doi: 10.3934/mfc.2022025
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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. AlbrecherP. 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. AlbrecherN. 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. AvanziJ.-L. PérezB. 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. AvramZ. 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. BayraktarA. 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. KuznetsovA. 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. AlbrecherP. 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. AlbrecherN. 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. AvanziJ.-L. PérezB. 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. AvramZ. 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. BayraktarA. 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. KuznetsovA. 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.

Figure 1.  The value function $ v_{a, b}(x) $ as a function of the initial state $ x $
Figure 2.  The value function $ v_{a, b}(x) $ as a function of $ a $ when $ x = 0.5;b = 5 $
Figure 3.  The value function $ v_{a, b}(x) $ as a function of $ b $ for $ x = 5 $
Figure 4.  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 $
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