American Institute of Mathematical Sciences

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May  2022, 18(3): 1541-1556. doi: 10.3934/jimo.2021032

The finite-time ruin probability of a risk model with a general counting process and stochastic return

Baoyin Xun 1, , Kam C. Yuen 2, and  Kaiyong Wang 1,,

1. 

School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, China
2. 

Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China

* Corresponding author: Kaiyong Wang

Received  July 2020 Revised  December 2020 Published  May 2022 Early access  February 2021

This paper considers a general risk model with stochastic return and a Brownian perturbation, where the claim arrival process is a general counting process and the price process of the investment portfolio is expressed as a geometric Lévy process. When the claim sizes are pairwise strong quasi-asymptotically independent random variables with heavy-tailed distributions, the asymptotics of the finite-time ruin probability of this risk model have been obtained.

Keywords: Finite-time ruin probability, stochastic return, Lévy process, dependence structure, general risk model.
Mathematics Subject Classification: Primary: 62P05, 62E10; Secondary: 60F05.
Citation: Baoyin Xun, Kam C. Yuen, Kaiyong Wang. The finite-time ruin probability of a risk model with a general counting process and stochastic return. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1541-1556. doi: 10.3934/jimo.2021032
References:
[1]

A. V. Asimit, E. Furman, Q. Tang and R. Vernic, Asymptotics for risk capital allocations based on conditional tail expectation, Insurance: Mathematics and Economics, 49 (2011), 310-324.  doi: 10.1016/j.insmatheco.2011.05.002.

[2] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511721434.
[3]

H. W. Block, T. H. Savits and M. Shaked, Some concepts of negative dependence, Annals of Probability, 10 (1982), 765-772.  doi: 10.1214/aop/1176993784.

[4]

D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Processes and Their Applications, 49 (1994), 75-98.  doi: 10.1016/0304-4149(94)90113-9.

[5]

N. Ebrahimi and M. Ghosh, Multivariate negative dependence, Communications in Statistics A—Theory Methods, 10 (1981), 307-337.  doi: 10.1080/03610928108828041.

[6]

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.

[7]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distribution, 2$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-7101-1.

[8]

K.-A. Fu, On joint ruin probability for a bidimensional Lévy-driven risk model with stochastic returns and heavy-tailed claims, Journal of Mathematical Analysis and Applications, 442 (2016), 17-30.  doi: 10.1016/j.jmaa.2016.04.042.

[9]

K.-A. Fu and C. Y. A. Ng, Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims, Insurance: Mathematics and Economics, 56 (2014), 80-87.  doi: 10.1016/j.insmatheco.2014.04.001.

[10]

K. Fu and C. Yu, On a two-dimensional risk model with time-dependent claim sizes and risky investments, Journal of Computational and Applied Mathematics, 344 (2018), 367-380.  doi: 10.1016/j.cam.2018.05.043.

[11]

Q. Gao and X. Liu, Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest, Statistics and Probability Letters, 83 (2013), 1527-1538.  doi: 10.1016/j.spl.2013.02.018.

[12]

J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, Journal of Theoretical Probability, 22 (2009), 871-882.  doi: 10.1007/s10959-008-0159-5.

[13]

C. Klüppelberg, Subexponential distribution and integrated tails, Journal of Applied Probability, 25 (1998), 132-141.  doi: 10.2307/3214240.

[14]

J. Li, Asymptotics in a time-dependent renewal risk model with stochastic return, Journal of Mathematical Analysis and Applications, 387 (2012), 1009-1023.  doi: 10.1016/j.jmaa.2011.10.012.

[15]

J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Statistics and Probability Letters, 83 (2013), 2081-2087.  doi: 10.1016/j.spl.2013.05.023.

[16]

J. Li, Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return, Insurance: Mathematics and Economics, 71 (2016), 195-204.  doi: 10.1016/j.insmatheco.2016.09.003.

[17]

J. Li, A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation, Statistics and Probability Letters, 127 (2017), 49-55.  doi: 10.1016/j.spl.2017.03.028.

[18]

X. Liu, Q. Gao and Y. Wang, A note on a dependent risk model with constant interest rate, Statistics and Probability Letters, 82 (2012), 707-712.  doi: 10.1016/j.spl.2011.12.016.

[19]

Y. Mao, K. Wang, L. Zhu and Y. Ren, Asymptotics for the finite-time ruin probability of a risk model with a general counting process, Japan Journal of Industrial and Applied Mathematics, 34 (2017), 243-252.  doi: 10.1007/s13160-017-0245-0.

[20]

K. Maulik and S. Resnick, Characterizations and examples of hidden regular variation, Extremes, 7 (2004), 31-67.  doi: 10.1007/s10687-004-4728-4.

[21]

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

[22]

J. Peng and D. Wang, Uniform asymptotics for ruin probabilities in a dependent renewal risk model with stochastic return on investments, Stochastics: An International Journal of Probability and Stochastic Processes, 90 (2018), 432-471.  doi: 10.1080/17442508.2017.1365077.

