# On the scaling property in fluctuation theory for stable Lévy processes

Abstract : We find an expression for the joint Laplace transform of the law of $(T_{[x,+\infty[},X_{T_{[x,+\infty[}})$ for a Lévy process $X$, where $T_{[x,+\infty[}$ is the first hitting time of $[x,+\infty[$ by $X$. When $X$ is an $\alpha$-stable Lévy process, with $1<\alpha<2$, we show how to recover from this formula the law of $X_{T_{[x,+\infty[}}$; this result was already obtained by D. Ray, in the symmetric case and by N. Bingham, in the case when $X$ is non spectrally negative. Then, we study the behaviour of the time of first passage $T_{[x,+\infty[},$ conditioned to $\{X_{T_{[x,+\infty[}} -x \leq h\}$ when $h$ tends to $0$. This study brings forward an asymptotic variable $T_x^0$, which seems to be related to the absolute continuity of the law of the supremum of $X$.
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https://hal.archives-ouvertes.fr/hal-00505184
Contributor : Fernando Cordero <>
Submitted on : Thursday, July 22, 2010 - 8:19:39 PM
Last modification on : Thursday, March 21, 2019 - 1:19:41 PM
Long-term archiving on: Monday, October 25, 2010 - 12:10:44 PM

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• HAL Id : hal-00505184, version 1
• ARXIV : 1007.3959

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Fernando Cordero. On the scaling property in fluctuation theory for stable Lévy processes. 2010. ⟨hal-00505184⟩

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