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September  2021, 17(5): 2639-2667. doi: 10.3934/jimo.2020087

Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes

Wei Zhong 1, , Yongxia Zhao 1,, and  Ping Chen 2,

1. 

School of Statistics, Qufu Normal University, Qufu, Shandong, 273165, China
2. 

Faculty of Business and Economics, The University of Melbourne, Melbourne, VIC 3010, Australia

* Corresponding author: Yongxia Zhao

Received  October 2019 Revised  December 2019 Published  September 2021 Early access  April 2020

In the dual risk model, we study the periodic dividend problem with a non-exponential discount function which results in a time-inconsistent control problem. Viewing it within the game theoretic framework, we extend the Hamilton-Jacobi-Bellman (HJB) system of equations from the fixed terminal to the time of ruin and derive the verification theorem, and we generalize the theory of classical optimal periodic dividend. Under two special non-exponential discount functions, we obtain the closed-form expressions of equilibrium strategy and the corresponding equilibrium value function in a compound Poisson dual model. Finally, some numerical examples are presented to illustrate the impact of some parameters.

Keywords: Time-inconsistent optimal control, Non-exponential discounting, Equilibrium periodic dividend strategy, Extended HJB system of equations, Lévy process.
Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 60G51.
Citation: 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
References:
[1]

I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Mathematical Control and Related Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.

[2] D. Applebaum, Lévy processes and Stochastic Calculus, 2$^nd$ edition, Cambridge university press, 2009.  doi: 10.1017/CBO9780511755323.
[3]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0.

[4]

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.

[5]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion, ASTIN Bulletin, 38 (2008), 653-667.  doi: 10.1017/S0515036100015324.

[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]

O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, Lévy Processes: Theory and Applications, Springer Science and Business Media, New York, 2012.

[8]

F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer Science and Business Media, London, 2008. doi: 10.1007/978-1-84628-797-8.

[9]

T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.

[10]

T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, Working paper, Stockholm School of Economics, 2010.

[11]

S. Chen, Z. Li and Y. Zeng, Optimal dividend strategy for a general diffusion process with time-inconsistent preferences and ruin penalty, SIAM Journal on Financial Mathematics, 9 (2018), 274-314.  doi: 10.1137/16M1088983.

[12]

S. Chen, X. Wang, Y. Deng and Y. Zeng, Optimal dividend-financing strategies in a dual risk model with time-inconsistent preferences, Insurance: Mathematics and Economics, 67 (2016), 27-37.  doi: 10.1016/j.insmatheco.2015.11.005.

[13]

A. Chunxiang, Z. Li and F. Wang, Optimal investment strategy under time-inconsistent preferences and high-water mark contract, Operations Research Letters, 44 (2016), 212-218.  doi: 10.1016/j.orl.2015.12.013.

[14]

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.

[15]

B. De Finetti, Su un'impostazione alternativa della teoria collectiva del rischio, Transactions of the 15th International Congress of Actuaries, 2 (1957), 433-443. 

[16]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.

[17]

C. Foucart, P. S. Li and X. Zhou, Time-changed spectrally positive Lévy processes starting from infinity, preprint, arXiv: 1901.10689.

[18]

S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, NBER Working Paper Series, (2006), 1–49. doi: 10.3386/w12042.

[19]

F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 3$^{nd}$ edition, World Scientific Publishing Company, 1999. doi: 10.1142/p821.

[20]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer Science and Business Media, Berlin, 2006. doi: 10.1007/978-3-540-31343-4.

[21]

Y. Li, Z. Li and Y. Zeng, Equilibrium dividend strategy with non-exponential discounting in a dual model, Journal of Optimization Theory and Applications, 168 (2016), 699-722.  doi: 10.1007/s10957-015-0742-8.

[22]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, The Quarterly Journal of Economics, 107 (1992), 578-596.  doi: 10.1017/CBO9780511803475.034.

[23]

E. G. J. Luttmer and T. Mariotti, Subjective discounting in an exchange economy, Journal of Political Economy, 111 (2003), 959-989.  doi: 10.1086/376954.

[24]

J. L. Pérez and K. Yamazaki, On the optimality of periodic barrier strategies for a spectrally positive Lévy process, Insurance: Mathematics and Economics, 77 (2017), 1-13.  doi: 10.1016/j.insmatheco.2017.08.001.

