The area of a spectrally positive stable process stopped at zero

TL;DR

This paper establishes a distributional identity for the area under a spectrally positive stable process stopped at zero, revealing its connection to inverse Beta variables and self-decomposability, with explicit density representations.

Contribution

It extends previous results for Brownian motion to spectrally positive stable processes, providing new distributional identities and density representations.

Findings

The stopped area is distributed as a perpetuity of a spectrally negative Lévy process.

The density has a convergent series representation with Fréchet-like behavior at zero.

The distribution involves an inverse Beta random variable and the square of a positive stable variable.

Abstract

An identity in law for the area of a spectrally positive L\'evy stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse Beta random variable and the square of a positive stable random variable. This identity entails that the stopped area is distributed as the perpetuity of a spectrally negative L\'evy process, and is hence self-decomposable. We also derive a convergent series representation for the density, whose behaviour at zero is shown to be Fr\'echet-like.