-
Previous Article
Exclusion sets in the Δ-type eigenvalue inclusion set for tensors
- JIMO Home
- This Issue
-
Next Article
Higher-order weak radial epiderivatives and non-convex set-valued optimization problems
Asymptotics for a bidimensional risk model with two geometric Lévy price processes
| 1. | Department of Statistics, Nanjing Audit University, Nanjing 211815, China |
| 2. | School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China |
| 3. | Department of Mathematical Sciences, Xi'an Jiaotong-Liverpool University, Suzhou 215123, China |
| 4. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
Consider a bidimensional risk model with two geometric Lévy price processes and dependent heavy-tailed claims, in which we allow arbitrary dependence structures between the two claim-number processes generated by two kinds of businesses, and between the two geometric Lévy price processes. Under the assumption that the claims have consistently varying tails, the asymptotics for the infinite-time and finite-time ruin probabilities are derived.
References:
| [1] |
N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. |
| [2] |
H. W. Block, T. H. Savits and M. Shaked,
Some concepts of negative dependence, Ann. Probab., 10 (1982), 765-772.
doi: 10.1214/aop/1176993784. |
| [3] |
Y. Chen and K. W. Ng,
The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims, Insurance Math. Econom., 40 (2007), 415-423.
doi: 10.1016/j.insmatheco.2006.06.004. |
| [4] |
Y. Chen, L. Wang and Y. Wang,
Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models, J. Math. Anal. Appl., 401 (2013), 114-129.
doi: 10.1016/j.jmaa.2012.11.046. |
| [5] |
Y. Chen, K. C. Yuen and K. W. Ng,
Asymptotics for 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. |
| [6] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
| [7] |
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. |
| [8] |
T. Jiang, Y. Wang, Y. Chen and H. Xu,
Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insurance Math. Econom., 64 (2015), 45-53.
doi: 10.1016/j.insmatheco.2015.04.006. |
| [9] |
K. Joag-Dev and F. Proschan,
Negative association of random variables with application, Ann. Statist., 11 (1983), 286-295.
doi: 10.1214/aos/1176346079. |
| [10] |
V. Kalashnikov and R. Norberg,
Power tailed ruin probabilities in the presence of risky investments, Stochastic Process. Appl, 98 (2002), 211-228.
doi: 10.1016/S0304-4149(01)00148-X. |
| [11] |
E. L. Lehmann,
Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153.
doi: 10.1214/aoms/1177699260. |
| [12] |
J. Li,
Asymptotics in a time-dependent renewal risk model with stochastic return, J. Math. Anal. Appl., 387 (2012), 1009-1023.
doi: 10.1016/j.jmaa.2011.10.012. |
| [13] |
J. Li, Z. Liu and Q. Tang,
On the ruin probabilities of a bidimensional perturbed risk model, Insurance Math. Econom., 41 (2007), 185-195.
doi: 10.1016/j.insmatheco.2006.10.012. |
| [14] |
J. Li and H. Yang,
Asymptotic ruin probabilities for a bidimensional renewal risk model with constant interest rate and dependent claims, J. Math. Anal. Appl., 426 (2015), 247-266.
doi: 10.1016/j.jmaa.2015.01.047. |
| [15] |
X. Liu, Q. Gao and Y. Wang,
A note on a dependent risk model with constant interest rate, Statist. Probab. Lett., 82 (2012), 707-712.
doi: 10.1016/j.spl.2011.12.016. |
| [16] |
K. Maulik and S. Resnick,
Characterizations and examples of hidden regular variation, Extremes, 7 (2004), 31-67.
doi: 10.1007/s10687-004-4728-4. |
| [17] |
J. Paulsen,
Risk theory in a stochastic economic environmen, Stochastic Process. Appl., 46 (1993), 327-361.
doi: 10.1016/0304-4149(93)90010-2. |
| [18] |
J. Paulsen,
On Cramér-like asymptotics for risk processes with stochastic return on investments, Ann. Appl. Probab., 12 (2002), 1247-1260.
doi: 10.1214/aoap/1037125862. |
| [19] |
J. Paulsen and H. K. Gjessing,
Ruin theory with stochastic return on investments, Adv. Appl. Probab., 29 (1997), 965-985.
doi: 10.2307/1427849. |
| [20] |
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 Math. Econom., 46 (2010), 362-370.
