American Institute of Mathematical Sciences

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July  2020, 16(4): 1967-1986. doi: 10.3934/jimo.2019038

On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy

Xuanhua Peng 1, , Wen Su 2, and  Zhimin Zhang 2,,

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

* Corresponding author: Zhimin Zhang

Received  July 2018 Revised  November 2018 Published  July 2020 Early access  May 2019

Fund Project: The research of Xuanhua Peng was supported by the Chongqing Social Science Planning Project (No. 2017YBGL151), the Chongqing Municipal Education Commission Humanities and Social Sciences Research Project (No. 18SKGH006) and the Southwest University of Political Science and Law Research Project (No. 2018XZQN-35). The research of Zhimin Zhang was supported by the National Natural Science Foundation of China (Nos. 11471058, 11871121), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 16YJC910005) and Fundamental Research Funds for the Central Universities (No. 2018CDQYST0016)

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.

Keywords: Dividend payments, Gerber-Shiu function, integro-differential equation, threshold-type dividend strategy, periodic observation.
Mathematics Subject Classification: Primary: 91B30; Secondary: 62P05.
Citation: Xuanhua Peng, Wen Su, Zhimin Zhang. On a perturbed compound Poisson risk model under a periodic threshold-type dividend strategy. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1967-1986. doi: 10.3934/jimo.2019038
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. 

[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. 

[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.

Figure 1.  $ V(u;b) $ as a function of $ u $: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Compound of exponentials
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Figure 2.  $ V(u;b) $ as a function of $ b $: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Compound of exponentials
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Figure 3.  $ \phi(u;b) $ as a function of $ u $: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Combination of exponentials
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Figure 4.  $ \phi(u;b) $ as a function of $ b $: (a) Exponential; (b) Erlang(2); (c) Mixture of exponentials; (d) Combination of exponentials
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Figure 5.  The Gerber-Shiu function $ \phi(u;b) $: (a) $ b = 5 $; (b) $ u = 5 $
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