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January  2017, 13(1): 1-21. doi: 10.3934/jimo.2016001

Optimal dividends and capital injections for a spectrally positive Lévy process

Yongxia Zhao a, , Rongming Wang b, and  Chuancun Yin c,

a, c. 

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

School of Finance and Statistics, East China Normal University, Shanghai 200241, China

Received  April 2015 Revised  June 2015 Published  March 2016

Fund Project: The authors acknowledge the financial support of National Natural Science Foundation of China (11231005,11201123,11501321), Promotive research fund for excellent young and middle-aged scientists of Shandong Province (BS2014SF006), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (15KJB110009) and Postdoctoral Foundation of Qufu Normal University. The authors would like to thank the anonymous referees for help.

This paper investigates an optimal dividend and capital injection problem for a spectrally positive Lévy process, where the dividend rate is restricted. Both the ruin penalty and the costs from the transactions of capital injection are considered. The objective is to maximize the total value of the expected discounted dividends, the penalized discounted capital injections before ruin, and the expected discounted ruin penalty. By the fluctuation theory of Lévy processes, the optimal dividend and capital injection strategy is obtained. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Besides, a series of numerical examples are provided to illustrate our consults.

Keywords: Dividend, capital injection, Lévy process, scale function.
Mathematics Subject Classification: Primary: 93E20, 60G51; Secondary: 91G80.
Citation: Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001
References:
[1]

B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41 (2007), 111-123.  doi: 10.1016/j.insmatheco.2006.10.002.  Google Scholar

[2]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion, Astin Bulletin, 38 (2008), 653-667.  doi: 10.2143/AST.38.2.2033357.  Google Scholar

[3]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bulletin, 41 (2011), 611-644.  doi: 10.2139/ssrn.1709174.  Google Scholar

[4]

E. Bayraktar, A. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin, 43 (2013), 359-372.  doi: 10.1017/asb.2013.17.  Google Scholar

[5]

E. Bayraktar, A. Kyprianou and K. Yamazaki, Optimal dividends in the dual model under transaction costs, Insurance: Mathematics and Economics, 54 (2014), 133-143.  doi: 10.1016/j.insmatheco.2013.11.007.  Google Scholar

[6]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, 1996.  Google Scholar

[7]

T. Chan, A. E. Kyprianou and M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probability Theory and Related Fields, 150 (2011), 129-143.  doi: 10.1007/s00440-010-0289-4.  Google Scholar

[8]

M. Egami and K. Yamazaki, Phase-type fitting of scale functions for spectrally negative Lévy process, Journal of Computational and Applied Mathematics, 264 (2014), 1-22.  doi: 10.1016/j.cam.2013.12.044.  Google Scholar

[9] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, 2 edition, Springer Verlag, New York, 2006.   Google Scholar
[10]

A. Kuznetsov, A. E. Kyprianou and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, Lévy Matters Ⅱ, Lecture Notes in Mathematics, (2013), 97-186.  doi: 10.1007/978-3-642-31407-0_2.  Google Scholar

[11] A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Universitext, Springer-Verlag, Berlin, 2006.   Google Scholar
[12]

Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005.  Google Scholar

[13]

X. Peng, M. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.  Google Scholar

[14]

N. Scheer and H. Schmidli, Optimal dividend strategies in a cramér-lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92.  doi: 10.1007/s13385-011-0007-3.  Google Scholar

[15]

D. Yao, H. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[16]

D. Yao, H. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576.  doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

[17]

D. Yao, R. Wang and L. Xu, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model, Journal of Industrial and Management Optimization, 10 (2014), 1235-1259.  doi: 10.3934/jimo.2014.10.1235.  Google Scholar

[18]

C. Yin, Y. Wen and Y. Zhao, On the optimal dividend problem for a spectrally positive Lévy process, ASTIN Bulletin, (2014), 635-651.  doi: 10.1017/asb.2014.12.  Google Scholar

[19]

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 (2014), 272-295.  doi: 10.1007/s10957-014-0653-0.  Google Scholar

show all references

References:
[1]

B. Avanzi, H. U. Gerber and E. S. W. Shiu, Optimal dividends in the dual model, Insurance: Mathematics and Economics, 41 (2007), 111-123.  doi: 10.1016/j.insmatheco.2006.10.002.  Google Scholar

[2]

B. Avanzi and H. U. Gerber, Optimal dividends in the dual model with diffusion, Astin Bulletin, 38 (2008), 653-667.  doi: 10.2143/AST.38.2.2033357.  Google Scholar

[3]

B. Avanzi, J. Shen and B. Wong, Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bulletin, 41 (2011), 611-644.  doi: 10.2139/ssrn.1709174.  Google Scholar

[4]

E. Bayraktar, A. Kyprianou and K. Yamazaki, On optimal dividends in the dual model, ASTIN Bulletin, 43 (2013), 359-372.  doi: 10.1017/asb.2013.17.  Google Scholar

[5]

