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January 2018, 14(1): 231-247. doi: 10.3934/jimo.2017044

## Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims

 1 Institute of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China 2 Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China 3 Institute of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China

* Corresponding author: Yang Yang

Received  January 2016 Revised  February 2017 Published  April 2017

Fund Project: The research of Yang Yang was supported by the National Natural Science Foundation of China (No. 71471090, 71671166, 11301278), the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 14YJCZH182), Natural Science Foundation of Jiangsu Province of China (No. BK20161578), the Major Research Plan of Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJA110001), Qing Lan Project, PAPD, Program of Excellent Science and Technology Innovation Team of the Jiangsu Higher Education Institutions of China, 333 Talent Training Project of Jiangsu Province, High Level Talent Project of Six Talents Peak of Jiangsu Province (No. JY-039), Project of Construction for Superior Subjects of Mathematics/Statistics of Jiangsu Higher Education Institutions, and Project of the Key Lab of Financial Engineering of Jiangsu Province (No. NSK2015-17). The research of Kam C. Yuen was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU17329216), and the CAE 2013 research grant from the Society of Actuaries. The research of Jun-feng Liu was supported by the National Natural Science Foundation of China (No. 11401313), and Natural Science Foundation of Jiangsu Province of China (No. BK20161579)

Consider a bivariate Lévy-driven risk model in which the loss process of an insurance company and the investment return process are two independent Lévy processes. Under the assumptions that the loss process has a Lévy measure of consistent variation and the return process fulfills a certain condition, we investigate the asymptotic behavior of the finite-time ruin probability. Further, we derive two asymptotic formulas for the finite-time and infinite-time ruin probabilities in a single Lévy-driven risk model, in which the loss process is still a Lévy process, whereas the investment return process reduces to a deterministic linear function. In such a special model, we relax the loss process with jumps whose common distribution is long tailed and of dominated variation.

