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April  2017, 13(2): 995-1007. doi: 10.3934/jimo.2016058

Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times

1. 

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China

2. 

Center of Financial Engineering, Nanjing Audit University, Nanjing 211815, China

* Corresponding author

Received  December 2015 Revised  June 2016 Published  August 2016

We investigate the infinite-time ruin probability of a renewal risk model with exponential Lévy process investment and dependent claims and inter-arrival times. Assume that claims and corresponding inter-arrival times form a sequence of independent and identically distributed copies of a random pair $(X,T)$ with dependent components. When the product of the claims and the discount factors of the corresponding inter-arrival times are heavy tailed, we establish an asymptotic formula for the infinite-time ruin probability without any restriction on the dependence structure of $(X,T)$.

Citation: Rongfei Liu, Dingcheng Wang, Jiangyan Peng. Infinite-time ruin probability of a renewal risk model with exponential Levy process investment and dependent claims and inter-arrival times. Journal of Industrial and Management Optimization, 2017, 13 (2) : 995-1007. doi: 10.3934/jimo.2016058
References:
[1]

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.

[2] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511721434.
[3]

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

[4]

Y. Chen, The finite-time ruin probability with dependent insurance and financial risks, Journal of Applied Probability, 48 (2011), 1035-1048.  doi: 10.1017/S0021900200008603.

[5]

D. B. 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.

[6]

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

[7]

S. EmmerC. Klüppelberg and R. Korn, Optimal portfolios with bounded capital at risk, Mathematical Finance, 11 (2001), 365-384.  doi: 10.1111/1467-9965.00121.

[8]

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.

[9]

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.

[10]

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.

[11]

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: 10.1017/S0001867800003190.

[12]

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.

[13]

R. Korn, Optimal Portfolios, World Scientific, Singapore, 1997.

[14]

J. LiQ. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Advances in Applied Probability, 42 (2010), 1126-1146.  doi: 10.1017/S0001867800004559.

[15]

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.

[16]

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.

[17] K. Sato, Lévy Processes and Infinite Divisibility, Cambridge University Press, Cambridge, 1999. 
[18]

D. WangC. Su and Y. Zeng, Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory, Science in China Series A: Mathematics, 48 (2005), 1379-1394.  doi: 10.1360/022004-16.

show all references

References:
[1]

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.

[2] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511721434.
[3]

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

[4]

Y. Chen, The finite-time ruin probability with dependent insurance and financial risks, Journal of Applied Probability, 48 (2011), 1035-1048.  doi: 10.1017/S0021900200008603.

[5]

D. B. 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.

[6]

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

[7]

S. EmmerC. Klüppelberg and R. Korn, Optimal portfolios with bounded capital at risk, Mathematical Finance, 11 (2001), 365-384.  doi: 10.1111/1467-9965.00121.

[8]

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.

[9]

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.

[10]

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.

[11]

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: 10.1017/S0001867800003190.

[12]

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.

[13]

R. Korn, Optimal Portfolios, World Scientific, Singapore, 1997.

[14]

J. LiQ. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Advances in Applied Probability, 42 (2010), 1126-1146.  doi: 10.1017/S0001867800004559.

[15]

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.

[16]

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.

[17] K. Sato, Lévy Processes and Infinite Divisibility, Cambridge University Press, Cambridge, 1999. 
[18]

D. WangC. Su and Y. Zeng, Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory, Science in China Series A: Mathematics, 48 (2005), 1379-1394.  doi: 10.1360/022004-16.

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