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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. AsmussenH. 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.  Google Scholar

[5]

A. L. BadescuE. 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.  Google Scholar

[6]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[7]

M. BoudreaultH. CossetteD. 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.  Google Scholar

[8]

L. Breiman, On some limit theorems similar to the arc-sin law, Theory of Probability and its Applications, 10 (1965), 351-360.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, London, 2004.  Google Scholar

[13]

H. CossetteE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Kotz and N. Balakrishnan, Continuous Multivariate Distributions, Wiley-Interscience, New York, 2000. doi: 10.1002/0471722065.  Google Scholar

[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.  Google Scholar

[25]

J. LiQ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Paulsen, Ruin models with investment income, Probability Surveys, 5 (2008), 416-434.  doi: 10.1214/08-PS134.  Google Scholar

[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.  Google Scholar

[31]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Q. Tang, R. Vernic and Z. Yuan, Risk analysis for insurance business in the presence of dependent extremal risks, work in progress. Google Scholar

[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.  Google Scholar

[36]

K. WangY. 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.  Google Scholar

[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.  Google Scholar

[38]

Y. YangK. 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.  Google Scholar

[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.  Google Scholar

[40]

K. C. YuenG. 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.  Google Scholar

[41]

K. C. YuenG. 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.  Google Scholar

[42]

M. ZhouK. 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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. AsmussenH. 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.  Google Scholar

[5]

A. L. BadescuE. 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.  Google Scholar

[6]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9780511721434.  Google Scholar

[7]

M. BoudreaultH. CossetteD. 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.  Google Scholar

[8]

L. Breiman, On some limit theorems similar to the arc-sin law, Theory of Probability and its Applications, 10 (1965), 351-360.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Cont and P. Tankov, Financial Modelling with Jump Processes, Chapman & Hall/CRC, London, 2004.  Google Scholar

[13]

H. CossetteE. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

S. Kotz and N. Balakrishnan, Continuous Multivariate Distributions, Wiley-Interscience, New York, 2000. doi: 10.1002/0471722065.  Google Scholar

[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.  Google Scholar

[25]

J. LiQ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

J. Paulsen, Ruin models with investment income, Probability Surveys, 5 (2008), 416-434.  doi: 10.1214/08-PS134.  Google Scholar

[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.  Google Scholar

[31]

K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Q. Tang, R. Vernic and Z. Yuan, Risk analysis for insurance business in the presence of dependent extremal risks, work in progress. Google Scholar

[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.  Google Scholar

[36]

K. WangY. 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.  Google Scholar

[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.  Google Scholar

[38]

Y. YangK. 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.  Google Scholar

[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.  Google Scholar

[40]

K. C. YuenG. 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.  Google Scholar

[41]

K. C. YuenG. 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.  Google Scholar

[42]

M. ZhouK. 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.  Google Scholar

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