Law of the first passage triple of a spectrally positive strictly stable process

TL;DR

This paper derives a series representation for the joint distribution of the first passage time, undershoot, and overshoot of a spectrally positive stable process, enabling precise sampling and analysis of related stochastic properties.

Contribution

It provides a novel series representation for the joint distribution of the first passage triple in spectrally positive stable processes, including corollaries for related laws.

Findings

Series representation for the joint distribution of the first passage triple

Exact sampling method for the first passage triple

Derived laws for the process's supremum and related quantities

Abstract

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 allows exact sampling of the first passage triple.