116. doi: 10.3934/dcdsb.2018127
[20]

Simone Göttlich, Stephan Martin, Thorsten Sickenberger. Time-continuous production networks with random breakdowns. Networks & Heterogeneous Media, 2011, 6 (4) : 695-714. doi: 10.3934/nhm.2011.6.695

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences