Series representations and asymptotic expansions for the density of the supremum of a stable process

TL;DR

This paper derives explicit asymptotic expansions and convergent series representations for the density of the supremum of a strictly stable process, especially when the process parameters satisfy certain rational relations.

Contribution

It provides new explicit asymptotic expansions and conditions under which these expansions are convergent series for the supremum density of stable processes.

Findings

Asymptotic expansions are derived for non-rational stability indices.

Convergent series representations are established under specific parameter relations.

Results enhance understanding of the supremum distribution of stable processes.

Abstract

We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some integers $k,l \ge 1$ we prove that these asymptotic expansions are in fact convergent series representations of the density of supremum.