# American Institute of Mathematical Sciences

October  2011, 7(4): 849-874. doi: 10.3934/jimo.2011.7.849

## Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums

 1 Department of Mathematics, Soochow University, Suzhou, 215006, China

Received  May 2010 Revised  May 2011 Published  August 2011

In this paper, we discuss a nonstandard renewal risk model, where the price process of the investment portfolio is modelled as a geometric Lévy process, the claim sizes and premium sizes form sequences of identically distributed and upper-tail independent random variables, respectively, the claim size and its corresponding inter-claim time satisfy a certain dependence structure described via a conditional tail probability of the claim size given the inter-claim time before the claim occurs, and there is a similar dependence structure between the premium size and the inter-arrival time before the premium is paid. When the claim-size distribution belongs to the extended-regular-varying class, we obtain a uniform tail asymptotics for stochastically discounted aggregate claims. Furthermore, assuming that the tail of the premium-size distribution is lighter than that of the claim-size distribution, the uniform estimates for the finite- and infinite-time ruin probabilities are presented respectively.
Citation: 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
##### References:
 [1] H. Albrecher and O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals, Insurance Math. Econom., 35 (2004), 245-254. doi: 10.1016/j.insmatheco.2003.09.009. [2] H. Albrecher and J. L. Teugels, Exponential behavior in the presence of dependence in risk theory, J. App. Probab., 43 (2006), 257-273. doi: 10.1239/jap/1143936258. [3] A. V. Asimit and A. L. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model,, Scand. Actuar. J., 2010 (): 93.  doi: 10.1080/03461230802700897. [4] A. L. Badescu, E. C. K. Cheung and D. Landriault, Dependent risk models with bivariate phase-type distributions, J. Appl. Probab., 46 (2009), 113-131. doi: 10.1239/jap/1238592120. [5] R. Biard, C. Lefévre and S. Loisel, Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationary assumptions are relaxed, Insurance Math. Econom., 43 (2008), 412-421. doi: 10.1016/j.insmatheco.2008.08.004. [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation," Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987. [7] A. V. Boĭkov, The Cramer-Lundberg model with stochastic premiums, Theory Probab. Appl., 47 (2003), 489-493. doi: 10.1137/S0040585X9797987. [8] M. Boudreault, H. Cossette, D. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes, Scand. Actuar. J., 5 (2006), 265-285. doi: 10.1080/03461230600992266. [9] R. J. Boucherie, O. J. Boxma and K. Sigman, A note on negative customers, GI/G/I workload, and risk processes, Prob. Eng. Inf. Sci., 11 (1997), 305-311. doi: 10.1017/S0269964800004848. [10] L. Breiman, On some limit theorms similar to the arc-sin law, Teor. Verojatnost. i Primenen, 10 (1965), 323-331. doi: 10.1137/1110037. [11] D. B. H. Cline, Intermediate regular and $\Pi$ variation, Proc. London Math. Soc., 68 (1994), 594-616. doi: 10.1112/plms/s3-68.3.594. [12] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stoch. Proc. Appl., 