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

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Abstract

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 Levy processes into itself. Secondly, some classical features of fluctuation theory for spectrally negative Levy processes (see, e.g., [In Seminaire de Probabalites XXXVIII (2005) 16-29 Springer]) as well as more recent fluctuation identities for positive self-similar Markov processes found in [Ann. Inst. H. Poincare Probab. Statist. 45 (2009) 667-684].
Original languageEnglish
Pages (from-to)917-928
Number of pages12
JournalAnnales de l'Institut Henri Poincaré: Probabilités et Statistiques
Volume47
Issue number3
DOIs
Publication statusPublished - Aug 2011

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Self-similar Processes
Markov Process
Lévy Process
Fluctuations (theory)
Laplace
Poincaré
Exponent
Fluctuations
Subset
Markov process
Lévy process

Keywords

  • Lamperti-stable processes
  • Bessel processes
  • stable processes
  • positive self-similar Markov process
  • spectrally negative Levy process
  • Ciesielski-Taylor identity

Cite this

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title = "A Ciesielski-Taylor type identity for positive self-similar Markov processes",
abstract = "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 Levy processes into itself. Secondly, some classical features of fluctuation theory for spectrally negative Levy processes (see, e.g., [In Seminaire de Probabalites XXXVIII (2005) 16-29 Springer]) as well as more recent fluctuation identities for positive self-similar Markov processes found in [Ann. Inst. H. Poincare Probab. Statist. 45 (2009) 667-684].",
keywords = "Lamperti-stable processes, Bessel processes, stable processes, positive self-similar Markov process, spectrally negative Levy process, Ciesielski-Taylor identity",
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AB - 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 Levy processes into itself. Secondly, some classical features of fluctuation theory for spectrally negative Levy processes (see, e.g., [In Seminaire de Probabalites XXXVIII (2005) 16-29 Springer]) as well as more recent fluctuation identities for positive self-similar Markov processes found in [Ann. Inst. H. Poincare Probab. Statist. 45 (2009) 667-684].

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