Some Results on a Double Compound Poisson-Geometric Risk Model with Interference

Author(s)
Dezhi Yan

ABSTRACT

In this paper, we study the actual operating of an insurance company with random income. A double compound Poisson-Geometric risk model with interference was established. By using the martingale method, the adjustment coefficient equation, the formula and the upper bound of ruin probability, the time to reach a given level in this new risk mo- del were obtained.

In this paper, we study the actual operating of an insurance company with random income. A double compound Poisson-Geometric risk model with interference was established. By using the martingale method, the adjustment coefficient equation, the formula and the upper bound of ruin probability, the time to reach a given level in this new risk mo- del were obtained.

Cite this paper

D. Yan, "Some Results on a Double Compound Poisson-Geometric Risk Model with Interference,"*Theoretical Economics Letters*, Vol. 2 No. 1, 2012, pp. 45-49. doi: 10.4236/tel.2012.21008.

D. Yan, "Some Results on a Double Compound Poisson-Geometric Risk Model with Interference,"

References

[1] J. Grandell, “Aspects of Risk Theory,” Springer, Berlin, 1991. doi:10.1007/978-1-4613-9058-9

[2] H. U. Gerber, “An Introduction to Mathematical Risk Theory,” Monograph Series, Vol. 8, S. S. Heubner Foundation, Philadelphia, 1979.

[3] F. Dufresne and H. U. Gerber, “Risk Theory for the Compound Poisson Process That Is Disturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 10, 1991, pp. 51-59. doi:10.1016/0167-6687(91)90023-Q

[4] N. Veraverbeke, “Asymptotic Estimations for the Probability of Ruin in a Pois-son Model with Diffusion,” Insurance: Mathematics and Economics, Vol. 13, 1993, pp. 57-62. doi:10.1016/0167-6687(93)90535-W

[5] H. U. Gerber and B. Landry, “On the Discounted Penalty at Ruin in a Jump-Diffusion and the Perturbed Put Option,” Insurance: Mathematics and Economics, Vol. 22, 1998, pp. 263-276. doi:10.1016/S0167-6687(98)00014-6

[6] H. U. Gerber and E. S. W. Shiu, “On the Time Value of Ruin,” North American Actuarial Journal, Vol. 2, No. 1, 1998, pp. 48-78.

[7] G. J. Wang and R. Wu, “Some Distributions for Classic Risk Process That Is Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 26, No. 1, 2000, pp. 15-24. doi:10.1016/S0167-6687(99)00035-9

[8] G. J. Wang, “A Decomposition of the Ruin Probability for the Risk Process Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 28, No. 1, 2001, pp. 49-59. doi:10.1016/S0167-6687(00)00065-2

[9] C. C.-L. Tsai, “On the Discounted Distribution Functions of the Surplus Process,” Insurance: Mathematics and Economics, Vol. 28, No. 3, 2001, pp. 401-419. doi:10.1016/S0167-6687(01)00067-1

[10] C. C.-L. Tsai, “A Generalized Defective Renewal Equation for the Surplus Process Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 30, No. 1, 2002, pp. 51-66. doi:10.1016/S0167-6687(01)00096-8

[11] C. C.-L. Tsai, “On the Expectations of the Present Values of the Time of Ruin Perturbed by Diffusion,” Insurance: Mathematics and Eco-nomics, Vol. 32, No. 3, 2003, pp. 413-429. doi:10.1016/S0167-6687(03)00130-6

[12] C. S. Zhang and G. J. Wang, “The Joint Density Function of Three Characteristics on Jump-Diffusion Risk Process,” Insurance: Mathematics and Economics, Vol. 32, No. 3, 2003, pp. 445-455. doi:10.1016/S0167-6687(03)00133-1

[13] S. N. Chiu and C. C. Yin, “The Time of Ruin, the Surplus Prior to Ruin and the Deficit at Ruin for the Classical Process Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 33, No. 1, 2003, pp. 59-66. doi:10.1016/S0167-6687(03)00143-4

[14] J. Paulsen, “Risk Theory in a Stochastic Environment,” Stochastic Process and Their Applications, Vol. 21, 1993, pp. 327-361.

[15] J. Paulsen, “Ruin Theory with Compounding Assets: A Survey,” Insurance: Mathematics and Economics, Vol. 22, No. 1, 1998, pp. 3-16. doi:10.1016/S0167-6687(98)00009-2

[16] J. Paulsen and H. K. Gjessing, “Ruin Theory with Stochastic Return on Invest-ments,” Advances in Applied Probability, Vol. 29, 1997, pp. 965-985. doi:10.2307/1427849

[17] V. Kalashnikov and R. Norberg, “Power Tailed Ruin Probabilities in the Presence of Risky Investments,” Stochastic Process and Their Applications, Vol. 98, 2002, pp. 221-228.

[18] V. E. Bening, V. Yu. Korolev and L. X. Liu, “Asymptotic Behavior of Generalized Risk Processes,” Acta Mathematica Sinica, English Series, Vol. 20, No. 2, 2004, pp. 349-356. doi:10.1007/s10114-003-0244-8

[19] J. Cai, “Ruin Probability and Penalty Functions with Stochastic Rates of Interest,” Stochastic Process and Their Applications, Vol. 112, No. 1, 2004, pp. 53-78.