49 (1994), 75-98. doi: 10.1016/0304-4149(94)90113-9. [13] R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. [14] H. Cossette, E. Marceau and F. Marri, On the compound Poisson risk model with dependence based on a generalized Falie-Gumbel-Morgenstern copula, Insurance Math. Econom., 43 (2008), 444-455. doi: 10.1016/j.insmatheco.2008.08.009. [15] Q. Gao and Y. Wang, Randomly weighted sums with dominantly varying-tailed increments and applications to risk theory, J. Korean Stat. Soc., 39 (2010), 305-314. doi: 10.1016/j.jkss.2010.02.004. [16] C. C. Heyde and D. Wang, Finite-time ruin probaility with an exponential Lévy process investment return and heavy-tailed claims, Adv. App. Probab., 41 (2009), 206-224. doi: 10.1239/aap/1240319582. [17] V. Kalashnikov and R. Norberg, Power tailed ruin probabilities in the presence of risky investments, Stochastic Proc. Appl., 98 (2002), 211-228. doi: 10.1016/S0304-4149(01)00148-X. [18] C. Klüppelberg and R. Kostadinova, Integrated insurance risk models with exponential Lévy investment, Insurance Math. Econom., 42 (2008), 560-577. [19] S. Kotz, N. Balakrishnan and N. L. Johnson, "Continuous Multivariate Distribution. Vol. I. Models and Applications," 2nd edition, Wiley Sereis in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2000. [20] E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153. doi: 10.1214/aoms/1177699260. [21] J. Li, Q. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv. Appl. Probab., 42 (2010), 1126-1146. doi: 10.1239/aap/1293113154. [22] R. B. Nelsen, "An Introduction to Copulas," 2nd edition, Springer Series in Statistics, Springer, New York, 2006. [23] J. Paulsen and H. K. Gjessing, Ruin theory with stochastic return on investments, Adv. Appl. Probab., 29 (1997), 965-985. [24] S. I. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extremes 5 (2002), 303-336. doi: 10.1023/A:1025148622954. [25] S. I. Resnick, "Extreme Values, Regular Variation and Point Processes," Reprint of the 1987 original, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2008. [26] X. M. Shen, Z. Y. Lin and Y. Zhang, Uniform estimate for maximum of randomly weighted sums with applications to ruin theory, Methodol. Comput. Appl. Probab., 11 (2009), 669-685. doi: 10.1007/s11009-008-9090-6. [27] Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and finanicial risks, Stochastic Proc. Appl., 108 (2003), 299-325. [28] Q. Tang, G. Wang and K. 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. [29] G. Temnov, Risk processes with random income, J. Math. Sci., 123 (2004), 3780-3794. doi: 10.1023/B:JOTH.0000036319.21285.22. [30] Y. Zhang, X. Shen and C. Weng, Approximation of the tail probability of randomly weighted sums and applications, Stochastic Proc. Appl., 119 (2009), 655-675. doi: 10.1016/j.spa.2008.03.004. [31] Z. Zhang and H. Yang, On a risk model with stochastic premiums income and dependence between income and loss, J. Comput. Appl. Math., 234 (2010), 44-57. doi: 10.1016/j.cam.2009.12.004. [32] M. Zhou and J. Cai, A perturbed risk model with dependence between premium rates and claim sizes, Insurance Math. Econom., 45 (2009), 382-392. doi: 10.1016/j.insmatheco.2009.08.008.