E. Bayraktar, A. Kyprianou and K. Yamazaki, Optimal dividends in the dual model under transaction costs, Insurance: Mathematics and Economics, 54 (2014), 133-143.  doi: 10.1016/j.insmatheco.2013.11.007.  Google Scholar

[6]

J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, 1996.  Google Scholar

[7]

T. Chan, A. E. Kyprianou and M. Savov, Smoothness of scale functions for spectrally negative Lévy processes, Probability Theory and Related Fields, 150 (2011), 129-143.  doi: 10.1007/s00440-010-0289-4.  Google Scholar

[8]

M. Egami and K. Yamazaki, Phase-type fitting of scale functions for spectrally negative Lévy process, Journal of Computational and Applied Mathematics, 264 (2014), 1-22.  doi: 10.1016/j.cam.2013.12.044.  Google Scholar

[9] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, 2 edition, Springer Verlag, New York, 2006.   Google Scholar
[10]

A. Kuznetsov, A. E. Kyprianou and V. Rivero, The theory of scale functions for spectrally negative Lévy processes, Lévy Matters Ⅱ, Lecture Notes in Mathematics, (2013), 97-186.  doi: 10.1007/978-3-642-31407-0_2.  Google Scholar

[11] A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Universitext, Springer-Verlag, Berlin, 2006.   Google Scholar
[12]

Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005.  Google Scholar

[13]

X. Peng, M. Chen and J. Guo, Optimal dividend and equity issuance problem with proportional and fixed transaction costs, Insurance: Mathematics and Economics, 51 (2012), 576-585.  doi: 10.1016/j.insmatheco.2012.08.004.  Google Scholar

[14]

N. Scheer and H. Schmidli, Optimal dividend strategies in a cramér-lundberg model with capital injections and administration costs, European Actuarial Journal, 1 (2011), 57-92.  doi: 10.1007/s13385-011-0007-3.  Google Scholar

[15]

D. Yao, H. Yang and R. Wang, Optimal risk and dividend control problem with fixed costs and salvage value: Variance premium principle, Economic Modelling, 37 (2014), 53-64.  doi: 10.1016/j.econmod.2013.10.026.  Google Scholar

[16]

D. Yao, H. Yang and R. Wang, Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, European Journal of Operational Research, 211 (2011), 568-576.  doi: 10.1016/j.ejor.2011.01.015.  Google Scholar

[17]

D. Yao, R. Wang and L. Xu, Optimal dividend and capital injection strategy with fixed costs and restricted dividend rate for a dual model, Journal of Industrial and Management Optimization, 10 (2014), 1235-1259.  doi: 10.3934/jimo.2014.10.1235.  Google Scholar

[18]

C. Yin, Y. Wen and Y. Zhao, On the optimal dividend problem for a spectrally positive Lévy process, ASTIN Bulletin, (2014), 635-651.  doi: 10.1017/asb.2014.12.  Google Scholar

[19]

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 (2014), 272-295.  doi: 10.1007/s10957-014-0653-0.  Google Scholar

Figure 1.  LEFT: The influence of $l_0$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $l_0$ on the value function
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Figure 2.  LEFT: The influence of $\delta$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $\delta$ on the value function
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Figure 3.  LEFT: The influence of $\sigma$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $\sigma$ on the value function
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Table 1.  The influence of P on xp* and x*
P↑ $\mathcal{I}$ -1 0 0.5 0.8380 1 1.4 1.5
xp*↑ 0 0.1601 1.0765 1.4922 1.7590 1.8830 2.1794 2.2509
xq*≡ 1.7590 1.7590 1.7590 1.7590 1.7590 1.7590 1.7590 1.7590
x*↑ xp* xp* xp* xp* xp*=xq* xq* xq* xq*
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P↑ $\mathcal{I}$ -1 0 0.5 0.8380 1 1.4 1.5
xp*↑ 0 0.1601 1.0765 1.4922 1.7590 1.8830 2.1794 2.2509
xq*≡ 1.7590 1.7590 1.7590 1.7590 1.7590 1.7590 1.7590 1.7590
x*↑ xp* xp* xp* xp* xp*=xq* xq* xq* xq*
Table 2.  The influences of ϕ and K on η, xq* and x*
ϕ = 1:1 K=0.1
K↑ 0.12 0.1256 0.14 ϕ↑ 1.12 1.1226 1.14
η ↑ 1.1753 1.2011 1.2649 ↓ 1.0623 1.0604 1.0481
xq* ↑ 1.8572 1.8830 1.9467 ↑ 1.8687 1.8830 1.9755
x* ↑ xq* xq*=xp* xp* ↑ xq* xq*=xp* xp*
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ϕ = 1:1 K=0.1
K↑ 0.12 0.1256 0.14 ϕ↑ 1.12 1.1226 1.14
η ↑ 1.1753 1.2011 1.2649 ↓ 1.0623 1.0604 1.0481
xq* ↑ 1.8572 1.8830 1.9467 ↑ 1.8687 1.8830 1.9755
x* ↑ xq* xq*=xp* xp* ↑ xq* xq*=xp* xp*
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