# 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 $X$.
Keywords :
Type de document :
Pré-publication, Document de travail
2010
Domaine :

Littérature citée [10 références]

https://hal.archives-ouvertes.fr/hal-00505184
Contributeur : Fernando Cordero <>
Soumis le : jeudi 22 juillet 2010 - 20:19:39
Dernière modification le : lundi 29 mai 2017 - 14:24:20
Document(s) archivé(s) le : lundi 25 octobre 2010 - 12:10:44

### Fichiers

On-the-scaling-property-in-flu...
Fichiers produits par l'(les) auteur(s)

### Identifiants

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

### Citation

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

### Métriques

Consultations de la notice

## 207

Téléchargements de fichiers