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 $ρ=\p(X_1>0)$ 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.

This paper has been withdrawn by the author. The results of this paper can now be found in Section 7 of the paper "On extrema of stable processes" http://arxiv.org/abs/1001.0991