# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021032

## 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

* Corresponding author: Kaiyong Wang

Received  July 2020 Revised  December 2020 Published  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.

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 & Management Optimization, 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.  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] 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.  Google Scholar [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.  Google Scholar [5] N. Ebrahimi and M. Ghosh, Multivariate negative dependence, Communications in Statistics A—Theory Methods, 10 (1981), 307-337.  doi: 10.1080/03610928108828041.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] C. Klüppelberg, Subexponential distribution and integrated tails, Journal of Applied Probability, 25 (1998), 132-141.  doi: 10.2307/3214240.  Google Scholar [14] J. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987. doi: 10.1007/978-0-387-75953-1.  Google Scholar [25] S. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extremes, 5 (2002), 303-336.  doi: 10.1023/A:1025148622954.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  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] 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.  Google Scholar [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.  Google Scholar [5] N. Ebrahimi and M. Ghosh, Multivariate negative dependence, Communications in Statistics A—Theory Methods, 10 (1981), 307-337.  doi: 10.1080/03610928108828041.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] C. Klüppelberg, Subexponential distribution and integrated tails, Journal of Applied Probability, 25 (1998), 132-141.  doi: 10.2307/3214240.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer, New York, 1987. doi: 10.1007/978-0-387-75953-1.  Google Scholar [25] S. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extremes, 5 (2002), 303-336.  doi: 10.1023/A:1025148622954.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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