For a spectrally positive and strictly stable process with index in (1, 2), a series representation is obtained for the joint distribution of the “first passage triple” that consists of the time of first passage and the undershoot and the overshoot at first passage. The result leads to several corollaries, including (1) the joint law of the first passage triple and the pre-passage running supremum, and (2) at a fixed time point, the joint law of the process’ value, running supremum, and the time of the running supremum. The representation can be decomposed as a sum of strictly positive functions that allow exact sampling of the first passage triple.

中文翻译：

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谱正态严格稳定过程的首通三重定律

对于一个指数为(1, 2)的谱正态严格稳定过程，得到了由第一次通过时间和下冲时间和第一次通过时过冲时间组成的“第一次通过三元组”的联合分布的级数表示. 结果导致了几个推论，包括（1）第一代三元组和预传运行最高的联合规律，以及（2）在固定时间点，过程值的联合规律，运行最高，和运行至高无上的时间。该表示可以分解为严格正函数的总和，允许对第一段三元组进行精确采样。