American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimal reinsurance and investment strategy with two piece utility function
  • JIMO Home
  • This Issue
  • Next Article
    Multiple common due-dates assignment and optimal maintenance activity scheduling with linear deteriorating jobs
April  2017, 13(2): 721-736. doi: 10.3934/jimo.2016043

On a perturbed compound Poisson model with varying premium rates

Zhimin Zhang 1, , Yang Yang 2, and  Chaolin Liu 1,,

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.

Keywords: Varying premium rates, Gerber-Shiu function, Laplace transform, ruin.
Mathematics Subject Classification: Primary: 91B30; Secondary: 62P05.
Citation: Zhimin Zhang, Yang Yang, Chaolin Liu. On a perturbed compound Poisson model with varying premium rates. Journal of Industrial & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer-Verlag, Berlin, 2006. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  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) $f_X(x)=\frac{1}{6} e^{-\frac{1}{2}x}+\frac{4}{3} e^{-2x}$
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  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
Table Options

Download as excel

$u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$
$\phi_1(u)$ 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
Table 2.  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
Table Options

Download as excel

$u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$
$\phi_1(u)$ 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
Table 3.  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
Table Options

Download as excel

$u$ $0$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$
$\phi_1(u)$ 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
[1]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039
[2]

Srdjan Stojanovic. Interest rates risk-premium and shape of the yield curve. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1603-1615. doi: 10.3934/dcdsb.2016013
[3]

Chihiro Matsuoka, Koichi Hiraide. Special functions created by Borel-Laplace transform of Hénon map. Electronic Research Announcements, 2011, 18: 1-11. doi: 10.3934/era.2011.18.1
[4]

Amir Khan, Asaf Khan, Tahir Khan, Gul Zaman. Extension of triple Laplace transform for solving fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 755-768. doi: 10.3934/dcdss.2020042
[5]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657
[6]

Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[7]

Rakesh Pilkar, Erik M. Bollt, Charles Robinson. Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1085-1097. doi: 10.3934/mbe.2011.8.1085
[8]

Xin Jiang, Kam Chuen Yuen, Mi Chen. Optimal investment and reinsurance with premium control. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019080
[9]

Bin Guo, Wenjie Gao. Finite-time blow-up and extinction rates of solutions to an initial Neumann problem involving the $p(x,t)-Laplace$ operator and a non-local term. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 715-730. doi: 10.3934/dcds.2016.36.715
[10]

Wing Yan Lee, Fangda Liu. Analysis of a dynamic premium strategy: From theoretical and marketing perspectives. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1545-1564. doi: 10.3934/jimo.2018020
[11]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[12]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325
[13]

Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111
[14]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
[15]

Xiaoqing Liang, Lihua Bai. Minimizing expected time to reach a given capital level before ruin. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1771-1791. doi: 10.3934/jimo.2017018
[16]

Yinghui Dong, Guojing Wang. Ruin probability for renewal risk model with negative risk sums. Journal of Industrial & Management Optimization, 2006, 2 (2) : 229-236. doi: 10.3934/jimo.2006.2.229
[17]

Emilija Bernackaitė, Jonas Šiaulys. The finite-time ruin probability for an inhomogeneous renewal risk model. Journal of Industrial & Management Optimization, 2017, 13 (1) : 207-222. doi: 10.3934/jimo.2016012
[18]

Bing Liu, Ming Zhou. Robust portfolio selection for individuals: Minimizing the probability of lifetime ruin. Journal of Industrial & Management Optimization, 2017, 13 (5) : 0-0. doi: 10.3934/jimo.2020005
[19]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649
[20]

Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (26)
  • HTML views (337)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences