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

Friedrich Hubalek, Alexey Kuznetsov

Published 2010-10-18, updated 2012-01-27Version 2

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.