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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 |
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, 93-104.
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. |
show all references
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, 93-104.
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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