
-
Previous Article
$ Z $-eigenvalue exclusion theorems for tensors
- JIMO Home
- This Issue
-
Next Article
Impact of risk aversion on two-echelon supply chain systems with carbon emission reduction constraints
On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy
1. | School of Economics, Southwest University of Political Science and Law, Chongqing 401120, China |
2. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
In this paper, we model the insurance company's surplus flow by a perturbed compound Poisson model. Suppose that at a sequence of random time points, the insurance company observes the surplus to decide dividend payments. If the observed surplus level is larger than the maximum of a threshold $ b>0 $ and the last observed level (after dividends payment if possible), then a fraction $ 0<\theta<1 $ of the excess amount is paid out as a lump sum dividend. We assume that the solvency is also discretely monitored at these observation times, so that the surplus process stops when the observed value becomes negative. Integro-differential equations for the expected discounted dividend payments before ruin and the Gerber-Shiu expected discounted penalty function are derived, and solutions are also analyzed by Laplace transform method. Numerical examples are given to illustrate the applicability of our results.
References:
[1] |
H. Albrecher, N. Bäuerle and S. Thonhauser,
Optimal dividend-payout in random discrete time, Statistics and Risk Modeling, 28 (2011a), 251-276.
doi: 10.1524/stnd.2011.1097. |
[2] |
H. Albrecher, E. C. K. Cheung and S. Thonhauser,
Randomized observation periods for the compound Poisson risk model: Dividends, ASTIN Bulletin, 41 (2011b), 645-672.
|
[3] |
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. |
[4] |
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. |
[5] |
B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo,
On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113.
doi: 10.1016/j.insmatheco.2012.10.008. |
[6] |
B. Avanzi, V. Tu and B. Wong,
On optimal periodic dividend strategies in the dual model with diffusion, Insurance: Mathematics and Economics, 55 (2014), 210-224.
doi: 10.1016/j.insmatheco.2014.01.005. |
[7] |
B. Avanzi, W. Tu and B. Wong,
Optimal dividends under Erlang(2) inter-dividend decision times, Insurance: Mathematics and Economics, 79 (2018), 225-242.
doi: 10.1016/j.insmatheco.2018.01.009. |
[8] |
E. C. K. Cheung and Z. Zhang,
Periodic threshold-type dividend strategy in the compound Poisson risk model, Sandinavian Actuarial Journal, 1 (2019), 1-31.
doi: 10.1080/03461238.2018.1481454. |
[9] |
B. de Finetti, Su un' impostazione alternativa dell teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. Google Scholar |
[10] |
D. C. M. Dickson and C. Hipp,
On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29 (2001), 333-344.
doi: 10.1016/S0167-6687(01)00091-9. |
[11] |
H. Dong, C. Yin and H. Dai,
Spectrally negative Lévy risk model under Erlangized barrier strategy, Journal of Computational and Applied Mathematics, 351 (2019), 101-116.
doi: 10.1016/j.cam.2018.11.001. |
[12] |
F. Dufresne and H. U. Gerber,
Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10 (1991), 51-59.
doi: 10.1016/0167-6687(91)90023-Q. |
[13] |
H. Gao and C. Yin,
The perturbed Sparre Andersen model with a threshold dividend strategy, Journal of Computational and Applied Mathematics, 220 (2008), 394-408.
doi: 10.1016/j.cam.2007.08.015. |
[14] |
H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Scandinavian Actuarial Journal, (1970), 205–210.
doi: 10.1080/03461238.1970.10405664. |
[15] |
H. U. Gerber and B. Landry,
On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: Mathematics and Economics, 22 (1998), 263-276.
doi: 10.1016/S0167-6687(98)00014-6. |
[16] |
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. |
[17] |
H. U. Gerber and E. S. W. Shiu,
Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.
doi: 10.1080/10920277.2004.10596125. |
[18] |
A. E. Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures, Second edition. Universitext. Springer, Heidelberg, 2014.
doi: 10.1007/978-3-642-37632-0. |
[19] |
D. Landriault, B. Li, J. T. Y. Wong and D. Xu,
Poissonian potential measures for Lévy risk models, Insurance: Mathematics and Economics, 82 (2018), 152-166.
