A Ciesielski-Taylor type identity for positive self-similar Markov processes

Kyprianou, A. E.; Patie, P.

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 917-928.

L'objectif principal de ce papier est de donner une preuve d'une version générale de l'identité de Ciesielski-Taylor pour la famille de processus positifs auto-similaires markoviens de type spectralement négatif, ce qui nous permet d'unifier l'ensemble des résultats déjà connus sur ce sujet. Notre preuve s'appuie sur trois concepts importants. Tout d'abord, nous introduisons une famille de transformations qui associe l'ensemble des exposants de Laplace de processus de Lévy spectralement négatifs à lui-même. Ensuite, nous combinons des résultats empruntés à la théorie des fluctuations des processus de Lévy spectralement négatifs (voir e.g., [In Séminaire de Probabalités XXXVIII (2005) 16-29 Springer]) et à celles des processus positifs auto-similaires markoviens spectralement négatifs élaborées plus récemment par [Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 667-684].

The aim of this note is to give a straightforward proof of a general version of the Ciesielski-Taylor identity for positive self-similar Markov processes of the spectrally negative type which umbrellas all previously known Ciesielski-Taylor identities within the latter class. The approach makes use of three fundamental features. Firstly, a new transformation which maps a subset of the family of Laplace exponents of spectrally negative Lévy processes into itself. Secondly, some classical features of fluctuation theory for spectrally negative Lévy processes (see, e.g., [In Séminaire de Probabalités XXXVIII (2005) 16-29 Springer]) as well as more recent fluctuation identities for positive self-similar Markov processes found in [Ann. Inst. H. Poincaré Probab. Statist. 45 (2009) 667-684].

Classification : 60G18, 60G51, 60B52
Mots clés : positive self-similar Markov process, Ciesielski-Taylor identity, spectrally negative Lévy process, Bessel processes, stable processes, lamperti-stable processes

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