Citation: Yang Yang, Kam C. Yuen, Jun-Feng Liu. Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims. Journal of Industrial & Management Optimization, 2018, 14 (1) : 231-247. doi: 10.3934/jimo.2017044
##### References:
 [1] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10. 1017/CBO9780511721434. [2] 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. [3] Y. Chen, K. W. Ng and Q. Tang, Weighted sums of subexponential random variables and their maxima, Adv. in Appl. Probab., 37 (2005), 510-522. doi: 10.1017/S0001867800000288. [4] Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25 (2009), 76-89. doi: 10.1080/15326340802641006. [5] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Process. Appl., 49 (1994), 75-98. doi: 10.1016/0304-4149(94)90113-9. [6] P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin, 1997. doi: 10. 1007/978-3-642-33483-2. [7] S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions, Springer-Verlag, New York, 2011. doi: 10. 1007/978-1-4419-9473-8. [8] A. Frolova, Y. Kabanov and S. Pergamenshchikov, In the insurance business risky investments are dangerous, Finance Stoch., 6 (2002), 227-235. doi: 10.1007/s007800100057. [9] Q. Gao and Y. Wang, Randomly weighted sums with dominated varying-tailed increments and application to risk theory, J. Korean Statist. Society, 39 (2010), 305-314. doi: 10.1016/j.jkss.2010.02.004. [10] H. K. Gjessing and J. Paulsen, Present value distributions with applications to ruin theory and stochastic equations, Stochastic Process. Appl., 71 (1997), 123-144. doi: 10.1016/S0304-4149(97)00072-0. [11] D. R. Grey, Regular variation in the tail behaviour of solutions of random difference equations, Ann. Appl. Probab., 4 (1994), 169-183. doi: 10.1214/aoap/1177005205. [12] F. Guo and D. Wang, Finite-and infinite-time ruin probabilities with general stochastic investment return processes and bivariate upper tail independent and heavy-tailed claims, Adv. in Appl. Probab., 45 (2013), 241-273. doi: 10.1017/S0001867800006261. [13] X. Hao and Q. Tang, A uniform asymptotic estimate for discounted aggregate claims with sunexponential tails, Insurance Math. Econom., 43 (2008), 116-120. doi: 10.1016/j.insmatheco.2008.03.009. [14] X. Hao and Q. Tang, Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments, J. Appl. Probab., 4 (2012), 939-953. [15] C. C. Heyde and D. Wang, Finite-time ruin probability with an exponential L´evy process investment return and heavy-tailed claims, Adv. in Appl. Probab., 41 (2009), 206-224. doi: 10.1017/S0001867800003190. [16] V. Kalashnikov and D. Konstantinides, Ruin under interest force and subexponential claims: A simple treatment, Insurance Math. Econom., 27 (2000), 145-149. doi: 10.1016/S0167-6687(00)00045-7. [17] 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. [18] C. Klüppelberg and R. Kostadinova, Integrated insurance risk models with exponential L´evy investment, Insurance Math. Econom., 42 (2008), 560-577. doi: 10.1016/j.insmatheco.2007.06.002. [19] C. Klüppelberg and U. Stadtmüller, Ruin probabilities in the presence of heavy-tails and interest rates, Scand. Actuar. J., 1 (1998), 49-58. doi: 10.1080/03461238.1998.10413991. [20] D. Konstantinides, Q. Tang and G. Tsitsiashvili, Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails, Insurance Math. Econom., 31 (2002), 447-460. doi: 10.1016/S0167-6687(02)00189-0. [21] 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. [22] 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. [23] J. Paulsen and H. K. Gjessing, Ruin theory with stochastic return on investments, Adv. in Appl. Probab., 29 (1997), 965-985. doi: 10.1017/S0001867800047972. [24] P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2003. doi: 10. 1007/978-3-662-10061-5. [25] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman & Hall, New York, 1994. [26] Q. Tang, The finite-time ruin probability of the compound Poisson model with constant interest force, J. Appl. Probab., 42 (2005), 608-619. doi: 10.1017/S0021900200000656. [27] Q. Tang, Heavy tails of discounted aggregate claims in the continuous-time renewal model, J. Appl. Probab., 44 (2007), 285-294. doi: 10.1017/S0021900200117826. [28] Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl., 108 (2003), 299-325. doi: 10.1016/j.spa.2003.07.001. [29] 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. [30] Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17 (2014), 467-493. doi: 10.1007/s10687-014-0191-z. [31] W. Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab., 11 (1979), 750-783. doi: 10.2307/1426858. [32] D. Wang, Finite-time ruin probability with heavy-tailed claims and constant interest rate, Stoch. Models, 24 (2008), 41-57. doi: 10.1080/15326340701826898. [33] K. Wang, Y. Wang and Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab., 15 (2013), 109-124. doi: 10.1007/s11009-011-9226-y. [34] Y. Yang, R. Leipus and J. Šiaulys, On the ruin probability in a dependent discrete time risk model with insurance and financial risks, J. Comput. Appl. Math., 236 (2012), 3286-3295. doi: 10.1016/j.cam.2012.02.030. [35] Y. Yang, J. Lin and Z. Tan, The finite-time ruin probability in the presence of Sarmanov dependent financial and insurance risks, Appl. Math. J. Chinese Univ., 29 (2014), 194-204. doi: 10.1007/s11766-014-3209-z. [36] Y. Yang, K. Wang and D. Konstantinides, Uniform asymptotics for discounted aggregate claims in dependent risk models, J. Appl. Probab., 51 (2014), 669-684. doi: 10.1017/S0021900200011591. [37] Y. Yang and Y. Wang, Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims, Statist. Probab. Letters, 80 (2010), 143-154. doi: 10.1016/j.spl.2009.09.023.

