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