Numerical method for a compound Poisson risk model with liquid reserves and proportional investment

Chunwei Wang; Shujing Wang; Jiaen Xu; Shaohua Li; Chunwei Wang; Shujing Wang; Jiaen Xu; Shaohua Li

doi:10.3934/math.2024532

AIMS Mathematics

2024, Volume 9, Issue 5: 10893-10910. doi: 10.3934/math.2024532

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Numerical method for a compound Poisson risk model with liquid reserves and proportional investment

Henan University of Science and Technology, Luoyang 471023, Henan, China
^† These authors contributed equally to this work.

Received: 14 December 2023 Revised: 28 January 2024 Accepted: 29 February 2024 Published: 19 March 2024
MSC : 65C30, 91B05, 91G05

In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples.
- liquid reserves,
- threshold strategy,
- proportional investment,
- dividends paid up to ruin,
- sinc numerical method
Citation: Chunwei Wang, Shujing Wang, Jiaen Xu, Shaohua Li. Numerical method for a compound Poisson risk model with liquid reserves and proportional investment[J]. AIMS Mathematics, 2024, 9(5): 10893-10910. doi: 10.3934/math.2024532

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Abstract

In this paper, a classical risk model with liquid reserves and proportional investment is considered, and the expected total discounted dividend before ruin of insurance companies under the threshold dividend strategy is studied. First, the integral differential equations of the expected total discounted dividend before ruin satisfying certain boundary conditions is derived. Second, since the explicit solutions of the equations cannot be obtained, the numerical approximation solutions are obtained by the sinc approximation method. Finally, we discuss the effects of parameters such as risk capital ratio and liquid reserve on the expected total discounted dividend before ruin by some examples.

