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April  2019, 15(2): 481-505. doi: 10.3934/jimo.2018053

Asymptotics for a bidimensional risk model with two geometric Lévy price processes

Yang Yang 1, , Kaiyong Wang 2, , Jiajun Liu 3, and  Zhimin Zhang 4,,

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

* Corresponding author: Zhimin Zhang

Received  August 2017 Revised  November 2017 Published  April 2019 Early access  April 2018

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.

Keywords: Asymptotics, consistently varying tail, long tail, dominatedly varying tail, dependence, bidimensional risk model, geometric Lévy price process, infinite-time and finite-time ruin probabilities.
Mathematics Subject Classification: Primary: 91B30; Secondary: 60G51, 60K05.
Citation: Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial and Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053
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.

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