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 $\alpha$ 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.