# American Institute of Mathematical Sciences

January  2017, 13(1): 155-185. doi: 10.3934/jimo.2016010

## Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns

 1 School of Mathematical Sciences, School of Management and Economics, University of Electronic Science and Technology of China, Chengdu, 611731, China 2 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

* Corresponding author: Dingcheng Wang

Received  February 2015 Revised  December 2015 Published  March 2016

Fund Project: The research of Jiangyan Peng is supported by the National Natural Science Foundation of China (project no: 71501025) and China Postdoctoral Science Foundation (project no: 2015M572467). The research of Dingcheng Wang is supported by the National Natural Science Foundation of China (project no: 71271042).

Consider a non-standard renewal risk model with dependence structures, where claim sizes follow a one-sided linear process with independent and identically distributed step sizes, the step sizes and inter-arrival times respectively form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure. An insurance company is allowed to make risk-free and risky investments, where the price process of the investment portfolio follows an exponential Lévy process. When the step-size distribution is dominatedly-varying-tailed, some asymptotic estimates for the finite-and infinite-time ruin probabilities are obtained.

Citation: Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial and Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
##### References:
 [1] H. Albreche and O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals, Insurance: Mathematics and Economics, 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, Journal of Applied Probability, 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, Scandinavian Actuarial Journal, 2010 (2010), 93-104.  doi: 10.1080/03461230802700897. [4] S. Asmussen, H. Schmidli and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps, Advances in Applied Probability, 31 (1999), 422-447.  doi: 10.1239/aap/1029955142. [5] A. L. Badescu, E. C. K. Cheung and D. Landriault, Dependent risk models with bivariate phase-type distributions, Journal of Applied Probability, 46 (2009), 113-131.  doi: 10.1239/jap/1238592120. [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434. [7] M. Boudreault, H. Cossette, D. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal, 2006 (2006), 265-285.  doi: 10.1080/03461230600992266. [8] L. Breiman, On some limit theorems similar to the arc-sin law, Theory of Probability and its Applications, 10 (1965), 351-360. [9] P. J. Brockwell and R. A. Davis, Time series: Theory and Methods, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4419-0320-4. [10] J. Cai, Ruin probabilities and penalty functions with stochastic rates of interest, Stochastic Processes and their Applications, 112 (2004), 53-78.  doi: 10.1016/j.spa.2004.01.007. [11] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Processes and their Applications, 49 (1994), 75-98.  doi: 10.1016/0304-4149(94)90113-9. [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, London, 2004. [13] H. Cossette, E. Marceau and F. Marri, On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance: Mathematics and Economics, 43 (2008), 444-455.  doi: 10.1016/j.insmatheco.2008.08.009. [14] 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. [15] S. Emmer and C. Klüppelberg, Optimal portfolios when stock prices follow an exponential Lévy process, Finance and Stochastics, 8 (2004), 17-44.  doi: 10.1007/s00780-003-0105-4. [16] K. A. Fu and C. Y. A. Ng, Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims, Insurance: Mathematics and Economics, 56 (2014), 80-87.  doi: 10.1016/j.insmatheco.2014.04.001. [17] F. Guo and D. Wang, Uniform asymptotic estimates for ruin probabilities of renewal risk models with exponential Lévy process investment returns and dependent claims, Applied Stochastic Models in Business and Industry, 29 (2013), 295-313.  doi: 10.1002/asmb.1925. [18] 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, Advances in Applied Probability, 45 (2013), 241-273.  doi: 10.1239/aap/1363354110. [19] X. Hao and Q. Tang, A uniform asymptotic estimate for discounted aggregate claims with subexponential tails, Insurance: Mathematics and Economics, 43 (2008), 116-120.  doi: 10.1016/j.insmatheco.2008.03.009. [20] C. C. Heyde and D. Wang, Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims, Advances in Applied Probability, 41 (2009), 206-224.  doi: 10.1239/aap/1240319582. [21] V. Kalashnikov and R. Norberg, Power tailed ruin probabilities in presence of risky investment, Stochastic Processes and their Applications, 98 (2002), 211-228.  doi: 10.1016/S0304-4149(01)00148-X. [22] C. Klüppelberg and R. Kostadinova, Integrated insurance risk models with exponential Lévy investment, Insurance: Mathematics and Economics, 42 (2008), 560-577.  doi: 10.1016/j.insmatheco.2007.06.002. [23] S. Kotz and N. Balakrishnan, Continuous Multivariate Distributions, Wiley-Interscience, New York, 2000. doi: 10.1002/0471722065. [24] J. Li, Asymptotics in a time-dependent renewal risk model with stochastic return, Journal of Mathematical Analysis and Applications, 387 (2012), 1009-1023.  doi: 10.1016/j.jmaa.2011.10.012. [25] J. Li, Q. Tang and R. Wu, Subexponential tails of discounted aggregate claims a time-dependent renewal risk model, Advances in Applied Probability, 42 (2010), 1126-1146.  doi: 10.1239/aap/1293113154. [26] K. Maulik and B. Zwart, Tail asymptotics for exponential functionals of Lévy processes, Stochastic Processes and their Applications, 116 (2006), 156-177.  doi: 10.1016/j.spa.2005.09.002. [27] T. Mikosch and G. Samorodnitsky, The supremum of a negative drift random walk with dependent heavy-tailed steps, Annals of Applied Probability, 10 (2000), 1025-1064.  doi: 10.1214/aoap/1019487517. [28] J. Peng and J. Huang, Ruin probability in a one-sided linear model with constant interest rate, Statistics & Probability Letters, 80 (2010), 662-669.  doi: 10.1016/j.spl.2009.12.024. [29] J. Paulsen, Ruin models with investment income, Probability Surveys, 5 (2008), 416-434.  doi: 10.1214/08-PS134. [30] J. Paulsen, On Cramér-like asymptotics for risk processes with stochastic return on investments, Annals of Applied Probability, 12 (2002), 1247-1260.  doi: 10.1214/aoap/1037125862. [31] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. [32] Q. Tang, Insensitivity to negative dependence of asymptotic tail probabilities of sums and maxima of sums, Stochastic Analysis and Applications, 26 (2008), 435-450.  doi: 10.1080/07362990802006964. [33] 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 Processes and their Applications, 108 (2003), 299-325.  doi: 10.1016/j.spa.2003.07.001. [34] Q. Tang, R. Vernic and Z. Yuan, Risk analysis for insurance business in the presence of dependent extremal risks, work in progress. [35] D. Wang and Q. Tang, Tail probabilities of randomly weighted sums of random variables with dominated variation, Stochastic Models, 22 (2006), 253-272.  doi: 10.1080/15326340600649029. [36] 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, Methodology and Computing in Applied Probability, 15 (2013), 109-124.  doi: 10.1007/s11009-011-9226-y. [37] H. Yang and L. Zhang, Martingale method for ruin probability in an autoregressive model with constant interest rate, Probability in the Engineering and Informational Sciences, 17 (2003), 183-198.  doi: 10.1017/S0269964803172026. [38] Y. Yang, K. Y. Wang and D. G. Konstantinides, Uniform asymptotics for discounted aggregate claims in dependent risk models, Journal of Applied Probability, 51 (2014), 669-684.  doi: 10.1239/jap/1409932666. [39] Y. Yang and Y. Wang, Tail behavior of the product of two dependent random variables with applications to risk theory, Extremes, 16 (2013), 55-74.  doi: 10.1007/s10687-012-0153-2. [40] K. C. Yuen, G. Wang and K. W. Ng, Ruin probabilities for a risk process with stochastic return on investments, Stochastic Processes and their Applications, 110 (2004), 259-274.  doi: 10.1016/j.spa.2003.10.007. [41] K. C. Yuen, G. Wang and R. Wu, On the renewal risk process with stochastic interest, Stochastic Processes and their Applications, 116 (2006), 1496-1510.  doi: 10.1016/j.spa.2006.04.012. [42] M. Zhou, K. Wang and Y. Wang, Estimates for the finite-time ruin probability with insurance and financial risks, Acta Mathematicae Applicatae Sinica-English Series, 28 (2012), 795-806.  doi: 10.1007/s10255-012-0189-8.

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##### References:
 [1] H. Albreche and O. J. Boxma, A ruin model with dependence between claim sizes and claim intervals, Insurance: Mathematics and Economics, 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, Journal of Applied Probability, 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, Scandinavian Actuarial Journal, 2010 (2010), 93-104.  doi: 10.1080/03461230802700897. [4] S. Asmussen, H. Schmidli and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps, Advances in Applied Probability, 31 (1999), 422-447.  doi: 10.1239/aap/1029955142. [5] A. L. Badescu, E. C. K. Cheung and D. Landriault, Dependent risk models with bivariate phase-type distributions, Journal of Applied Probability, 46 (2009), 113-131.  doi: 10.1239/jap/1238592120. [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434. [7] M. Boudreault, H. Cossette, D. Landriault and E. Marceau, On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian Actuarial Journal, 2006 (2006), 265-285.  doi: 10.1080/03461230600992266. [8] L. Breiman, On some limit theorems similar to the arc-sin law, Theory of Probability and its Applications, 10 (1965), 351-360. [9] P. J. Brockwell and R. A. Davis, Time series: Theory and Methods, 2nd edition, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4419-0320-4. [10] J. Cai, Ruin probabilities and penalty functions with stochastic rates of interest, Stochastic Processes and their Applications, 112 (2004), 53-78.  doi: 10.1016/j.spa.2004.01.007. [11] D. B. H. Cline and G. Samorodnitsky, Subexponentiality of the product of independent random variables, Stochastic Processes and their Applications, 49 (1994), 75-98.  doi: 10.1016/0304-4149(94)90113-9. [12] R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, London, 2004. [13] H. Cossette, E. Marceau and F. Marri, On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern copula, Insurance: Mathematics and Economics, 43 (2008), 444-455.  doi: 10.1016/j.insmatheco.2008.08.009. [14] 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. [15] S. Emmer and C. Klüppelberg, Optimal portfolios when stock prices follow an exponential Lévy process, Finance and Stochastics, 8 (2004), 17-44.  doi: 10.1007/s00780-003-0105-4. [16] K. A. Fu and C. Y. A. Ng, Asymptotics for the ruin probability of a time-dependent renewal risk model with geometric Lévy process investment returns and dominatedly-varying-tailed claims, Insurance: Mathematics and Economics, 56 (2014), 80-87.  doi: 10.1016/j.insmatheco.2014.04.001. [17] F. Guo and D. Wang, Uniform asymptotic estimates for ruin probabilities of renewal risk models with exponential Lévy process investment returns and dependent claims, Applied Stochastic Models in Business and Industry, 29 (2013), 295-313.  doi: 10.1002/asmb.1925. [18] 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, Advances in Applied Probability, 45 (2013), 241-273.  doi: 10.1239/aap/1363354110. [19] X. Hao and Q. Tang, A uniform asymptotic estimate for discounted aggregate claims with subexponential tails, Insurance: Mathematics and Economics, 43 (2008), 116-120.  doi: 10.1016/j.insmatheco.2008.03.009. [20] C. C. Heyde and D. Wang, Finite-time ruin probability with an exponential Lévy process investment return and heavy-tailed claims, Advances in Applied Probability, 41 (2009), 206-224.  doi: 10.1239/aap/1240319582. [21] V. Kalashnikov and R. Norberg, Power tailed ruin probabilities in presence of risky investment, Stochastic Processes and their Applications, 98 (2002), 211-228.  doi: 10.1016/S0304-4149(01)00148-X. [22] C. Klüppelberg and R. Kostadinova, Integrated insurance risk models with exponential Lévy investment, Insurance: Mathematics and Economics, 42 (2008), 560-577.  doi: 10.1016/j.insmatheco.2007.06.002. [23] S. Kotz and N. Balakrishnan, Continuous Multivariate Distributions, Wiley-Interscience, New York, 2000. doi: 10.1002/0471722065. [24] J. Li, Asymptotics in a time-dependent renewal risk model with stochastic return, Journal of Mathematical Analysis and Applications, 387 (2012), 1009-1023.  doi: 10.1016/j.jmaa.2011.10.012. [25] J. Li, Q. Tang and R. Wu, Subexponential tails of discounted aggregate claims a time-dependent renewal risk model, Advances in Applied Probability, 42 (2010), 1126-1146.  doi: 10.1239/aap/1293113154. [26] K. Maulik and B. Zwart, Tail asymptotics for exponential functionals of Lévy processes, Stochastic Processes and their Applications, 116 (2006), 156-177.  doi: 10.1016/j.spa.2005.09.002. [27] T. Mikosch and G. Samorodnitsky, The supremum of a negative drift random walk with dependent heavy-tailed steps, Annals of Applied Probability, 10 (2000), 1025-1064.  doi: 10.1214/aoap/1019487517. [28] J. Peng and J. Huang, Ruin probability in a one-sided linear model with constant interest rate, Statistics & Probability Letters, 80 (2010), 662-669.  doi: 10.1016/j.spl.2009.12.024. [29] J. Paulsen, Ruin models with investment income, Probability Surveys, 5 (2008), 416-434.  doi: 10.1214/08-PS134. [30] J. Paulsen, On Cramér-like asymptotics for risk processes with stochastic return on investments, Annals of Applied Probability, 12 (2002), 1247-1260.  doi: 10.1214/aoap/1037125862. [31] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. [32] Q. Tang, Insensitivity to negative dependence of asymptotic tail probabilities of sums and maxima of sums, Stochastic Analysis and Applications, 26 (2008), 435-450.  doi: 10.1080/07362990802006964. [33] 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 Processes and their Applications, 108 (2003), 299-325.  doi: 10.1016/j.spa.2003.07.001. [34] Q. Tang, R. Vernic and Z. Yuan, Risk analysis for insurance business in the presence of dependent extremal risks, work in progress. [35] D. Wang and Q. Tang, Tail probabilities of randomly weighted sums of random variables with dominated variation, Stochastic Models, 22 (2006), 253-272.  doi: 10.1080/15326340600649029. [36] 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, Methodology and Computing in Applied Probability, 15 (2013), 109-124.  doi: 10.1007/s11009-011-9226-y. [37] H. Yang and L. Zhang, Martingale method for ruin probability in an autoregressive model with constant interest rate, Probability in the Engineering and Informational Sciences, 17 (2003), 183-198.  doi: 10.1017/S0269964803172026. [38] Y. Yang, K. Y. Wang and D. G. Konstantinides, Uniform asymptotics for discounted aggregate claims in dependent risk models, Journal of Applied Probability, 51 (2014), 669-684.  doi: 10.1239/jap/1409932666. [39] Y. Yang and Y. Wang, Tail behavior of the product of two dependent random variables with applications to risk theory, Extremes, 16 (2013), 55-74.  doi: 10.1007/s10687-012-0153-2. [40] K. C. Yuen, G. Wang and K. W. Ng, Ruin probabilities for a risk process with stochastic return on investments, Stochastic Processes and their Applications, 110 (2004), 259-274.  doi: 10.1016/j.spa.2003.10.007. [41] K. C. Yuen, G. Wang and R. Wu, On the renewal risk process with stochastic interest, Stochastic Processes and their Applications, 116 (2006), 1496-1510.  doi: 10.1016/j.spa.2006.04.012. [42] M. Zhou, K. Wang and Y. Wang, Estimates for the finite-time ruin probability with insurance and financial risks, Acta Mathematicae Applicatae Sinica-English Series, 28 (2012), 795-806.  doi: 10.1007/s10255-012-0189-8.
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