Abstract

In the context of maximizing cumulative dividends under barrier policies, generalized Azéma–Yor (draw-down) stopping times receive increasing attention during these past years. Based on Pontryagin’s maximality principle, we illustrate the necessity of such generalizations under the framework of spectrally negative Markov processes. Roughly speaking, starting from the explicit expression of the optimal value of discounted dividends in terms of the scale functions, we write down the optimality conditions (via Pontryagin’s principle). The use of generalized draw-downs is then quantified through a structure term (linked to the existence of non bang-bang optimal controls). We thoroughly study several classes of Lévy processes (Bertoin, Lévy Processes, vol. 121. Cambridge University Press, 1998; Kyprianou, Fluctuations of Lévy Processes with Applications: Introductory Lectures. Springer Science & Business Media, 2014) constituting the usual models of insurance claims and a particular piece-wise deterministic Markov model (extending the premium rate to reserve-dependent settings). In all these models, we disprove the consistency of the aforementioned structure equation, thus denying the necessity of such generalizations. We end the paper with some heuristics on possible non-trivial cases for general Markov models.

中文翻译：

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普遍的提取时间会带来更好的红利吗？基于Pontryagin原理的答案

摘要

在障碍政策最大程度地提高累积红利的背景下，过去几年中，通用的Azéma-Yor（提款）停止时间越来越受到关注。基于庞特里亚金的极大原理，我们说明了在频谱负马尔可夫过程框架下进行这种概括的必要性。粗略地讲，我们从规模函数的折扣红利最优值的明确表达开始，写下了最优性条件（通过Pontryagin原理）。然后，通过结构项（与非爆炸性最优控件的存在相关联）对广义下拉菜单的使用进行量化。我们彻底研究了几类Lévy过程（Bertoin，《Lévy过程》，第121卷，剑桥大学出版社，1998年； Kyprianou，Lévy过程的波动及其应用：入门讲座。Springer Science＆Business Media，2014年）构成了保险索赔的通常模型和特定的分段确定性马尔可夫模型（将保费率扩展至依赖储备的设置）。在所有这些模型中，我们都证明了上述结构方程的一致性，因此否认了这种概括的必要性。我们以一些启发式方法结束了本文，这些启发式方法适用于一般Markov模型的可能非平凡案例。