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

Dezhi Yan

Department of Economic, Shandong Jiaotong University, Jinan, China.

DOI: 10.4236/tel.[phone]                

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.

Keywords

Ruin Probability; Compound Poisson-Geometric Risk Model; Martingale; Stopping Time; Moment Generating Function; Laplace Transform; Adjustment Coefficient Equation

Conflicts of Interest

The authors declare no conflicts of interest.

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