Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes
Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes
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DOI：
10.1239/jap/1331216839
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发表时间：
2010-08
影响因子：
1
通讯作者：
A. Kyprianou;R. Loeffen;José-Luis Pérez
中科院分区：
文献类型：
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作者：
A. Kyprianou;R. Loeffen;José-Luis Pérez
在过去的几年里，人们对de Finetti(1957年)的经典控制问题重新产生了兴趣，因为在这种情况下，随机性的潜在来源是谱负的Lévy过程。特别是，Loeffen(2008年)向前迈出了重要的一步，他证明了关于Lévy过程的一个自然且非常一般的条件是它的Lévy测度是绝对连续的，具有完全单调的密度，这使得人们可以继续分析相关的Hamilton-Jacobi-Bellman方程。本文考虑了De Finetti的控制问题，但有这样的限制，即控制策略相对于勒贝格测度是绝对连续的。Asmussen和Taksar(1997)，JeanBlc-Picqué和Shiryaev(1995)和Bogusavskaya(2006)在扩散情形，Gerber和Shiu(2006)在具有指数分布跳跃的Cramér-Lundberg过程的情形中考虑了这个问题。我们证明了Lévy测度具有完全单调密度的条件的稳健性，并建立了这种情况下的显式最优策略，该策略涵盖了上述已有结果。所讨论的显式最优策略是所谓的折射策略。
In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.