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Article Contents

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

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.

Mathematics Subject Classification: Primary: 93E20, 60G51; Secondary: 91G80.

 Citation:

• 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

Figure 2.  LEFT: The influence of $\delta$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $\delta$ on the value function

Figure 3.  LEFT: The influence of $\sigma$ on $\eta$, $x_p^*$, $x_q^*$ and $x^*$. RIGHT: The influence of $\sigma$ on the value function

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*

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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