A note on the supremum of a stable process

Abstract

If $X$ is a spectrally positive stable process of index $\alpha \in (1,2)$ whose Lévy measure has density $c{x}^{-\alpha -1}$ on $(0,\infty ),$ and $S_1={\sup }_{0<t\le 1}{X}_t,$ it is known that $P(S_1>x)\backsim c{\alpha }^{-1}{x}^{-\alpha }$ as $x\to \infty .$ It is also known that $S_1$has a continuous density, $s$ say. The point of this note is to show that $s(x)\backsim c{x}^{-(\alpha +1)}$ as $x\to \infty .$

To appear in a Special Volume of Stochastics: An International Journal of Probability and Stochastic Processes (http://www.informaworld.com/openurl?genre=journal%26issn=1744-2508) edited by N.H. Bingham and I.V. Evstigneev which will be reprinted as Volume 57 of the IMS Lecture Notes Monograph Series (http://imstat.org/publications/lecnotes.htm)