[25]

W. Schoutens, Lévy processes in Finance: Pricing Financial Derivatives, Wiley, New York, 2003. doi: 10.1002/0470870230.

[26]

R. Thaler, Some empirical evidence on dynamic inconsistency, Insurance: Mathematics and Economics, 8 (1981), 201-207.  doi: 10.1016/0165-1765(81)90067-7.

[27]

Y. Tian, Optimal capital structure and investment decisions under time-inconsistent preferences, Journal of Economic Dynamics and Control, 65 (2016), 83-104.  doi: 10.1016/j.jedc.2016.02.001.

[28]

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.

[29]

Q. Zhao, J. Wei and R. Wang, On dividend strategies with non-exponential discounting, Insurance: Mathematics and Economics, 58 (2014), 1-13.  doi: 10.1016/j.insmatheco.2014.06.001.

[30]

Y. Zhao, P. Chen and H. Yang, Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 74 (2017), 135-146.  doi: 10.1016/j.insmatheco.2017.03.006.

[31]

Y. Zhao, R. Wang and D. Yao, Optimal dividend and equity issuance in the perturbed dual model under a penalty for ruin, Communications in Statistics-Theory and Methods, 45 (2016), 365–384. doi: 10.1080/03610926.2013.810269.

[32]

Y. Zhao, R. Wang, D. Yao and P. Chen, Optimal dividends and capital injections in the dual model with a random time horizon, Journal of Optimization Theory and Applications, 167 (2015), 272-295.  doi: 10.1007/s10957-014-0653-0.

show all references

References:
[1]

I. Alia, A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion, Mathematical Control and Related Fields, 9 (2019), 541-570.  doi: 10.3934/mcrf.2019025.

[2] D. Applebaum, Lévy processes and Stochastic Calculus, 2$^nd$ edition, Cambridge university press, 2009.  doi: 10.1017/CBO9780511755323.
[3]

S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0.

[4]

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.

[5]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion, ASTIN Bulletin, 38 (2008), 653-667.  doi: 10.1017/S0515036100015324.

[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]

O. E. Barndorff-Nielsen, T. Mikosch and S. I. Resnick, Lévy Processes: Theory and Applications, Springer Science and Business Media, New York, 2012.

[8]

F. Biagini, Y. Hu, B. Øksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer Science and Business Media, London, 2008. doi: 10.1007/978-1-84628-797-8.

[9]

T. Björk, M. Khapko and A. Murgoci, On time-inconsistent stochastic control in continuous time, Finance and Stochastics, 21 (2017), 331-360.  doi: 10.1007/s00780-017-0327-5.

[10]

T. Björk and A. Murgoci, A General Theory of Markovian Time Inconsistent Stochastic Control Problems, Working paper, Stockholm School of Economics, 2010.

[11]

S. Chen, Z. Li and Y. Zeng, Optimal dividend strategy for a general diffusion process with time-inconsistent preferences and ruin penalty, SIAM Journal on Financial Mathematics, 9 (2018), 274-314.  doi: 10.1137/16M1088983.

[12]

S. Chen, X. Wang, Y. Deng and Y. Zeng, Optimal dividend-financing strategies in a dual risk model with time-inconsistent preferences, Insurance: Mathematics and Economics, 67 (2016), 27-37.  doi: 10.1016/j.insmatheco.2015.11.005.

[13]

A. Chunxiang, Z. Li and F. Wang, Optimal investment strategy under time-inconsistent preferences and high-water mark contract, Operations Research Letters, 44 (2016), 212-218.  doi: 10.1016/j.orl.2015.12.013.

[14]

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.

[15]

B. De Finetti, Su un'impostazione alternativa della teoria collectiva del rischio, Transactions of the 15th International Congress of Actuaries, 2 (1957), 433-443. 

[16]

I. Ekeland and T. A. Pirvu, Investment and consumption without commitment, Mathematics and Financial Economics, 2 (2008), 57-86.  doi: 10.1007/s11579-008-0014-6.

[17]

C. Foucart, P. S. Li and X. Zhou, Time-changed spectrally positive Lévy processes starting from infinity, preprint, arXiv: 1901.10689.

[18]

S. R. Grenadier and N. Wang, Investment under uncertainty and time-inconsistent preferences, NBER Working Paper Series, (2006), 1–49. doi: 10.3386/w12042.

