# 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 & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001
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##### References:
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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