This paper studies de Finetti's optimal dividend problem with capital injection. We confirm the optimality of a double barrier strategy when the underlying risk model follows a L\'evy process that may have positive and negative jumps. The main result in this paper is a generalization of Theorem 3 in Avram et al.(2007), which is the spectrally negative case, and Theorem 3.1 in Bayraktar et al.(2013), which is the spectrally positive case. In contrast with the spectrally one-sided cases, double barrier strategies cannot be handled by using scale functions to obtain some properties of the expected net present values (NPVs) of dividends and capital injections. Instead, to obtain these properties, we observe changes in the sample path (and the associated NPV) when there is a slight change to the initial value or the barrier value.

中文翻译：

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关于 Lévy 过程双屏障策略的优化

本文研究了 de Finetti 的资本注入最优分红问题。当潜在风险模型遵循可能具有正跳和负跳的 L\'evy 过程时，我们确认了双屏障策略的最优性。本文的主要结果是 Avram 等人 (2007) 中的定理 3 的推广，这是频谱负情况，以及 Bayraktar 等人 (2013) 中的定理 3.1，这是频谱正情况。与光谱片面的情况相比，双壁垒策略不能通过使用尺度函数来获得股息和资本注入的预期净现值 (NPV) 的某些属性来处理。相反，为了获得这些属性，当初始值或障碍值发生轻微变化时，我们会观察样本路径（以及相关的 NPV）的变化。