# American Institute of Mathematical Sciences

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

## Optimal dividends and capital injections for a spectrally positive Lévy process

 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.

Citation: 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
##### 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. [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. [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. [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. [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. [6] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, 1996. [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. [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. [9] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, 2 edition, Springer Verlag, New York, 2006. [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. [11] A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Universitext, Springer-Verlag, Berlin, 2006. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [6] J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, 1996. [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. [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. [9] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, 2 edition, Springer Verlag, New York, 2006. [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. [11] A.E. Kyprianou, Introductory Lectures on Fluctuations of Lévy Processes with Applications, Universitext, Springer-Verlag, Berlin, 2006. [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. [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. [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. [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. [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. [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. [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. [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.
LEFT: The influence of $l_0$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $l_0$ on the value function
LEFT: The influence of $\delta$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $\delta$ on the value function
LEFT: The influence of $\sigma$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $\sigma$ on the value function
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*
 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*
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*
 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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