A note on the supremum of a stable process

Details A note on the supremum of a stable process A note on the supremum of a stable process

2007-12-20

arxiv.org/abs/0712.3414v1

Mathematics

Probability

To appear in a Special Volume of Stochastics: An International Journal of Probability and Stochastic Processes (http://www.i

Scientific paper

If $X$ is a spectrally positive stable process of index $\alpha\in(1,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty),$ and $S_1=\sup_{0x)\backsim c\alpha^{-1}x^{-\alpha}$ 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 $s(x)\backsim cx^{-(\alpha+1)}$ as $x\to\infty.$

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