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Optimal dividends and capital injections for a spectrally positive Lévy process. (English) Zbl 1362.93171
Summary: 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.

##### MSC:
 93E20 Optimal stochastic control 60G51 Processes with independent increments; Lévy processes 91G80 Financial applications of other theories
##### Keywords:
dividend; capital injection; Lévy process; scale function
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