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Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. (English) Zbl 1394.91243
Summary: In this paper, we investigate an optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. We assume that the periodic dividend strategy has exponential inter-dividend-decision times and continuous monitoring of solvency. Both proportional and fixed transaction costs from capital injection are considered. The objective is to maximize the total value of the expected discounted dividends and the penalized discounted capital injections until the time of ruin. By the fluctuation theory of Lévy processes in [H. Albrecher et al., Bernoulli 22, No. 3, 1364–1382 (2016; Zbl 1338.60125)], the optimal periodic dividend and capital injection strategies are derived. We also find that the optimal return function can be expressed in terms of the scale functions of Lévy processes. Finally, numerical examples are studied to illustrate our results.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60G51 Processes with independent increments; Lévy processes 93E20 Optimal stochastic control
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