[23]

V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, American Mathematical Society, Providence, Rhode Island, 1996. doi: 10.1090/mmono/148.

[24]

S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987. doi: 10.1007/978-0-387-75953-1.

[25]

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

[26]

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 Processes and Their Applications, 108 (2003), 299-325.  doi: 10.1016/j.spa.2003.07.001.

[27]

Q. Tang, G. Wang and K. C. Yuen, Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model, Insurance: Mathematics and Economics, 46 (2010), 362-370.  doi: 10.1016/j.insmatheco.2009.12.002.

[28]

Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17 (2014), 467-393.  doi: 10.1007/s10687-014-0191-z.

[29]

D. Wang, Finite-time ruin probability with heavy-tailed claims and constant interest rate, Stochastic Models, 24 (2008), 41-57.  doi: 10.1080/15326340701826898.

[30]

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, Methodology and Computing in Applied Probability, 15 (2013), 109-124.  doi: 10.1007/s11009-011-9226-y.

[31]

K. Wang, L. Chen, Y. Yang and M. Gao, The finite-time ruin probability of a risk model with stochastic return and Brownian perturbation, Japan Journal of Industrial and Applied Mathematics, 35 (2018), 1173-1189.  doi: 10.1007/s13160-018-0321-0.

[32]

K. Wang, Y. Cui and Y. Mao, Estimates for the finite-time ruin probability of a time-dependent risk model with a Brownian perturbation, Mathematical Problems in Engineering, 2020, Art. ID 7130243, 5 pp. doi: 10.1155/2020/7130243.

[33]

Y. Wang, Z. Cui, K. Wang and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, Journal of Mathematical Analysis and Applications, 390 (2012), 208-223.  doi: 10.1016/j.jmaa.2012.01.025.

[34]

Y. Yang and E. Hashorva, Extremes and products of multivariate AC-product risks, Insurance: Mathematics and Economics, 52 (2013), 312-319.  doi: 10.1016/j.insmatheco.2013.01.005.

[35]

Y. Yang, K. Wang and D. G. Konstantinides, Uniform asymptotics for discounted aggregate claims in dependent risk models, Journal of Applied Probability, 51 (2014), 669-684.  doi: 10.1239/jap/1409932666.

[36]

Y. Yang, K. Wang, J. Liu and Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, Journal of Industrial and Management Optimization, 15 (2019), 481-505.  doi: 10.3934/jimo.2018053.

[37]

Y. Yang, Z. Zhang, T. Jiang and D. Cheng, Uniformly asymptotic behavior of ruin probabilities in a time-dependent renewal risk model with stochastic return, Journal of Computational and Applied Mathematics, 287 (2015), 32-43.  doi: 10.1016/j.cam.2015.03.020.

[38]

Y. Zhang and W. Wang, Ruin probabilities of a bidimensional risk model with investment, Statistics and Probability Letters, 82 (2012), 130-138.  doi: 10.1016/j.spl.2011.09.010.

show all references

References:
[1]

A. V. Asimit, E. Furman, Q. Tang and R. Vernic, Asymptotics for risk capital allocations based on conditional tail expectation, Insurance: Mathematics and Economics, 49 (2011), 310-324.  doi: 10.1016/j.insmatheco.2011.05.002.

[2] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511721434.
[3]

H. W. Block, T. H. Savits and M. Shaked, Some concepts of negative dependence, Annals of Probability, 10 (1982), 765-772.  doi: 10.1214/aop/1176993784.

[4]

D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Processes and Their Applications, 49 (1994), 75-98.  doi: 10.1016/0304-4149(94)90113-9.

[5]

N. Ebrahimi and M. Ghosh, Multivariate negative dependence, Communications in Statistics A—Theory Methods, 10 (1981), 307-337.  doi: 10.1080/03610928108828041.

[6]

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.

[7]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distribution, 2$^{nd}$ edition, Springer, New York, 2013. doi: 10.1007/978-1-4614-7101-1.

[8]

K.-A. Fu, On joint ruin probability for a bidimensional Lévy-driven risk model with stochastic returns and heavy-tailed claims, Journal of Mathematical Analysis and Applications, 442 (2016), 17-30.  doi: 10.1016/j.jmaa.2016.04.042.

[9]

K.-A. Fu and C. Y. A. Ng, Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims, Insurance: Mathematics and Economics, 56 (2014), 80-87.  doi: 10.1016/j.insmatheco.2014.04.001.

[10]

K. Fu and C. Yu, On a two-dimensional risk model with time-dependent claim sizes and risky investments, Journal of Computational and Applied Mathematics, 344 (2018), 367-380.  doi: 10.1016/j.cam.2018.05.043.

[11]

Q. Gao and X. Liu, Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest, Statistics and Probability Letters, 83 (2013), 1527-1538.  doi: 10.1016/j.spl.2013.02.018.

[12]

J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, Journal of Theoretical Probability, 22 (2009), 871-882.  doi: 10.1007/s10959-008-0159-5.