show all references

##### References:
 [1] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10. 1017/CBO9780511721434. [2] 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. [3] Y. Chen, K. W. Ng and Q. Tang, Weighted sums of subexponential random variables and their maxima, Adv. in Appl. Probab., 37 (2005), 510-522. doi: 10.1017/S0001867800000288. [4] Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stoch. Models, 25 (2009), 76-89. doi: 10.1080/15326340802641006. [5] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Process. Appl., 49 (1994), 75-98. doi: 10.1016/0304-4149(94)90113-9. [6] P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin, 1997. doi: 10. 1007/978-3-642-33483-2. [7] S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions, Springer-Verlag, New York, 2011. doi: 10. 1007/978-1-4419-9473-8. [8] A. Frolova, Y. Kabanov and S. Pergamenshchikov, In the insurance business risky investments are dangerous, Finance Stoch., 6 (2002), 227-235. doi: 10.1007/s007800100057. [9] Q. Gao and Y. Wang, Randomly weighted sums with dominated varying-tailed increments and application to risk theory, J. Korean Statist. Society, 39 (2010), 305-314. doi: 10.1016/j.jkss.2010.02.004. [10] H. K. Gjessing and J. Paulsen, Present value distributions with applications to ruin theory and stochastic equations, Stochastic Process. Appl., 71 (1997), 123-144. doi: 10.1016/S0304-4149(97)00072-0. [11] D. R. Grey, Regular variation in the tail behaviour of solutions of random difference equations, Ann. Appl. Probab., 4 (1994), 169-183. doi: 10.1214/aoap/1177005205. [12] F. Guo and D. Wang, Finite-and infinite-time ruin probabilities with general stochastic investment return processes and bivariate upper tail independent and heavy-tailed claims, Adv. in Appl. Probab., 45 (2013), 241-273. doi: 10.1017/S0001867800006261. [13] X. Hao and Q. Tang, A uniform asymptotic estimate for discounted aggregate claims with sunexponential tails, Insurance Math. Econom., 43 (2008), 116-120. doi: 10.1016/j.insmatheco.2008.03.009. [14] X. Hao and Q. Tang, Asymptotic ruin probabilities for a bivariate Lévy-driven risk model with heavy-tailed claims and risky investments, J. Appl. Probab., 4 (2012), 939-953. [15] C. C. Heyde and D. Wang, Finite-time ruin probability with an exponential L´evy process investment return and heavy-tailed claims, Adv. in Appl. Probab., 41 (2009), 206-224. doi: 10.1017/S0001867800003190. [16] V. Kalashnikov and D. Konstantinides, Ruin under interest force and subexponential claims: A simple treatment, Insurance Math. Econom., 27 (2000), 145-149. doi: 10.1016/S0167-6687(00)00045-7. [17] 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. [18] C. Klüppelberg and R. Kostadinova, Integrated insurance risk models with exponential L´evy investment, Insurance Math. Econom., 42 (2008), 560-577. doi: 10.1016/j.insmatheco.2007.06.002. [19] C. Klüppelberg and U. Stadtmüller, Ruin probabilities in the presence of heavy-tails and interest rates, Scand. Actuar. J., 1 (1998), 49-58. doi: 10.1080/03461238.1998.10413991. [20] D. Konstantinides, Q. Tang and G. Tsitsiashvili, Estimates for the ruin probability in the classical risk model with constant interest force in the presence of heavy tails, Insurance Math. Econom., 31 (2002), 447-460. doi: 10.1016/S0167-6687(02)00189-0. [21] 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. [22] 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. [23] J. Paulsen and H. K. Gjessing, Ruin theory with stochastic return on investments, Adv. in Appl. Probab., 29 (1997), 965-985. doi: 10.1017/S0001867800047972. [24] P. E. Protter, Stochastic Integration and Differential Equations, 2nd edition, Springer-Verlag, Berlin, 2003. doi: 10. 1007/978-3-662-10061-5. [25] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman & Hall, New York, 1994. [26] Q. Tang, The finite-time ruin probability of the compound Poisson model with constant interest force, J. Appl. Probab., 42 (2005), 608-619. doi: 10.1017/S0021900200000656. [27] Q. Tang, Heavy tails of discounted aggregate claims in the continuous-time renewal model, J. Appl. Probab., 44 (2007), 285-294. doi: 10.1017/S0021900200117826. [28] Q. Tang and G. Tsitsiashvili, Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic Process. Appl., 108 (2003), 299-325. doi: 10.1016/j.spa.2003.07.001. [29] 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. [30] Q. Tang and Z. Yuan, Randomly weighted sums of subexponential random variables with application to capital allocation, Extremes, 17 (2014), 467-493. doi: 10.1007/s10687-014-0191-z. [31] W. Vervaat, On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables, Adv. in Appl. Probab., 11 (1979), 750-783. doi: 10.2307/1426858. [32] D. Wang, Finite-time ruin probability with heavy-tailed claims and constant interest rate, Stoch. Models, 24 (2008), 41-57. doi: 10.1080/15326340701826898. [33] K. Wang, Y. Wang and Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab., 15 (2013), 109-124. doi: 10.1007/s11009-011-9226-y. [34] Y. Yang, R. Leipus and J. Šiaulys, On the ruin probability in a dependent discrete time risk model with insurance and financial risks, J. Comput. Appl. Math., 236 (2012), 3286-3295. doi: 10.1016/j.cam.2012.02.030. [35] Y. Yang, J. Lin and Z. Tan, The finite-time ruin probability in the presence of Sarmanov dependent financial and insurance risks, Appl. Math. J. Chinese Univ., 29 (2014), 194-204. doi: 10.1007/s11766-014-3209-z. [36] Y. Yang, K. Wang and D. Konstantinides, Uniform asymptotics for discounted aggregate claims in dependent risk models, J. Appl. Probab., 51 (2014), 669-684. doi: 10.1017/S0021900200011591. [37] Y. Yang and Y. Wang, Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varying-tailed claims, Statist. Probab. Letters, 80 (2010), 143-154. doi: 10.1016/j.spl.2009.09.023.
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