doi: 10.1016/j.insmatheco.2018.07.004. |
[20] |
S. Li,
The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion, Scandinavian Actuarial Journal, 2 (2006), 73-85.
doi: 10.1080/03461230600589237. |
[21] |
S. Li and J. Garrido,
On ruin for the Eralng(n) risk model, Insurance: Mathematics and Economics, 34 (2004), 391-408.
doi: 10.1016/j.insmatheco.2004.01.002. |
[22] |
X. S. Lin and K. P. Pavlova,
The compound Poisson risk model with a threshold dividend strategy, Insurance: Mathematics and Economics, 38 (2006), 57-80.
doi: 10.1016/j.insmatheco.2005.08.001. |
[23] |
X. S. Lin, G. E. Willmot and S. Drekic,
The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 33 (2003), 551-566.
doi: 10.1016/j.insmatheco.2003.08.004. |
[24] |
W. Su, Y. Yong and Z. Zhang,
Estimating the Gerber-Shiu function in the perturbed compound Poisson model by Laguerre series expansion, Journal of Mathematical Analysis and Applications, 469 (2019), 705-729.
doi: 10.1016/j.jmaa.2018.09.033. |
[25] |
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. |
[26] |
N. Wan, Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion, Insurance: Mathematics and Economics, 40 (2007), 509–523.
doi: 10.1016/j.insmatheco.2006.08.002. |
[27] |
H. Yang and Z. Zhang,
The perturbed compound Poisson risk model with multi-layer dividend strategy, Statistics and Probability Letters, 79 (2009), 70-78.
doi: 10.1016/j.spl.2008.07.017. |
[28] |
C. Yin, Y. Shen and Y. Wen,
Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52.
doi: 10.1016/j.cam.2012.12.004. |
[29] |
C. Yin and Y. Wen,
Optimal dividend problem with a terminal value for spectrally positive Levy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.
doi: 10.1016/j.insmatheco.2013.09.019. |
[30] |
C. Yin, Y. Wen and Y. Zhao,
On the optimal dividend problem for a spectrally positive Lévy process, ASTIN Bulletin, 44 (2014), 635-651.
doi: 10.1017/asb.2014.12. |
[31] |
Z. Zhang,
On a risk model with randomized dividend-decision times, Journal of Insdustrial and Management Optimization, 10 (2014), 1041-1058.
doi: 10.3934/jimo.2014.10.1041. |
[32] |
Z. Zhang,
Estimating the Gerber-Shiu function by Fourier-Sinc series expansion, Sandinavian Actuarial Journal, 10 (2017), 1-22.
doi: 10.1080/03461238.2016.1268541. |
[33] |
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. |
[34] |
Z. Zhang and E. C. K. Cheung,
A note on a Lévy insurance risk model under periodic dividend decisions, Journal of Industrial and Management Optimization, 14 (2018), 35-63.
doi: 10.3934/jimo.2017036. |
[35] |
Z. Zhang, E. C. K. Cheung and H. Yang,
Lévy insurance risk process with Poissonian taxation, Scandinavian Actuarial Journal, 1 (2017), 51-87.
doi: 10.1080/03461238.2015.1062042. |
[36] |
Z. Zhang, E. C. K. Cheung and H. Yang,
On the compound Poisson risk model with periodic capital injection, ASTIN Bulletin, 48 (2018), 435-477.
doi: 10.1017/asb.2017.22. |
[37] |
Z. Zhang and C. Liu,
Moments of discounted dividend payments in a risk model with randomized dividend-decision times, Frontiers of Mathematics in China, 12 (2017), 493-513.
doi: 10.1007/s11464-016-0609-9. |
[38] |
Y. Zhao, R. Wang and C. Yin,
Optimal dividends and capital injections for a spectrally positive Lévy process, Journal of Industrial and Management Optimization, 13 (2017), 1-21.
doi: 10.3934/jimo.2016001. |
show all references
References:
[1] |
H. Albrecher, N. Bäuerle and S. Thonhauser,
Optimal dividend-payout in random discrete time, Statistics and Risk Modeling, 28 (2011a), 251-276.