doi: 10.1016/j.insmatheco.2009.12.002. |
| [21] |
G. Wang and R. Wu,
Distributions for the risk process with a stochastic return on investments, Stochastic Process. Appl., 95 (2001), 329-341.
doi: 10.1016/S0304-4149(01)00102-8. |
| [22] |
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. |
| [23] |
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. |
| [24] |
Y. Yang, K. Wang and D. G. Konstantinides,
Uniform asymptotics for discounted aggregate claims in dependent risk models, J. Appl. Probab., 51 (2014), 669-684.
doi: 10.1239/jap/1409932666. |
| [25] |
Y. Yang and K. C. Yuen,
Finite-time and infinite-time ruin probabilities in a two-dimensional delayed renewal risk model with Sarmanov dependent claims, J. Math. Anal. Appl., 442 (2016), 600-626.
doi: 10.1016/j.jmaa.2016.04.068. |
| [26] |
K. C. Yuen, J. Guo and X. Wu,
On the first time of ruin in the bivariate compound Poisson model, Insurance Math. Econom., 38 (2006), 298-308.
doi: 10.1016/j.insmatheco.2005.08.011. |
show all references
References:
| [1] |
N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. |
| [2] |
H. W. Block, T. H. Savits and M. Shaked,
Some concepts of negative dependence, Ann. Probab., 10 (1982), 765-772.
doi: 10.1214/aop/1176993784. |
| [3] |
Y. Chen and K. W. Ng,
The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims, Insurance Math. Econom., 40 (2007), 415-423.
doi: 10.1016/j.insmatheco.2006.06.004. |
| [4] |
Y. Chen, L. Wang and Y. Wang,
Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models, J. Math. Anal. Appl., 401 (2013), 114-129.
doi: 10.1016/j.jmaa.2012.11.046. |
| [5] |
Y. Chen, K. C. Yuen and K. W. Ng,
Asymptotics for 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. |
| [6] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. |
| [7] |
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. |
| [8] |
T. Jiang, Y. Wang, Y. Chen and H. Xu,
Uniform asymptotic estimate for finite-time ruin probabilities of a time-dependent bidimensional renewal model, Insurance Math. Econom., 64 (2015), 45-53.
doi: 10.1016/j.insmatheco.2015.04.006. |
| [9] |
K. Joag-Dev and F. Proschan,
Negative association of random variables with application, Ann. Statist., 11 (1983), 286-295.
doi: 10.1214/aos/1176346079. |
| [10] |
V. Kalashnikov and R. Norberg,
Power tailed ruin probabilities in the presence of risky investments, Stochastic Process. Appl, 98 (2002), 211-228.
doi: 10.1016/S0304-4149(01)00148-X. |
| [11] |
E. L. Lehmann,
Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153.
doi: 10.1214/aoms/1177699260. |
| [12] |
J. Li,
Asymptotics in a time-dependent renewal risk model with stochastic return, J. Math. Anal. Appl., 387 (2012), 1009-1023.
doi: 10.1016/j.jmaa.2011.10.012. |
| [13] |
J. Li, Z. Liu and Q. Tang,
On the ruin probabilities of a bidimensional perturbed risk model, Insurance Math. Econom., 41 (2007), 185-195.
doi: 10.1016/j.insmatheco.2006.10.012. |
| [14] |
J. Li and H. Yang,
Asymptotic ruin probabilities for a bidimensional renewal risk model with constant interest rate and dependent claims, J. Math. Anal. Appl., 426 (2015), 247-266.
doi: 10.1016/j.jmaa.2015.01.047. |
| [15] |
X. Liu, Q. Gao and Y. Wang,
A note on a dependent risk model with constant interest rate, Statist. Probab. Lett., 82 (2012), 707-712.
doi: 10.1016/j.spl.2011.12.016. |
| [16] |
K. Maulik and S. Resnick,
Characterizations and examples of hidden regular variation, Extremes, 7 (2004), 31-67.
doi: 10.1007/s10687-004-4728-4. |
| [17] |
J. Paulsen,
Risk theory in a stochastic economic environmen, Stochastic Process. Appl., 46 (1993), 327-361.
doi: 10.1016/0304-4149(93)90010-2. |
| [18] |
J. Paulsen,
On Cramér-like asymptotics for risk processes with stochastic return on investments, Ann. Appl. Probab., 12 (2002), 1247-1260.
doi: 10.1214/aoap/1037125862. |
| [19] |
J. Paulsen and H. K. Gjessing,
Ruin theory with stochastic return on investments, Adv. Appl. Probab., 29 (1997), 965-985.
doi: 10.2307/1427849. |
| [20] |
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 Math. Econom., 46 (2010), 362-370.
doi: 10.1016/j.insmatheco.2009.12.002. |
| [21] |
G. Wang and R. Wu,
Distributions for the risk process with a stochastic return on investments, Stochastic Process. Appl., 95 (2001), 329-341.
doi: 10.1016/S0304-4149(01)00102-8. |
| [22] |
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. |
| [23] |
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. |
| [24] |
Y. Yang, K. Wang and D. G. Konstantinides,
Uniform asymptotics for discounted aggregate claims in dependent risk models, J. Appl. Probab., 51 (2014), 669-684.
doi: 10.1239/jap/1409932666. |
| [25] |
Y. Yang and K. C. Yuen,
Finite-time and infinite-time ruin probabilities in a two-dimensional delayed renewal risk model with Sarmanov dependent claims, J. Math. Anal. Appl., 442 (2016), 600-626.
doi: 10.1016/j.jmaa.2016.04.068. |
| [26] |
K. C. Yuen, J. Guo and X. Wu,
On the first time of ruin in the bivariate compound Poisson model, Insurance Math. Econom., 38 (2006), 298-308.
doi: 10.1016/j.insmatheco.2005.08.011. |
| [1] |
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, 2022 doi: 10.3934/jimo.2022036 |
| [2] |
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 and Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010 |
| [3] |
Yinghua Dong, Yuebao Wang. Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums. Journal of Industrial and Management Optimization, 2011, 7 (4) : 849-874. doi: 10.3934/jimo.2011.7.849 |
| [4] |
Drew Fudenberg, David K. Levine. Tail probabilities for triangular arrays. Journal of Dynamics and Games, 2014, 1 (1) : 45-56. doi: 10.3934/jdg.2014.1.45 |
| [5] |
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 and Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058 |
| [6] |
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 |
| [7] |
Yutaka Sakuma, Atsushi Inoie, Ken’ichi Kawanishi, Masakiyo Miyazawa. Tail asymptotics for waiting time distribution of an M/M/s queue with general impatient time. Journal of Industrial and Management Optimization, 2011, 7 (3) : 593-606. doi: 10.3934/jimo.2011.7.593 |
| [8] |
Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial and Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012 |
| [9] |
Yang Yang, Kam C. Yuen, Jun-Feng Liu. Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. Journal of Industrial and Management Optimization, 2018, 14 (1) : 231-247. doi: 10.3934/jimo.2017044 |
| [10] |
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 and Management Optimization, 2016, 12 (1) : 31-43. doi: 10.3934/jimo.2016.12.31 |
| [11] |
Byeongchan Lee, Jonghun Yoon, Yang Woo Shin, Ganguk Hwang. Tail asymptotics of fluid queues in a distributed server system fed by a heavy-tailed ON-OFF flow. Journal of Industrial and Management Optimization, 2016, 12 (2) : 637-652. doi: 10.3934/jimo.2016.12.637 |
| [12] |
Jon Aaronson, Omri Sarig, Rita Solomyak. Tail-invariant measures for some suspension semiflows. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 725-735. doi: 10.3934/dcds.2002.8.725 |
| [13] |
María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2065-2078. doi: 10.3934/dcds.2018084 |
| [14] |
Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241 |
| [15] |
Arno Berger. On finite-time hyperbolicity. Communications on Pure and Applied Analysis, 2011, 10 (3) : 963-981. doi: 10.3934/cpaa.2011.10.963 |
| [16] |
Po-Chung Yang, Hui-Ming Wee, Shen-Lian Chung, Yong-Yan Huang. Pricing and replenishment strategy for a multi-market deteriorating product with time-varying and price-sensitive demand. Journal of Industrial and Management Optimization, 2013, 9 (4) : 769-787. doi: 10.3934/jimo.2013.9.769 |
| [17] |
Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298 |
| [18] |
Arno Berger, Doan Thai Son, Stefan Siegmund. Nonautonomous finite-time dynamics. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 463-492. doi: 10.3934/dcdsb.2008.9.463 |
| [19] |
Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 |
| [20] |
Robert G. McLeod, John F. Brewster, Abba B. Gumel, Dean A. Slonowsky. Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs. Mathematical Biosciences & Engineering, 2006, 3 (3) : 527-544. doi: 10.3934/mbe.2006.3.527 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]