A few remarks on the supremum of stable processes

Bernyk et al. [Bernyk, V., Dalang, R.C., Peskir, G., 2008. The law of the supremum of a stable Lévy process with no negative jumps. Ann. Probab. 36, 1777–1789] offer a power series and an integral representation for the density of S1, the maximum up to time 1, of a regular spectrally positive αα-stable Lévy process. They also state the asymptotic behavior for large values of the density. A fact which was proved by Doney [Doney R., 2008. A note on the supremum of a stable process. Stochastics 80(2–3), 151–155], by investigating the integral representation. In this note, we provide the asymptotic expansion of the density and of its successive derivatives from the power series representation. We also show that the density of the positive random variable S1−α is the Laplace transform of a function which takes negative values on R+R+ and thus it is not completely monotone.