# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2022025
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## A Lévy risk model with ratcheting and barrier dividend strategies

 School of Statistics and Data Science, Qufu Normal University, Qufu, Shandong 273165, China

*Corresponding author: Chuancun Yin

Received  March 2022 Early access July 2022

Fund Project: This research was supported by the National Natural Science Foundation of China (No. 12071251, 11571198, 11701319) and the Shandong Provincial Natural Science Foundation of China (ZR2020MA035)

The expected present value of dividends is one of the classical stability criteria in actuarial risk theory. In this paper, we consider the two-layer $(a, b)$ dividend strategy when the risk process is modeled by a spectrally negative Lévy process, such a strategy has an increasing dividend rate when the surplus exceeds level $a>0$, and all of the excess over $b>a$ as lump sum dividend payments. Using fluctuation identities and scale functions, we obtain explicit formulas for the expected net present value of dividends until ruin and the Laplace transform of the time to ruin. Finally, numerical illustrations are present to show the impacts of parameters on the expected net present value.

Citation: Hui Gao, Chuancun Yin. A Lévy risk model with ratcheting and barrier dividend strategies. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022025
##### References:
 [1] H. Albrecher, P. Azcue and N. Müler, Optimal ratcheting of dividends in a Brownian risk model, SIAM J. Financial Math., 13 (2022), 657–701, arXiv: 2012.10632. doi: 10.1137/20M1387171. [2] H. Albrecher, P. Azcue and N. Müler, Optimal ratcheting of dividends in insurance, SIAM Journal on Control and Optimization, 58 (2020), 1822-1845.  doi: 10.1137/19M1304878. [3] H. Albrecher, N. Bäuerle and M. Bladt, Dividends: From refracting to ratcheting, Insurance Math. Econom., 83 (2018), 47-58.  doi: 10.1016/j.insmatheco.2018.09.003. [4] H. Albrecher and J. Hartinger, A risk model with multilayer dividend strategy, American Actuarial Journal, 11 (2007), 43-64.  doi: 10.1080/10920277.2007.10597447. [5] B. Avanzi, J.-L. Pérez, B. Wong and K. Yamazaki, On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models, Insurance Math. Econom., 72 (2017), 148-162.  doi: 10.1016/j.insmatheco.2016.10.010. [6] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709. [7] E. Bayraktar, A. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin, 43 (2013), 359-373.  doi: 10.1017/asb.2013.17. [8] A. Kuznetsov, A. E. Kyprianou and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, Lévy Matters II, Lecture Notes in Math., Lévy Matters, Springer, Heidelberg, 2061 (2012), 97-186.  doi: 10.1007/978-3-642-31407-0_2. [9] A. E. Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2$^{nd}$ edition, Springer Science and Business Media, 2014. doi: 10.1007/978-3-642-37632-0. [10] R. L. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, The Annals of Applied Probability, 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504. [11] R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, Journal of Applied Probability, 46 (2009), 85-98.  doi: 10.1017/S0021900200005246. [12] R. L. Loeffen and J.-F. Renaud, De Finetti's optimal dividends problem with an affine penalty function at ruin, Insurance Math. Econom., 46 (2010), 98-108.  doi: 10.1016/j.insmatheco.2009.09.006. [13] S. Loisel and H. U. Gerber, Why ruin theory should be of interest for insurance practitioners and risk managers nowadays, In Actuarial and Financial Mathematics, (2012), 17–21. [14] C. Y. Ng, On the upcrossing and downcrossing probabilities of a dual risk model with phase-type gains, ASTIN Bulletin, 40 (2010), 281-306.  doi: 10.2143/AST.40.1.2049230. [15] Z.-J. Song and F.-Y. Sun, The dual risk model under a mixed ratcheting and periodic dividend strategy, Communications in Statistics-Theory and Methods, (2021), 1–15. doi: 10.1080/03610926.2021.1974483. [16] S. Thonhauser and H. Albrecher, Dividend maximization under consideration of the time value of ruin, Insurance Math. Econom., 41 (2007), 163-184.  doi: 10.1016/j.insmatheco.2006.10.013. [17] K. C. Yuen and C. Yin, On optimality of the barrier strategy for a general Lévy risk process, Mathematical and Computer Modelling, 53 (2011), 1700-1707.  doi: 10.1016/j.mcm.2010.12.042. [18] C. Yin and C. Wang, Optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes: An alternative approach, Journal of Computational and Applied Mathematics, 233 (2009), 482-491.  doi: 10.1016/j.cam.2009.07.051. [19] C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance Math. Econom., 53 (2013), 769-773.  doi: 10.1016/j.insmatheco.2013.09.019. [20] A. Zhang and Z. Liu, A Lévy risk model with ratcheting dividend strategy and historic high-related stopping, Mathematical Problems in Engineering, 2020 (2020), Art. ID 6282869, 12 pp. doi: 10.1155/2020/6282869.

