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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 |
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.
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. |

0.64215 | 0.33207 | 0.15448 | 0.06979 | 0.03119 | 0.01388 | 0.00617 | 0.00274 | 0.00121 | 0.00054 | |
0.41629 | 0.19545 | 0.08830 | 0.03946 | 0.01757 | 0.00781 | 0.00347 | 0.00154 | 0.00068 | 0.00030 |
0.64215 | 0.33207 | 0.15448 | 0.06979 | 0.03119 | 0.01388 | 0.00617 | 0.00274 | 0.00121 | 0.00054 | |
0.41629 | 0.19545 | 0.08830 | 0.03946 | 0.01757 | 0.00781 | 0.00347 | 0.00154 | 0.00068 | 0.00030 |
0.70195 | 0.42231 | 0.23766 | 0.13018 | 0.07048 | 0.03797 | 0.02041 | 0.01096 | 0.00588 | 0.00316 | |
0.48671 | 0.27859 | 0.15357 | 0.08339 | 0.04500 | 0.02421 | 0.01301 | 0.00699 | 0.00375 | 0.00201 |
0.70195 | 0.42231 | 0.23766 | 0.13018 | 0.07048 | 0.03797 | 0.02041 | 0.01096 | 0.00588 | 0.00316 | |
0.48671 | 0.27859 | 0.15357 | 0.08339 | 0.04500 | 0.02421 | 0.01301 | 0.00699 | 0.00375 | 0.00201 |
0.75332 | 0.55726 | 0.40147 | 0.28352 | 0.19789 | 0.13718 | 0.09472 | 0.06524 | 0.04488 | 0.03084 | |
0.59185 | 0.43080 | 0.30693 | 0.21544 | 0.14989 | 0.10374 | 0.07158 | 0.04929 | 0.03390 | 0.02330 |
0.75332 | 0.55726 | 0.40147 | 0.28352 | 0.19789 | 0.13718 | 0.09472 | 0.06524 | 0.04488 | 0.03084 | |
0.59185 | 0.43080 | 0.30693 | 0.21544 | 0.14989 | 0.10374 | 0.07158 | 0.04929 | 0.03390 | 0.02330 |
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