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doi: 10.3934/jimo.2022036
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## Asymptotic estimates for finite-time ruin probabilities in a generalized dependent bidimensional risk model with CMC simulations

 School of Mathematical Sciences, Soochow University, Suzhou 215006, China

*Corresponding author: Dongya Cheng

Received  August 2021 Revised  January 2022 Early access March 2022

Fund Project: The fourth author is supported by National Natural Science Foundation of China (No. 11401415)

This paper studies ruin probabilities of a generalized bidimensional risk model with dependent and heavy-tailed claims and additional net loss processes. When the claim sizes have long-tailed and dominated-varying-tailed distributions, precise asymptotic formulae for two kinds of finite-time ruin probabilities are derived, where the two claim-number processes from different lines of business are almost arbitrarily dependent. Under some extra conditions on the independence relation of claim inter-arrival times, the class of the claim-size distributions is extended to the subexponential distribution class. In order to verify the accuracy of the obtained theoretical result, a simulation study is performed via the crude Monte Carlo method.

Citation: Xinru Ji, Bingjie Wang, Jigao Yan, Dongya Cheng. Asymptotic estimates for finite-time ruin probabilities in a generalized dependent bidimensional risk model with CMC simulations. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022036
##### References:
 [1] Y. Chen, Y. Yang and T. Jiang, Uniform asymptotics for finite-time ruin probability of a bidimensional risk model, J. Math. Anal. Appl., 469 (2019), 525-536. doi: 10.1016/j.jmaa.2018.09.025. [2] Y. Chen, K. C. Yuen and K. W. Ng, Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims, Appl. Stochastic Models Bus. Ind., 27 (2011), 290-300. doi: 10.1002/asmb.834. [3] D. Cheng, Uniform asymptotics for the finite-time ruin probability of a generalized bidimensional risk model with Brownian perturbations, Stochastics, 93 (2021), 56-71. doi: 10.1080/17442508.2019.1708362. [4] D. Cheng and C. Yu, Uniform asymptotics for the ruin probabilities in a bidimensional renewal risk model with strongly subexponential claims, Stochastics, 91 (2019), 643-656. doi: 10.1080/17442508.2018.1539088. [5] D. Cheng, Y. Yang and X. Wang, Asymptotic finite-time ruin probabilities in a dependent bidimensional renewal risk model with subexponential claims, Japan J. Indust. Appl. Math., 37 (2020), 657-675. doi: 10.1007/s13160-020-00418-y. [6] D.B. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stoch. Process. Appl., 49 (1994), 75–98. doi: 10.1016/0304-4149(94)90113-9. [7] 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. [8] S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions, Springer, New York, 2011. doi: 10.1007/978-1-4614-7101-1. [9] H. Hult, F. Lindskog, T. Mikosch and G. Samorodnitsky, Functional large deviations for multivariate regularly varying random walks, Ann. Appl. Probab., 15 (2005), 2651-2680. doi: 10.1214/105051605000000502. [10] Z. Hu, and B. Jiang, On joint ruin probabilities of a two-dimensional risk model with constant interest rate, J. Appl. Prob., 50 (2013), 309-322. doi: 10.1239/jap/1371648943. [11] J. Li, The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims, Comm. Stat. Theory Methods., 46 (2017a), 1959-1971. doi: 10.1080/03610926.2015.1030428. [12] J. Li, A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation, Statist. Probab. Lett., 127 (2017b), 49-55. doi: 10.1016/j.spl.2017.03.028. [13] J. Li, A revisit to asymptotic ruin probabilities for a bidimensional renewal risk model, Statist. Probab. Lett., 140 (2018), 23-32. doi: 10.1016/j.spl.2018.04.003. [14] O. V. Sarmanov, Generalized normal correlation and two-dimensional Fréchet classes, Dokl. Akad. Nauk., 168 (1966), 32-35. [15] C. Stein, A note on cumulative sums, Ann. Math. Statist., 17 (1946), 498-499. doi: 10.1214/aoms/1177730890. [16] S. Wang, H. Qian, H. Sun and Geng, B., Uniform asymptotics for ruin probabilities of a non standard bidimensional perturbed risk model with subexponential claims, Comm. Stat., Theory Methods, 2021, 1-16. doi: 10.1080/03610926.2021.1882496. [17] H. Yang and J. Li, Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims, Insurance Math. Econom., 58 (2014), 185-192. doi: 10.1016/j.insmatheco.2014.07.007. [18] H. Yang and J. Li, Asymptotic ruin probabilities for a bidimensional renewal risk model, Stochastics, 89 (2017), 687-708. doi: 10.1080/17442508.2016.1276909. [19] R. B. Nelsen, An Introduction to Copulas, Springer Science & Business Media, 2006. doi: 10.1007/s11229-005-3715-x. [20] Y. Yang, K. Wang, J. Liu and Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, J. Ind. Manag. Optim., 15 (2019), 481-505. doi: 10.3934/jimo.2018053. [21] Y. Yang, T. Zhang and K. C. Yuen, Approximations for finite-time ruin probability in a dependent discrete-time risk model with CMC simulations, J. Comput. Appl. Math., 321 (2017), 143-159. doi: 10.1016/j.cam.2017.02.004.