[13]

C. Klüppelberg, Subexponential distribution and integrated tails, Journal of Applied Probability, 25 (1998), 132-141.  doi: 10.2307/3214240.

[14]

J. Li, Asymptotics in a time-dependent renewal risk model with stochastic return, Journal of Mathematical Analysis and Applications, 387 (2012), 1009-1023.  doi: 10.1016/j.jmaa.2011.10.012.

[15]

J. Li, On pairwise quasi-asymptotically independent random variables and their applications, Statistics and Probability Letters, 83 (2013), 2081-2087.  doi: 10.1016/j.spl.2013.05.023.

[16]

J. Li, Uniform asymptotics for a multi-dimensional time-dependent risk model with multivariate regularly varying claims and stochastic return, Insurance: Mathematics and Economics, 71 (2016), 195-204.  doi: 10.1016/j.insmatheco.2016.09.003.

[17]

J. Li, A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation, Statistics and Probability Letters, 127 (2017), 49-55.  doi: 10.1016/j.spl.2017.03.028.

[18]

X. Liu, Q. Gao and Y. Wang, A note on a dependent risk model with constant interest rate, Statistics and Probability Letters, 82 (2012), 707-712.  doi: 10.1016/j.spl.2011.12.016.

[19]

Y. Mao, K. Wang, L. Zhu and Y. Ren, Asymptotics for the finite-time ruin probability of a risk model with a general counting process, Japan Journal of Industrial and Applied Mathematics, 34 (2017), 243-252.  doi: 10.1007/s13160-017-0245-0.

[20]

K. Maulik and S. Resnick, Characterizations and examples of hidden regular variation, Extremes, 7 (2004), 31-67.  doi: 10.1007/s10687-004-4728-4.

[21]

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

[22]

J. Peng and D. Wang, Uniform asymptotics for ruin probabilities in a dependent renewal risk model with stochastic return on investments, Stochastics: An International Journal of Probability and Stochastic Processes, 90 (2018), 432-471.  doi: 10.1080/17442508.2017.1365077.

[23]

V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields, American Mathematical Society, Providence, Rhode Island, 1996. doi: 10.1090/mmono/148.

[24]

S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987. doi: 10.1007/978-0-387-75953-1.

[25]

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

[26]

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 Processes and Their Applications, 108 (2003), 299-325.  doi: 10.1016/j.spa.2003.07.001.

[27]

Q. Tang, G. Wang and K. C. Yuen, Uniform tail asymptotics for the stochastic present value of aggregate claims in the renewal risk model, Insurance: Mathematics and Economics, 46 (2010), 362-370.  doi: 10.1016/j.insmatheco.2009.12.002.

[28]

Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17 (2014), 467-393.  doi: 10.1007/s10687-014-0191-z.

[29]

D. Wang, Finite-time ruin probability with heavy-tailed claims and constant interest rate, Stochastic Models, 24 (2008), 41-57.  doi: 10.1080/15326340701826898.

[30]

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, Methodology and Computing in Applied Probability, 15 (2013), 109-124.  doi: 10.1007/s11009-011-9226-y.

[31]

K. Wang, L. Chen, Y. Yang and M. Gao, The finite-time ruin probability of a risk model with stochastic return and Brownian perturbation, Japan Journal of Industrial and Applied Mathematics, 35 (2018), 1173-1189.  doi: 10.1007/s13160-018-0321-0.

[32]

K. Wang, Y. Cui and Y. Mao, Estimates for the finite-time ruin probability of a time-dependent risk model with a Brownian perturbation, Mathematical Problems in Engineering, 2020, Art. ID 7130243, 5 pp. doi: 10.1155/2020/7130243.

[33]

Y. Wang, Z. Cui, K. Wang and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, Journal of Mathematical Analysis and Applications, 390 (2012), 208-223.  doi: 10.1016/j.jmaa.2012.01.025.

[34]

Y. Yang and E. Hashorva, Extremes and products of multivariate AC-product risks, Insurance: Mathematics and Economics, 52 (2013), 312-319.  doi: 10.1016/j.insmatheco.2013.01.005.

[35]

Y. Yang, K. Wang and D. G. Konstantinides, Uniform asymptotics for discounted aggregate claims in dependent risk models, Journal of Applied Probability, 51 (2014), 669-684.  doi: 10.1239/jap/1409932666.

[36]

Y. Yang, K. Wang, J. Liu and Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, Journal of Industrial and Management Optimization, 15 (2019), 481-505.  doi: 10.3934/jimo.2018053.

[37]

Y. Yang, Z. Zhang, T. Jiang and D. Cheng, Uniformly asymptotic behavior of ruin probabilities in a time-dependent renewal risk model with stochastic return, Journal of Computational and Applied Mathematics, 287 (2015), 32-43.  doi: 10.1016/j.cam.2015.03.020.

[38]

Y. Zhang and W. Wang, Ruin probabilities of a bidimensional risk model with investment, Statistics and Probability Letters, 82 (2012), 130-138.  doi: 10.1016/j.spl.2011.09.010.

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