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##### References:
 [1] H. Albrecher and O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals, Insurance Math. Econom., 35 (2004), 245-254. doi: 10.1016/j.insmatheco.2003.09.009. [2] H. Albrecher and J. L. Teugels, Exponential behavior in the presence of dependence in risk theory, J. App. Probab., 43 (2006), 257-273. doi: 10.1239/jap/1143936258. [3] A. V. Asimit and A. L. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model,, Scand. Actuar. J., 2010 (): 93.  doi: 10.1080/03461230802700897. [4] A. L. Badescu, E. C. K. Cheung and D. Landriault, Dependent risk models with bivariate phase-type distributions, J. Appl. Probab., 46 (2009), 113-131. doi: 10.1239/jap/1238592120. [5] R. Biard, C. Lefévre and S. Loisel, Impact of correlation crises in risk theory: Asymptotics of finite-time ruin probabilities for heavy-tailed claim amounts when some independence and stationary assumptions are relaxed, Insurance Math. Econom., 43 (2008), 412-421. doi: 10.1016/j.insmatheco.2008.08.004. [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation," Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1987. [7] A. V. Boĭkov, The Cramer-Lundberg model with stochastic premiums, Theory Probab. Appl., 47 (2003), 489-493. doi: 10.1137/S0040585X9797987. [8] M. Boudreault, H. Cossette, D. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes, Scand. Actuar. J., 5 (2006), 265-285. doi: 10.1080/03461230600992266. [9] R. J. Boucherie, O. J. Boxma and K. Sigman, A note on negative customers, GI/G/I workload, and risk processes, Prob. Eng. Inf. Sci., 11 (1997), 305-311. doi: 10.1017/S0269964800004848. [10] L. Breiman, On some limit theorms similar to the arc-sin law, Teor. Verojatnost. i Primenen, 10 (1965), 323-331. doi: 10.1137/1110037. [11] D. B. H. Cline, Intermediate regular and $\Pi$ variation, Proc. London Math. Soc., 68 (1994), 594-616. doi: 10.1112/plms/s3-68.3.594. [12] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stoch. Proc. Appl., 49 (1994), 75-98. doi: 10.1016/0304-4149(94)90113-9. [13] R. Cont and P. Tankov, "Financial Modelling with Jump Processes," Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. [14] H. Cossette, E. Marceau and F. Marri, On the compound Poisson risk model with dependence based on a generalized Falie-Gumbel-Morgenstern copula, Insurance Math. Econom., 43 (2008), 444-455. doi: 10.1016/j.insmatheco.2008.08.009. [15] Q. Gao and Y. Wang, Randomly weighted sums with dominantly varying-tailed increments and applications to risk theory, J. Korean Stat. Soc., 39 (2010), 305-314. doi: 10.1016/j.jkss.2010.02.004. [16] C. C. Heyde and D. Wang, Finite-time ruin probaility with an exponential Lévy process investment return and heavy-tailed claims, Adv. App. Probab., 41 (2009), 206-224. doi: 10.1239/aap/1240319582. [17] V. Kalashnikov and R. Norberg, Power tailed ruin probabilities in the presence of risky investments, Stochastic Proc. Appl., 98 (2002), 211-228. doi: 10.1016/S0304-4149(01)00148-X. [18] C. Klüppelberg and R. Kostadinova, Integrated insurance risk models with exponential Lévy investment, Insurance Math. Econom., 42 (2008), 560-577. [19] S. Kotz, N. Balakrishnan and N. L. Johnson, "Continuous Multivariate Distribution. Vol. I. Models and Applications," 2nd edition, Wiley Sereis in Probability and Statistics: Applied Probability and Statistics, Wiley-Interscience, New York, 2000. [20] E. L. Lehmann, Some concepts of dependence, Ann. Math. Statist., 37 (1966), 1137-1153. doi: 10.1214/aoms/1177699260. [21] J. Li, Q. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Adv. Appl. Probab., 42 (2010), 1126-1146. doi: 10.1239/aap/1293113154. [22] R. B. Nelsen, "An Introduction to Copulas," 2nd edition, Springer Series in Statistics, Springer, New York, 2006. [23] J. Paulsen and H. K. Gjessing, Ruin theory with stochastic return on investments, Adv. Appl. Probab., 29 (1997), 965-985. [24] S. I. Resnick, Hidden regular variation, second order regular variation and asymptotic independence, Extremes 5 (2002), 303-336. doi: 10.1023/A:1025148622954. [25] S. I. Resnick, "Extreme Values, Regular Variation and Point Processes," Reprint of the 1987 original, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2008. [26] X. M. Shen, Z. Y. Lin and Y. Zhang, Uniform estimate for maximum of randomly weighted sums with applications to ruin theory, Methodol. Comput. Appl. Probab., 11 (2009), 669-685. doi: 10.1007/s11009-008-9090-6. [27] Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and finanicial risks, Stochastic Proc. Appl., 108 (2003), 299-325. [28] Q. Tang, G. Wang and K. 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. [29] G. Temnov, Risk processes with random income, J. Math. Sci., 123 (2004), 3780-3794. doi: 10.1023/B:JOTH.0000036319.21285.22. [30] Y. Zhang, X. Shen and C. Weng, Approximation of the tail probability of randomly weighted sums and applications, Stochastic Proc. Appl., 119 (2009), 655-675. doi: 10.1016/j.spa.2008.03.004. [31] Z. Zhang and H. Yang, On a risk model with stochastic premiums income and dependence between income and loss, J. Comput. Appl. Math., 234 (2010), 44-57. doi: 10.1016/j.cam.2009.12.004. [32] M. Zhou and J. Cai, A perturbed risk model with dependence between premium rates and claim sizes, Insurance Math. Econom., 45 (2009), 382-392. doi: 10.1016/j.insmatheco.2009.08.008.
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