show all references

##### References:
 [1] H. Albrecher, P. Azcue and N. Müler, Optimal ratcheting of dividends in a Brownian risk model, SIAM J. Financial Math., 13 (2022), 657–701, arXiv: 2012.10632. doi: 10.1137/20M1387171. [2] H. Albrecher, P. Azcue and N. Müler, Optimal ratcheting of dividends in insurance, SIAM Journal on Control and Optimization, 58 (2020), 1822-1845.  doi: 10.1137/19M1304878. [3] H. Albrecher, N. Bäuerle and M. Bladt, Dividends: From refracting to ratcheting, Insurance Math. Econom., 83 (2018), 47-58.  doi: 10.1016/j.insmatheco.2018.09.003. [4] H. Albrecher and J. Hartinger, A risk model with multilayer dividend strategy, American Actuarial Journal, 11 (2007), 43-64.  doi: 10.1080/10920277.2007.10597447. [5] B. Avanzi, J.-L. Pérez, B. Wong and K. Yamazaki, On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models, Insurance Math. Econom., 72 (2017), 148-162.  doi: 10.1016/j.insmatheco.2016.10.010. [6] F. Avram, Z. Palmowski and M. R. Pistorius, On the optimal dividend problem for a spectrally negative Lévy process, The Annals of Applied Probability, 17 (2007), 156-180.  doi: 10.1214/105051606000000709. [7] E. Bayraktar, A. E. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin, 43 (2013), 359-373.  doi: 10.1017/asb.2013.17. [8] A. Kuznetsov, A. E. Kyprianou and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, Lévy Matters II, Lecture Notes in Math., Lévy Matters, Springer, Heidelberg, 2061 (2012), 97-186.  doi: 10.1007/978-3-642-31407-0_2. [9] A. E. Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures, 2$^{nd}$ edition, Springer Science and Business Media, 2014. doi: 10.1007/978-3-642-37632-0. [10] R. L. Loeffen, On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes, The Annals of Applied Probability, 18 (2008), 1669-1680.  doi: 10.1214/07-AAP504. [11] R. L. Loeffen, An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, Journal of Applied Probability, 46 (2009), 85-98.  doi: 10.1017/S0021900200005246. [12] R. L. Loeffen and J.-F. Renaud, De Finetti's optimal dividends problem with an affine penalty function at ruin, Insurance Math. Econom., 46 (2010), 98-108.  doi: 10.1016/j.insmatheco.2009.09.006. [13] S. Loisel and H. U. Gerber, Why ruin theory should be of interest for insurance practitioners and risk managers nowadays, In Actuarial and Financial Mathematics, (2012), 17–21. [14] C. Y. Ng, On the upcrossing and downcrossing probabilities of a dual risk model with phase-type gains, ASTIN Bulletin, 40 (2010), 281-306.  doi: 10.2143/AST.40.1.2049230. [15] Z.-J. Song and F.-Y. Sun, The dual risk model under a mixed ratcheting and periodic dividend strategy, Communications in Statistics-Theory and Methods, (2021), 1–15. doi: 10.1080/03610926.2021.1974483. [16] S. Thonhauser and H. Albrecher, Dividend maximization under consideration of the time value of ruin, Insurance Math. Econom., 41 (2007), 163-184.  doi: 10.1016/j.insmatheco.2006.10.013. [17] K. C. Yuen and C. Yin, On optimality of the barrier strategy for a general Lévy risk process, Mathematical and Computer Modelling, 53 (2011), 1700-1707.  doi: 10.1016/j.mcm.2010.12.042. [18] C. Yin and C. Wang, Optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes: An alternative approach, Journal of Computational and Applied Mathematics, 233 (2009), 482-491.  doi: 10.1016/j.cam.2009.07.051. [19] C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance Math. Econom., 53 (2013), 769-773.  doi: 10.1016/j.insmatheco.2013.09.019. [20] A. Zhang and Z. Liu, A Lévy risk model with ratcheting dividend strategy and historic high-related stopping, Mathematical Problems in Engineering, 2020 (2020), Art. ID 6282869, 12 pp. doi: 10.1155/2020/6282869.