doi: 10.1524/stnd.2011.1097. |
[2] |
H. Albrecher, E. C. K. Cheung and S. Thonhauser,
Randomized observation periods for the compound Poisson risk model: Dividends, ASTIN Bulletin, 41 (2011b), 645-672.
|
[3] |
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. |
[4] |
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. |
[5] |
B. Avanzi, E. C. K. Cheung, B. Wong and J.-K. Woo,
On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance: Mathematics and Economics, 52 (2013), 98-113.
doi: 10.1016/j.insmatheco.2012.10.008. |
[6] |
B. Avanzi, V. Tu and B. Wong,
On optimal periodic dividend strategies in the dual model with diffusion, Insurance: Mathematics and Economics, 55 (2014), 210-224.
doi: 10.1016/j.insmatheco.2014.01.005. |
[7] |
B. Avanzi, W. Tu and B. Wong,
Optimal dividends under Erlang(2) inter-dividend decision times, Insurance: Mathematics and Economics, 79 (2018), 225-242.
doi: 10.1016/j.insmatheco.2018.01.009. |
[8] |
E. C. K. Cheung and Z. Zhang,
Periodic threshold-type dividend strategy in the compound Poisson risk model, Sandinavian Actuarial Journal, 1 (2019), 1-31.
doi: 10.1080/03461238.2018.1481454. |
[9] |
B. de Finetti, Su un' impostazione alternativa dell teoria collettiva del rischio, Transactions of the XVth International Congress of Actuaries, 2 (1957), 433-443. Google Scholar |
[10] |
D. C. M. Dickson and C. Hipp,
On the time to ruin for Erlang(2) risk processes, Insurance: Mathematics and Economics, 29 (2001), 333-344.
doi: 10.1016/S0167-6687(01)00091-9. |
[11] |
H. Dong, C. Yin and H. Dai,
Spectrally negative Lévy risk model under Erlangized barrier strategy, Journal of Computational and Applied Mathematics, 351 (2019), 101-116.
doi: 10.1016/j.cam.2018.11.001. |
[12] |
F. Dufresne and H. U. Gerber,
Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: Mathematics and Economics, 10 (1991), 51-59.
doi: 10.1016/0167-6687(91)90023-Q. |
[13] |
H. Gao and C. Yin,
The perturbed Sparre Andersen model with a threshold dividend strategy, Journal of Computational and Applied Mathematics, 220 (2008), 394-408.
doi: 10.1016/j.cam.2007.08.015. |
[14] |
H. U. Gerber, An extension of the renewal equation and its application in the collective theory of risk, Scandinavian Actuarial Journal, (1970), 205–210.
doi: 10.1080/03461238.1970.10405664. |
[15] |
H. U. Gerber and B. Landry,
On the discounted penalty at ruin in a jump-diffusion and the perpetual put option, Insurance: Mathematics and Economics, 22 (1998), 263-276.
doi: 10.1016/S0167-6687(98)00014-6. |
[16] |
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. |
[17] |
H. U. Gerber and E. S. W. Shiu,
Optimal dividends: Analysis with Brownian motion, North American Actuarial Journal, 8 (2004), 1-20.
doi: 10.1080/10920277.2004.10596125. |
[18] |
A. E. Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures, Second edition. Universitext. Springer, Heidelberg, 2014.
doi: 10.1007/978-3-642-37632-0. |
[19] |
D. Landriault, B. Li, J. T. Y. Wong and D. Xu,
Poissonian potential measures for Lévy risk models, Insurance: Mathematics and Economics, 82 (2018), 152-166.
doi: 10.1016/j.insmatheco.2018.07.004. |
[20] |
S. Li,
The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion, Scandinavian Actuarial Journal, 2 (2006), 73-85.
doi: 10.1080/03461230600589237. |
[21] |
S. Li and J. Garrido,
On ruin for the Eralng(n) risk model, Insurance: Mathematics and Economics, 34 (2004), 391-408.
doi: 10.1016/j.insmatheco.2004.01.002. |
[22] |
X. S. Lin and K. P. Pavlova,
The compound Poisson risk model with a threshold dividend strategy, Insurance: Mathematics and Economics, 38 (2006), 57-80.
doi: 10.1016/j.insmatheco.2005.08.001. |
[23] |
X. S. Lin, G. E. Willmot and S. Drekic,
The classical risk model with a constant dividend barrier: Analysis of the Gerber-Shiu discounted penalty function, Insurance: Mathematics and Economics, 33 (2003), 551-566.
doi: 10.1016/j.insmatheco.2003.08.004. |
[24] |
W. Su, Y. Yong and Z. Zhang,
Estimating the Gerber-Shiu function in the perturbed compound Poisson model by Laguerre series expansion, Journal of Mathematical Analysis and Applications, 469 (2019), 705-729.
doi: 10.1016/j.jmaa.2018.09.033. |
[25] |
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. |
[26] |
N. Wan, Dividend payments with a threshold strategy in the compound Poisson risk model perturbed by diffusion, Insurance: Mathematics and Economics, 40 (2007), 509–523.
doi: 10.1016/j.insmatheco.2006.08.002. |
[27] |
H. Yang and Z. Zhang,
The perturbed compound Poisson risk model with multi-layer dividend strategy, Statistics and Probability Letters, 79 (2009), 70-78.
doi: 10.1016/j.spl.2008.07.017. |
[28] |
C. Yin, Y. Shen and Y. Wen,
Exit problems for jump processes with applications to dividend problems, Journal of Computational and Applied Mathematics, 245 (2013), 30-52.
doi: 10.1016/j.cam.2012.12.004. |
[29] |
C. Yin and Y. Wen,
Optimal dividend problem with a terminal value for spectrally positive Levy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.
doi: 10.1016/j.insmatheco.2013.09.019. |
[30] |
C. Yin, Y. Wen and Y. Zhao,
On the optimal dividend problem for a spectrally positive Lévy process, ASTIN Bulletin, 44 (2014), 635-651.
doi: 10.1017/asb.2014.12. |
[31] |
Z. Zhang,
On a risk model with randomized dividend-decision times, Journal of Insdustrial and Management Optimization, 10 (2014), 1041-1058.
doi: 10.3934/jimo.2014.10.1041. |
[32] |
Z. Zhang,
Estimating the Gerber-Shiu function by Fourier-Sinc series expansion, Sandinavian Actuarial Journal, 10 (2017), 1-22.
doi: 10.1080/03461238.2016.1268541. |
[33] |
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. |
[34] |
Z. Zhang and E. C. K. Cheung,
A note on a Lévy insurance risk model under periodic dividend decisions, Journal of Industrial and Management Optimization, 14 (2018), 35-63.
doi: 10.3934/jimo.2017036. |
[35] |
Z. Zhang, E. C. K. Cheung and H. Yang,
Lévy insurance risk process with Poissonian taxation, Scandinavian Actuarial Journal, 1 (2017), 51-87.
doi: 10.1080/03461238.2015.1062042. |
[36] |
Z. Zhang, E. C. K. Cheung and H. Yang,
On the compound Poisson risk model with periodic capital injection, ASTIN Bulletin, 48 (2018), 435-477.
doi: 10.1017/asb.2017.22. |
[37] |
Z. Zhang and C. Liu,
Moments of discounted dividend payments in a risk model with randomized dividend-decision times, Frontiers of Mathematics in China, 12 (2017), 493-513.
doi: 10.1007/s11464-016-0609-9. |
[38] |
Y. Zhao, R. Wang and C. Yin,
Optimal dividends and capital injections for a spectrally positive Lévy process, Journal of Industrial and Management Optimization, 13 (2017), 1-21.
doi: 10.3934/jimo.2016001. |




[1] |
Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277 |
[2] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[3] |
Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 |
[4] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[5] |
Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709 |
[6] |
Martial Agueh, Reinhard Illner, Ashlin Richardson. Analysis and simulations of a refined flocking and swarming model of Cucker-Smale type. Kinetic & Related Models, 2011, 4 (1) : 1-16. doi: 10.3934/krm.2011.4.1 |
[7] |
Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363 |
[8] |
Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018 |
[9] |
Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 |
[10] |
Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 |
[11] |
V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153 |
[12] |
Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475 |
[13] |
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 |
[14] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[15] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[16] |
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 |
[17] |
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 |
[18] |
Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094 |
[19] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[20] |
Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]