American Institute of Mathematical Sciences

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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

Yinghua Dong 1, and  Yuebao Wang 1,

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.
Keywords: upper-tail independence, extended-regular variation, dependence, Asymptotics, uniformity., Lévy process, ruin probabilities.
Mathematics Subject Classification: Primary: 91B30; Secondary: 60G70, 91B2.
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, 93-104. 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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