The value function $v_{a, b}(x)$ as a function of the initial state $x$
The value function $v_{a, b}(x)$ as a function of $a$ when $x = 0.5;b = 5$
The value function $v_{a, b}(x)$ as a function of $b$ for $x = 5$
The value function $v_{a, b}(x)$ as a function of $(c_1, c_2)$ for $(c_1, c_2)\in[0, 0.3]\times[0, 0.3]$ for the three cases: (a) $x = 10; a = 3; b = 5$; (b) $x = 10; a = 3; b = 20$; (c) $x = 2; a = 3; b = 5$
 [1] Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial and Management Optimization, 2022, 18 (2) : 795-823. doi: 10.3934/jimo.2020179 [2] Gongpin Cheng, Lin Xu. Optimal size of business and dividend strategy in a nonlinear model with refinancing and liquidation value. Mathematical Control and Related Fields, 2017, 7 (1) : 1-19. doi: 10.3934/mcrf.2017001 [3] Wei Zhong, Yongxia Zhao, Ping Chen. Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2639-2667. doi: 10.3934/jimo.2020087 [4] Dingjun Yao, Rongming Wang, Lin Xu. Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1235-1259. doi: 10.3934/jimo.2014.10.1235 [5] Zhimin Zhang, Eric C. K. Cheung. A note on a Lévy insurance risk model under periodic dividend decisions. Journal of Industrial and Management Optimization, 2018, 14 (1) : 35-63. doi: 10.3934/jimo.2017036 [6] Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial and Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001 [7] 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 [8] Gongpin Cheng, Rongming Wang, Dingjun Yao. Optimal dividend and capital injection strategy with excess-of-loss reinsurance and transaction costs. Journal of Industrial and Management Optimization, 2018, 14 (1) : 371-395. doi: 10.3934/jimo.2017051 [9] Manman Li, George Yin. Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model. Journal of Industrial and Management Optimization, 2019, 15 (2) : 517-535. doi: 10.3934/jimo.2018055 [10] Yin Li, Xuerong Mao, Yazhi Song, Jian Tao. Optimal investment and proportional reinsurance strategy under the mean-reverting Ornstein-Uhlenbeck process and net profit condition. Journal of Industrial and Management Optimization, 2022, 18 (1) : 75-93. doi: 10.3934/jimo.2020143 [11] Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2141-2157. doi: 10.3934/jimo.2019047 [12] Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic and Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53 [13] Yiling Chen, Baojun Bian. optimal investment and dividend policy in an insurance company: A varied bound for dividend rates. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5083-5105. doi: 10.3934/dcdsb.2019044 [14] Hongjun Gao, Fei Liang. On the stochastic beam equation driven by a Non-Gaussian Lévy process. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 1027-1045. doi: 10.3934/dcdsb.2014.19.1027 [15] Zhimin Zhang. On a risk model with randomized dividend-decision times. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1041-1058. doi: 10.3934/jimo.2014.10.1041 [16] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [17] Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137 [18] Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241 [19] Hamza Ruzayqat, Ajay Jasra. Unbiased parameter inference for a class of partially observed Lévy-process models. Foundations of Data Science, 2022, 4 (2) : 299-322. doi: 10.3934/fods.2022008 [20] Yong Zhang, Huifen Zhong, Yue Liu, Menghu Huang. Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1617-1630. doi: 10.3934/jimo.2018114

Impact Factor: