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April  2017, 13(2): 721-736. doi: 10.3934/jimo.2016043

## On a perturbed compound Poisson model with varying premium rates

 1 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 2 Department of Statistics, Nanjing Audit University, Nanjing 211815, China

* Corresponding author: Chaolin Liu

Received  November 2015 Revised  January 2016 Published  August 2016

Fund Project: Z.M. Zhang was supported by the National Natural Science Foundation of China [11471058,11101451,11301303] and the Natural Science Foundation Project of CQ CSTC of China [cstc2014jcyjA00007]. The research of Y. Yang was supported by National Natural Science Foundation of China (No. 71471090), the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 14YJCZH182), China Postdoctoral Science Foundation (No. 2014T70449,2012M520964), Natural Science Foundation of Jiangsu Province of China (No. BK20131339), the Major Research Plan of Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJA110001), Qing Lan Project, PAPD, Program of Excellent Science and Technology Innovation Team of the Jiangsu Higher Education Institu-tions of China, Project of Construction for Superior Subjects of Statistics of Jiangsu Higher Education Institutions, Project of the Key Lab of Financial Engineering of Jiangsu Province. The research of C.L. Liu was supported by the Fundamental Research Funds for the Central Universities(No. 106112015CDJXY100006).

In this paper, we consider a perturbed compound Poisson model with varying premium rates. The surplus process is observed at a sequence of review times. The effective premium rate is adjusted according to the surplus increment between the inter-review times. We study the Gerber-Shiu functions by Laplace transform method. When the claim size density is a combination of exponentials, the explicit expressions for the Laplace transforms of ruin time are derived.

Citation: Zhimin Zhang, Yang Yang, Chaolin Liu. On a perturbed compound Poisson model with varying premium rates. Journal of Industrial and Management Optimization, 2017, 13 (2) : 721-736. doi: 10.3934/jimo.2016043
##### References:
 [1] H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randommized observation times for the compound Poisson risk model: Dividends, Astin Bulletin, 41 (2011), 645-672. [2] H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scandinavian Actuarial Journal, 6 (2013), 424-452.  doi: 10.1080/03461238.2011.624686. [3] S. Asmussen, F. Avram and M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281.  doi: 10.2143/AST.32.2.1029. [4] S. Chadjiconstantinidis and A. D. Papaioannou, On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy, Journal of Computational and Applied Mathematics, 253 (2013), 26-50.  doi: 10.1016/j.cam.2013.02.014. [5] H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, 1970 (1970), 205-210. [6] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.  doi: 10.1080/10920277.1998.10595671. [7] V. Klimenok, On the modification of Rouche's theorem for the queuing theory problems, ueuing Systems, 38 (2001), 431-434.  doi: 10.1023/A:1010999928701. [8] A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, 2006. [9] S. Li, D. Landriault and C. Lemieux, A risk model with varying premiums: Its risk management implications, Insurance: Mathematics and Economics, 60 (2015), 38-46.  doi: 10.1016/j.insmatheco.2014.10.010. [10] C. Liu and Z. Zhang, On a generalized Gerber-Shiu function in a compound Poisson model perturbed by diffusion, Advances in Difference Equations, 2015 (2015), 1-20.  doi: 10.1186/s13662-015-0378-x. [11] D. A. Stanford, F. Avram, A. L. Badescu, L. Breuer and A. Da Silva Soares, Phase-type approximations to finite-time ruin probabilities in the Sparre-Anderson and stationary renewal risk models, ASTIN Bulletin, 35 (2005), 131-144.  doi: 10.2143/AST.35.1.583169. [12] D. A. Stanford, K. Yu and J. Ren, Erlangian approximation to finite time ruin probabilities in perturbed risk models, Scandinavian Actuarial Journal, 2011 (2011), 38-58.  doi: 10.1080/03461230903421492. [13] C. C. L. Tsai, On the discounted distribution functions of the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 28 (2001), 401-419.  doi: 10.1016/S0167-6687(01)00067-1. [14] C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66.  doi: 10.1016/S0167-6687(01)00096-8. [15] C. Yang and K. P. Sendova, The ruin time under the Sparre-Andersen dual model, Insurance: Mathematics and Economics, 54 (2014), 28-40.  doi: 10.1016/j.insmatheco.2013.10.012. [16] Z. Zhang and E. C. K. Cheung, The Markov additive risk process under an Erlangized dividend barrier strategy, Methodology and Computing in Applied Probability, 18 (2016), 275-306.  doi: 10.1007/s11009-014-9414-7. [17] Z. Zhang and H. Yang, Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times, Journal of Computational and Applied Mathematics, 235 (2011), 1189-1204.  doi: 10.1016/j.cam.2010.08.003. [18] Z. Zhang, H. Yang and H. Yang, On a Sparre Andersen risk model with time-dependent claim sizes and jump-diffusion perturbation, Methodology and Computing in Applied Probability, 14 (2012), 973-995.  doi: 10.1007/s11009-011-9215-1. [19] M. Zhou and J. Cai, A perturbed risk model with dependence between premium rates and claim sizes, Insurance: Mathematics and Economics, 45 (2009), 382-392.  doi: 10.1016/j.insmatheco.2009.08.008.

show all references

##### References:
 [1] H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randommized observation times for the compound Poisson risk model: Dividends, Astin Bulletin, 41 (2011), 645-672. [2] H. Albrecher, E. C. K. Cheung and S. Thonhauser, Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scandinavian Actuarial Journal, 6 (2013), 424-452.  doi: 10.1080/03461238.2011.624686. [3] S. Asmussen, F. Avram and M. Usabel, Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bulletin, 32 (2002), 267-281.  doi: 10.2143/AST.32.2.1029. [4] S. Chadjiconstantinidis and A. D. Papaioannou, On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy, Journal of Computational and Applied Mathematics, 253 (2013), 26-50.  doi: 10.1016/j.cam.2013.02.014. [5] H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Skandinavisk Aktuarietidskrift, 1970 (1970), 205-210. [6] H. U. Gerber and E. S. W. Shiu, On the time value of ruin, North American Actuarial Journal, 2 (1998), 48-78.  doi: 10.1080/10920277.1998.10595671. [7] V. Klimenok, On the modification of Rouche's theorem for the queuing theory problems, ueuing Systems, 38 (2001), 431-434.  doi: 10.1023/A:1010999928701. [8] A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, 2006. [9] S. Li, D. Landriault and C. Lemieux, A risk model with varying premiums: Its risk management implications, Insurance: Mathematics and Economics, 60 (2015), 38-46.  doi: 10.1016/j.insmatheco.2014.10.010. [10] C. Liu and Z. Zhang, On a generalized Gerber-Shiu function in a compound Poisson model perturbed by diffusion, Advances in Difference Equations, 2015 (2015), 1-20.  doi: 10.1186/s13662-015-0378-x. [11] D. A. Stanford, F. Avram, A. L. Badescu, L. Breuer and A. Da Silva Soares, Phase-type approximations to finite-time ruin probabilities in the Sparre-Anderson and stationary renewal risk models, ASTIN Bulletin, 35 (2005), 131-144.  doi: 10.2143/AST.35.1.583169. [12] D. A. Stanford, K. Yu and J. Ren, Erlangian approximation to finite time ruin probabilities in perturbed risk models, Scandinavian Actuarial Journal, 2011 (2011), 38-58.  doi: 10.1080/03461230903421492. [13] C. C. L. Tsai, On the discounted distribution functions of the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 28 (2001), 401-419.  doi: 10.1016/S0167-6687(01)00067-1. [14] C. C. L. Tsai and G. E. Willmot, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: Mathematics and Economics, 30 (2002), 51-66.  doi: 10.1016/S0167-6687(01)00096-8. [15] C. Yang and K. P. Sendova, The ruin time under the Sparre-Andersen dual model, Insurance: Mathematics and Economics, 54 (2014), 28-40.  doi: 10.1016/j.insmatheco.2013.10.012. [16] Z. Zhang and E. C. K. Cheung, The Markov additive risk process under an Erlangized dividend barrier strategy, Methodology and Computing in Applied Probability, 18 (2016), 275-306.  doi: 10.1007/s11009-014-9414-7. [17] Z. Zhang and H. Yang, Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times, Journal of Computational and Applied Mathematics, 235 (2011), 1189-1204.  doi: 10.1016/j.cam.2010.08.003. [18] Z. Zhang, H. Yang and H. Yang, On a Sparre Andersen risk model with time-dependent claim sizes and jump-diffusion perturbation, Methodology and Computing in Applied Probability, 14 (2012), 973-995.  doi: 10.1007/s11009-011-9215-1. [19] M. Zhou and J. Cai, A perturbed risk model with dependence between premium rates and claim sizes, Insurance: Mathematics and Economics, 45 (2009), 382-392.  doi: 10.1016/j.insmatheco.2009.08.008.
Ruin probabilities for Erlang(2) inter-review times. (a) $f_X(x)=3e^{-1.5x}-3 e^{-3x}$; (b) $f_X(x)=e^{-x}$; (c) $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$
Exact values of ruin probabilities when $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$
 $u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $\phi_1(u)$ 0.64215 0.33207 0.15448 0.06979 0.03119 0.01388 0.00617 0.00274 0.00121 0.00054 $\phi_2(u)$ 0.41629 0.19545 0.08830 0.03946 0.01757 0.00781 0.00347 0.00154 0.00068 0.00030
 $u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $\phi_1(u)$ 0.64215 0.33207 0.15448 0.06979 0.03119 0.01388 0.00617 0.00274 0.00121 0.00054 $\phi_2(u)$ 0.41629 0.19545 0.08830 0.03946 0.01757 0.00781 0.00347 0.00154 0.00068 0.00030
Exact values of ruin probabilities when $f_X(x)=e^{-x}$
 $u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $\phi_1(u)$ 0.70195 0.42231 0.23766 0.13018 0.07048 0.03797 0.02041 0.01096 0.00588 0.00316 $\phi_2(u)$ 0.48671 0.27859 0.15357 0.08339 0.04500 0.02421 0.01301 0.00699 0.00375 0.00201
 $u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $\phi_1(u)$ 0.70195 0.42231 0.23766 0.13018 0.07048 0.03797 0.02041 0.01096 0.00588 0.00316 $\phi_2(u)$ 0.48671 0.27859 0.15357 0.08339 0.04500 0.02421 0.01301 0.00699 0.00375 0.00201
Exact values of ruin probabilities when $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$
 $u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $\phi_1(u)$ 0.75332 0.55726 0.40147 0.28352 0.19789 0.13718 0.09472 0.06524 0.04488 0.03084 $\phi_2(u)$ 0.59185 0.43080 0.30693 0.21544 0.14989 0.10374 0.07158 0.04929 0.03390 0.02330
 $u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $\phi_1(u)$ 0.75332 0.55726 0.40147 0.28352 0.19789 0.13718 0.09472 0.06524 0.04488 0.03084 $\phi_2(u)$ 0.59185 0.43080 0.30693 0.21544 0.14989 0.10374 0.07158 0.04929 0.03390 0.02330
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