show all references

##### References:
 [1] Y. Chen, Y. Yang and T. Jiang, Uniform asymptotics for finite-time ruin probability of a bidimensional risk model, J. Math. Anal. Appl., 469 (2019), 525-536. doi: 10.1016/j.jmaa.2018.09.025. [2] Y. Chen, K. C. Yuen and K. W. Ng, Asymptotics for the ruin probabilities of a two-dimensional renewal risk model with heavy-tailed claims, Appl. Stochastic Models Bus. Ind., 27 (2011), 290-300. doi: 10.1002/asmb.834. [3] D. Cheng, Uniform asymptotics for the finite-time ruin probability of a generalized bidimensional risk model with Brownian perturbations, Stochastics, 93 (2021), 56-71. doi: 10.1080/17442508.2019.1708362. [4] D. Cheng and C. Yu, Uniform asymptotics for the ruin probabilities in a bidimensional renewal risk model with strongly subexponential claims, Stochastics, 91 (2019), 643-656. doi: 10.1080/17442508.2018.1539088. [5] D. Cheng, Y. Yang and X. Wang, Asymptotic finite-time ruin probabilities in a dependent bidimensional renewal risk model with subexponential claims, Japan J. Indust. Appl. Math., 37 (2020), 657-675. doi: 10.1007/s13160-020-00418-y. [6] D.B. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stoch. Process. Appl., 49 (1994), 75–98. doi: 10.1016/0304-4149(94)90113-9. [7] 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. [8] S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions, Springer, New York, 2011. doi: 10.1007/978-1-4614-7101-1. [9] H. Hult, F. Lindskog, T. Mikosch and G. Samorodnitsky, Functional large deviations for multivariate regularly varying random walks, Ann. Appl. Probab., 15 (2005), 2651-2680. doi: 10.1214/105051605000000502. [10] Z. Hu, and B. Jiang, On joint ruin probabilities of a two-dimensional risk model with constant interest rate, J. Appl. Prob., 50 (2013), 309-322. doi: 10.1239/jap/1371648943. [11] J. Li, The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims, Comm. Stat. Theory Methods., 46 (2017a), 1959-1971. doi: 10.1080/03610926.2015.1030428. [12] J. Li, A note on the finite-time ruin probability of a renewal risk model with Brownian perturbation, Statist. Probab. Lett., 127 (2017b), 49-55. doi: 10.1016/j.spl.2017.03.028. [13] J. Li, A revisit to asymptotic ruin probabilities for a bidimensional renewal risk model, Statist. Probab. Lett., 140 (2018), 23-32. doi: 10.1016/j.spl.2018.04.003. [14] O. V. Sarmanov, Generalized normal correlation and two-dimensional Fréchet classes, Dokl. Akad. Nauk., 168 (1966), 32-35. [15] C. Stein, A note on cumulative sums, Ann. Math. Statist., 17 (1946), 498-499. doi: 10.1214/aoms/1177730890. [16] S. Wang, H. Qian, H. Sun and