References

[1]	N. U. Prabhu, On the ruin problem of collective risk theory, Ann. Math. Stat., 32 (1961), 757–764. https://doi.org/10.1214/aoms/1177704970 doi: 10.1214/aoms/1177704970
[2]	M. I. Taksar, Optimal risk and dividend distribution control models for an insur- ance company, Math. Methods Oper. Res., 51 (2000), 1–42. https://doi.org/10.1007/s001860050001 doi: 10.1007/s001860050001
[3]	W. Yu, P. Guo, Q. Wang, G. Guan, Q. Yang, Y. Huang, et al., On a periodic capital injection and barrier dividend strategy in the compound Poisson risk model, Mathematics, 8 (2020), 511. https://doi.org/10.3390/math8040511 doi: 10.3390/math8040511
[4]	H. U. Gerber, E. S. W. Shiu, On optimal dividend strategies in the compound Poisson model, N. Am. Actuar. J., 10 (2006), 76–93. https://doi.org/10.1080/10920277.2006.10596249 doi: 10.1080/10920277.2006.10596249
[5]	H. Yuan, Y. Hu, Optimal investment for an insurer under liquid reserves, J. Ind. Manage. Optim., 17 (2020), 339–355. https://doi.org/10.3934/jimo.2019114 doi: 10.3934/jimo.2019114
[6]	B. Sundt, J. L. Teugels, Ruin estimates under interest force, Insur. Math. Econ., 16 (1995), 7–22. https://doi.org/10.1016/0167-6687(94)00023-8 doi: 10.1016/0167-6687(94)00023-8
[7]	Y. Fang, R. Wu, Optimal dividend strategy in the compound poisson model with constant interest, Stoch. Models, 23 (2007), 149–166. https://doi.org/10.1080/15326340601142271 doi: 10.1080/15326340601142271
[8]	J. Cai, R. Feng, G. E. Willmot, Analysis of the compound Poisson surplus model with liquid reserves, interest and dividends, ASTIN Bull., 39 (2009), 225–247. https://doi.org/10.2143/AST.39.1.2038063 doi: 10.2143/AST.39.1.2038063
[9]	L. Yang, C. He, Absolute ruin in the compound Poisson model with credit and debit interests and liquid reserves, Appl. Stoch. Models Bus. Ind., 30 (2014), 157–171. https://doi.org/10.1002/asmb.1953 doi: 10.1002/asmb.1953
[10]	X. Chen, H. Ou, A compound Poisson risk model with proportional investment, J. Comput. Appl. Math., 242 (2013), 248–260. https://doi.org/10.1016/j.cam.2012.10.027 doi: 10.1016/j.cam.2012.10.027
[11]	C. Yin, K. C. Yuen, Optimality of the threshold dividend strategy for the compound poisson model, Stat. Probab. Lett., 81 (2011), 1841–1846. https://doi.org/10.1016/j.spl.2011.07.022 doi: 10.1016/j.spl.2011.07.022
[12]	X. S. Lin, K. P. Pavlova, The compound Poisson risk model with a threshold dividend strategy, Insur. Math. Econ., 38 (2006), 57–80. https://doi.org/10.1016/j.insmatheco.2005.08.001 doi: 10.1016/j.insmatheco.2005.08.001
[13]	N. Wan, Dividend payments with a threshold strategy in the compound poisson risk model perturbed by diffusion, Insur. Math. Econ., 40 (2007), 509–523. https://doi.org/10.1016/j.insmatheco.2006.08.002 doi: 10.1016/j.insmatheco.2006.08.002
[14]	B. De Finetti, Su un'impostazione alternativa della teoria collettiva del rischio, In: Transactions of the XVth international congress of Actuaries, 2 (1957), 433–443.
[15]	J. Paulsen, H. K. Gjessing, Ruin theory with stochastic return on investments, Adv. Appl. Probab., 29 (1997), 965–985. https://doi.org/10.2307/1427849 doi: 10.2307/1427849
[16]	X. S. Lin, K. P. Sendova, The compound Poisson risk model with multiple thresholds, Insur. Math. Econ., 42 (2008), 617–627. https://doi.org/10.1016/j.insmatheco.2007.06.008 doi: 10.1016/j.insmatheco.2007.06.008
[17]	Z. Zhang, X. Han, The compound Poisson risk model under a mixed dividend strategy, Appl. Math. Comput., 315 (2017), 1–12. https://doi.org/10.1016/j.amc.2017.07.048 doi: 10.1016/j.amc.2017.07.048
[18]	Y. Zhang, L. Mao, B. Kou, A perturbed risk model with liquid reserves, credit and debit interests and dividends under absolute ruin, In: Advances in computational science and computing, 2019. https://doi.org/10.1007/978-3-030-02116-0_40
[19]	D. Peng, D. Liu, Z. Hou, Absolute ruin problems in a compound Poisson risk model with constant dividend barrier and liquid reserves, Adv. Differ. Equ., 2016 (2016), 72. https://doi.org/10.1186/s13662-016-0746-1 doi: 10.1186/s13662-016-0746-1
[20]	F. Dufresne, H. U. Gerber, Risk theory for the compound Poisson process that is perturbed by diffusion, Insur. Math. Econ., 10 (1991), 51–59. https://doi.org/10.1016/0167-6687(91)90023-Q doi: 10.1016/0167-6687(91)90023-Q
[21]	E. T. Whitaker, On the functions which are represented by the expansion of interpolating theory, Proc. Roy. Soc. Edinb., 35 (1915), 181–194. https://doi.org/10.1017/S0370164600017806 doi: 10.1017/S0370164600017806
[22]	T. Carlson, J. Dockery, J. Lund, A sinc-collocation method for initial value problem, Math. Comput., 66 (1997), 215–235.
[23]	T. Okayama, Error estimates with explicit constants for the Sinc approximation over infinite intervals, Appl. Math. Comput., 319 (2018), 125–137. https://doi.org/10.1016/j.amc.2017.02.02 doi: 10.1016/j.amc.2017.02.02
[24]	K. Maleknejad, K. Nedaiasl, Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations, Appl. Comput. Math. Appl., 62 (2011), 3292–3303. https://doi.org/10.1016/j.camwa.2011.08.045 doi: 10.1016/j.camwa.2011.08.045
[25]	C. Wang, N. Deng, S. Shen, Numerical method for a perturbed risk model with proportional investment, Mathematics, 11 (2022), 43. https://doi.org/10.1016/j.camwa.2011.08.045 doi: 10.1016/j.camwa.2011.08.045
[26]	F. Stenger, Numerical methods based on sinc and analytic functions, New York: Springer-Verlag, 1993. https://doi.org/10.1007/978-1-4612-2706-9
[27]	F. Stenger, Handbook of sinc numerical methods, Boca Raton: CRC Press, 2011. https://doi.org/10.1201/b10375
[28]	J. Lund, K. L. Bowers, Sinc methods for quadrature and differential equations, SIAM, 1992.

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