[20] K. C. Yuen, G. J. Wang and W. Ng Kai, “Ruin Probabilities for a Risk Process with Sto-chastic Return on Investments,” Stochastic Process and Their Applications, Vol. 110, 2004, pp. 259-274

[21] K. C. Yuen, G. J. Wang and R. Wu, “On the Renewal Risk Process with Stochastic Interest,” Stochastic process and Their Applications, Vol. 116, No. 10, 2006, pp. 1496-1510.

[22] G. Temnov, “Risk Process with Random Income,” Journal of Mathematical Sciences, Vol. 123, No. 1, 2004, pp. 3780-3794. doi:10.1023/B:JOTH.0000036319.21285.22

[1] J. Grandell, “Aspects of Risk Theory,” Springer, Berlin, 1991. doi:10.1007/978-1-4613-9058-9

[2] H. U. Gerber, “An Introduction to Mathematical Risk Theory,” Monograph Series, Vol. 8, S. S. Heubner Foundation, Philadelphia, 1979.

[3] F. Dufresne and H. U. Gerber, “Risk Theory for the Compound Poisson Process That Is Disturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 10, 1991, pp. 51-59. doi:10.1016/0167-6687(91)90023-Q

[4] N. Veraverbeke, “Asymptotic Estimations for the Probability of Ruin in a Pois-son Model with Diffusion,” Insurance: Mathematics and Economics, Vol. 13, 1993, pp. 57-62. doi:10.1016/0167-6687(93)90535-W

[5] H. U. Gerber and B. Landry, “On the Discounted Penalty at Ruin in a Jump-Diffusion and the Perturbed Put Option,” Insurance: Mathematics and Economics, Vol. 22, 1998, pp. 263-276. doi:10.1016/S0167-6687(98)00014-6

[6] H. U. Gerber and E. S. W. Shiu, “On the Time Value of Ruin,” North American Actuarial Journal, Vol. 2, No. 1, 1998, pp. 48-78.

[7] G. J. Wang and R. Wu, “Some Distributions for Classic Risk Process That Is Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 26, No. 1, 2000, pp. 15-24. doi:10.1016/S0167-6687(99)00035-9

[8] G. J. Wang, “A Decomposition of the Ruin Probability for the Risk Process Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 28, No. 1, 2001, pp. 49-59. doi:10.1016/S0167-6687(00)00065-2

[9] C. C.-L. Tsai, “On the Discounted Distribution Functions of the Surplus Process,” Insurance: Mathematics and Economics, Vol. 28, No. 3, 2001, pp. 401-419. doi:10.1016/S0167-6687(01)00067-1

[10] C. C.-L. Tsai, “A Generalized Defective Renewal Equation for the Surplus Process Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 30, No. 1, 2002, pp. 51-66. doi:10.1016/S0167-6687(01)00096-8

[11] C. C.-L. Tsai, “On the Expectations of the Present Values of the Time of Ruin Perturbed by Diffusion,” Insurance: Mathematics and Eco-nomics, Vol. 32, No. 3, 2003, pp. 413-429. doi:10.1016/S0167-6687(03)00130-6

[12] C. S. Zhang and G. J. Wang, “The Joint Density Function of Three Characteristics on Jump-Diffusion Risk Process,” Insurance: Mathematics and Economics, Vol. 32, No. 3, 2003, pp. 445-455. doi:10.1016/S0167-6687(03)00133-1

[13] S. N. Chiu and C. C. Yin, “The Time of Ruin, the Surplus Prior to Ruin and the Deficit at Ruin for the Classical Process Perturbed by Diffusion,” Insurance: Mathematics and Economics, Vol. 33, No. 1, 2003, pp. 59-66. doi:10.1016/S0167-6687(03)00143-4

[14] J. Paulsen, “Risk Theory in a Stochastic Environment,” Stochastic Process and Their Applications, Vol. 21, 1993, pp. 327-361.

[15] J. Paulsen, “Ruin Theory with Compounding Assets: A Survey,” Insurance: Mathematics and Economics, Vol. 22, No. 1, 1998, pp. 3-16. doi:10.1016/S0167-6687(98)00009-2

[16] J. Paulsen and H. K. Gjessing, “Ruin Theory with Stochastic Return on Invest-ments,” Advances in Applied Probability, Vol. 29, 1997, pp. 965-985. doi:10.2307/1427849

[17] V. Kalashnikov and R. Norberg, “Power Tailed Ruin Probabilities in the Presence of Risky Investments,” Stochastic Process and Their Applications, Vol. 98, 2002, pp. 221-228.

[18] V. E. Bening, V. Yu. Korolev and L. X. Liu, “Asymptotic Behavior of Generalized Risk Processes,” Acta Mathematica Sinica, English Series, Vol. 20, No. 2, 2004, pp. 349-356. doi:10.1007/s10114-003-0244-8

[19] J. Cai, “Ruin Probability and Penalty Functions with Stochastic Rates of Interest,” Stochastic Process and Their Applications, Vol. 112, No. 1, 2004, pp. 53-78.

[20] K. C. Yuen, G. J. Wang and W. Ng Kai, “Ruin Probabilities for a Risk Process with Sto-chastic Return on Investments,” Stochastic Process and Their Applications, Vol. 110, 2004, pp. 259-274

[21] K. C. Yuen, G. J. Wang and R. Wu, “On the Renewal Risk Process with Stochastic Interest,” Stochastic process and Their Applications, Vol. 116, No. 10, 2006, pp. 1496-1510.

[22] G. Temnov, “Risk Process with Random Income,” Journal of Mathematical Sciences, Vol. 123, No. 1, 2004, pp. 3780-3794. doi:10.1023/B:JOTH.0000036319.21285.22