Geng, B., Uniform asymptotics for ruin probabilities of a non standard bidimensional perturbed risk model with subexponential claims, Comm. Stat., Theory Methods, 2021, 1-16. doi: 10.1080/03610926.2021.1882496. [17] H. Yang and J. Li, Asymptotic finite-time ruin probability for a bidimensional renewal risk model with constant interest force and dependent subexponential claims, Insurance Math. Econom., 58 (2014), 185-192. doi: 10.1016/j.insmatheco.2014.07.007. [18] H. Yang and J. Li, Asymptotic ruin probabilities for a bidimensional renewal risk model, Stochastics, 89 (2017), 687-708. doi: 10.1080/17442508.2016.1276909. [19] R. B. Nelsen, An Introduction to Copulas, Springer Science & Business Media, 2006. doi: 10.1007/s11229-005-3715-x. [20] Y. Yang, K. Wang, J. Liu and Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, J. Ind. Manag. Optim., 15 (2019), 481-505. doi: 10.3934/jimo.2018053. [21] Y. Yang, T. Zhang and K. C. Yuen, Approximations for finite-time ruin probability in a dependent discrete-time risk model with CMC simulations, J. Comput. Appl. Math., 321 (2017), 143-159. doi: 10.1016/j.cam.2017.02.004.
Sample paths of the two discounted values of the surplus processes $R_1(t)$ (the blue lines) and $R_2(t)$ (the orange lines) (obtained based on $0\le t\le50$, $(x, y) = (20, 20)$, $r = 0.03$ and $c = 50$)
Results of simulations
 x y $\frac{N_{\text{sim}}}{N}$ $\frac{N_{\text{and}}}{N}$ $R_T(x, y)$ $E_{\text{sim}}$ $E_{\text{and}}$ $1000$ $1000$ $1.50\times 10^{-5}$ $1.54\times 10^{-5}$ $1.5222\times 10^{-5}$ $0.01458$ $-0.01169$ $1500$ $1500$ $7.10\times 10^{-6}$ $7.30\times 10^{-6}$ $6.7656\times 10^{-6}$ $-0.04943$ $-0.07899$ $2000$ $2000$ $3.90\times 10^{-6}$ $4.00\times 10^{-6}$ $3.8056\times 10^{-6}$ $-0.02481$ $-0.05108$ $2500$ $2500$ $2.50\times 10^{-6}$ $2.60\times 10^{-6}$ $2.4360\times 10^{-6}$ $-0.02627$ $-0.06732$ $3000$ $3000$ $1.70\times 10^{-6}$ $1.80\times 10^{-6}$ $1.6914\times 10^{-6}$ $-0.00508$ $-0.06421$
 x y $\frac{N_{\text{sim}}}{N}$ $\frac{N_{\text{and}}}{N}$ $R_T(x, y)$ $E_{\text{sim}}$ $E_{\text{and}}$ $1000$ $1000$ $1.50\times 10^{-5}$ $1.54\times 10^{-5}$ $1.5222\times 10^{-5}$ $0.01458$ $-0.01169$ $1500$ $1500$ $7.10\times 10^{-6}$ $7.30\times 10^{-6}$ $6.7656\times 10^{-6}$ $-0.04943$ $-0.07899$ $2000$ $2000$ $3.90\times 10^{-6}$ $4.00\times 10^{-6}$ $3.8056\times 10^{-6}$ $-0.02481$ $-0.05108$ $2500$ $2500$ $2.50\times 10^{-6}$ $2.60\times 10^{-6}$ $2.4360\times 10^{-6}$ $-0.02627$ $-0.06732$ $3000$ $3000$ $1.70\times 10^{-6}$ $1.80\times 10^{-6}$ $1.6914\times 10^{-6}$ $-0.00508$ $-0.06421$
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