The area under a spectrally positive stable excursion and other related processes

TL;DR

This paper investigates the distribution of areas under various excursions of spectrally positive stable Lévy processes, revealing connections to Wright's function and generalizing classical Airy function results.

Contribution

It introduces a novel analysis of the area distributions for spectrally positive stable Lévy excursions, extending known results involving Airy functions to a broader class of processes.

Findings

Derived the distribution of the area under the normalized excursion

Connected the results to Wright's function as a generalization of Airy functions

Extended classical Brownian area results to spectrally positive stable Lévy processes

Abstract

We study the distribution of the area under the normalized excursion of a spectrally positive stable L{\'e}vy process L, as well as the area under its meander, and under L conditioned to stay positive. Our results involve a special case of Wright's function, which may be seen as a generalization of the classic Airy function appearing in similar Brownian's areas.