For a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

中文翻译：

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某些正自相似马尔可夫过程的最大时间分解

对于谱负自相似马尔可夫过程$[0,\infty)$对于一个有限的整体上界，我们提供了一种易于处理的细节，在零吸收时间的最大值处进行了一种条件 Wiener-Hopf 分解，条件是总体上界和总体上界的跳跃。在一个伴随的结果中，这个吸收时间的拉普拉斯变换（在过程没有超过给定水平的情况下）在没有其他假设（例如过程允许循环扩展和/或连续达到零）下被识别，概括文献中的一些现有结果。