A convergent series representation for the density of the supremum of a stable process

TL;DR

This paper proves that for most stability indices, the asymptotic series representing the density of the supremum of a stable process are actually absolutely convergent series, providing a new representation.

Contribution

It establishes the absolute convergence of series representations for the supremum density of stable processes for almost all stability indices, extending previous asymptotic results.

Findings

Series are absolutely convergent for most stability indices.

Provides a new convergent series representation for the supremum density.

Extends previous asymptotic series results to convergent series.

Abstract

We study the density of the supremum of a strictly stable L\'evy process. We prove that for almost all values of the index $\alpha$ -- except for a dense set of Lebesgue measure zero -- the asymptotic series which were obtained in A. Kuznetsov (2010) "On extrema of stable processes" are in fact absolutely convergent series representations for the density of the supremum.