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

Abstract

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