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

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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 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 & 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.Google Scholar
[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. Google Scholar
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[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. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[11]	E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153. doi: 10.1214/aoms/1177699260. Google Scholar
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[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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References:

[1]	N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[6]	R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, Boca Raton, FL, 2004. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153. doi: 10.1214/aoms/1177699260. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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