[19]

F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 3$^{nd}$ edition, World Scientific Publishing Company, 1999. doi: 10.1142/p821.

[20]

A. E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Springer Science and Business Media, Berlin, 2006. doi: 10.1007/978-3-540-31343-4.

[21]

Y. Li, Z. Li and Y. Zeng, Equilibrium dividend strategy with non-exponential discounting in a dual model, Journal of Optimization Theory and Applications, 168 (2016), 699-722.  doi: 10.1007/s10957-015-0742-8.

[22]

G. Loewenstein and D. Prelec, Anomalies in intertemporal choice: Evidence and an interpretation, The Quarterly Journal of Economics, 107 (1992), 578-596.  doi: 10.1017/CBO9780511803475.034.

[23]

E. G. J. Luttmer and T. Mariotti, Subjective discounting in an exchange economy, Journal of Political Economy, 111 (2003), 959-989.  doi: 10.1086/376954.

[24]

J. L. Pérez and K. Yamazaki, On the optimality of periodic barrier strategies for a spectrally positive Lévy process, Insurance: Mathematics and Economics, 77 (2017), 1-13.  doi: 10.1016/j.insmatheco.2017.08.001.

[25]

W. Schoutens, Lévy processes in Finance: Pricing Financial Derivatives, Wiley, New York, 2003. doi: 10.1002/0470870230.

[26]

R. Thaler, Some empirical evidence on dynamic inconsistency, Insurance: Mathematics and Economics, 8 (1981), 201-207.  doi: 10.1016/0165-1765(81)90067-7.

[27]

Y. Tian, Optimal capital structure and investment decisions under time-inconsistent preferences, Journal of Economic Dynamics and Control, 65 (2016), 83-104.  doi: 10.1016/j.jedc.2016.02.001.

[28]

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.

[29]

Q. Zhao, J. Wei and R. Wang, On dividend strategies with non-exponential discounting, Insurance: Mathematics and Economics, 58 (2014), 1-13.  doi: 10.1016/j.insmatheco.2014.06.001.

[30]

Y. Zhao, P. Chen and H. Yang, Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 74 (2017), 135-146.  doi: 10.1016/j.insmatheco.2017.03.006.

[31]

Y. Zhao, R. Wang and D. Yao, Optimal dividend and equity issuance in the perturbed dual model under a penalty for ruin, Communications in Statistics-Theory and Methods, 45 (2016), 365–384. doi: 10.1080/03610926.2013.810269.

[32]

Y. Zhao, R. Wang, D. Yao and P. Chen, Optimal dividends and capital injections in the dual model with a random time horizon, Journal of Optimization Theory and Applications, 167 (2015), 272-295.  doi: 10.1007/s10957-014-0653-0.

Figure 1.  Influence of parameters $ \rho_1 $ and $ \rho_2 $ to equilibrium value function and equilibrium dividend barrier
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Figure 2.  Contour plot of $ c $ as a function of $ \beta $ and $ \lambda $
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Figure 3.  Influence of parameters $ \eta $ and $ c $ to equilibrium value function
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Table 1.  Influences of $ \beta $ and $ \lambda $ on $ b $
$ c=0.7 $ $ \gamma=1 $ $ \omega_1=0.7 $ $ \rho_1=0.1 $ $ \rho_2=0.3 $
$ \beta=1.5 $ $ \lambda=1.5 $
$ \lambda $$ \uparrow $ 1.3 1.8 $ \mathit{\boldsymbol{2.1920}} $ 3 $ \beta $$ \uparrow $ 0.8 1 1.3 1.7
$ b $$ \curvearrowright $ 0.5606 1.5520 $ \mathit{\boldsymbol{1.6794}} $ 1.5256 $ \downarrow $ 2.2197 1.9983 1.5353 0.6472
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$ c=0.7 $ $ \gamma=1 $ $ \omega_1=0.7 $ $ \rho_1=0.1 $ $ \rho_2=0.3 $
$ \beta=1.5 $ $ \lambda=1.5 $
$ \lambda $$ \uparrow $ 1.3 1.8 $ \mathit{\boldsymbol{2.1920}} $ 3 $ \beta $$ \uparrow $ 0.8 1 1.3 1.7
$ b $$ \curvearrowright $ 0.5606 1.5520 $ \mathit{\boldsymbol{1.6794}} $ 1.5256 $ \downarrow $ 2.2197 1.9983 1.5353 0.6472
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