Real self-similar processes started from the origin

Steffen Dereich, Leif Döring, Andreas E. Kyprianou

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

Since the seminal work of Lamperti, there is a lot of interest in the understanding of the general structure of self-similar Markov processes. Lamperti gave a representation of positive self-similar Markov processes with initial condition strictly larger than 0 which subsequently was extended to zero initial condition. For real self-similar Markov processes (rssMps), there is a generalization of Lamperti's representation giving a one-to-one correspondence between Markov additive processes and rssMps with initial condition different from the origin. We develop fluctuation theory for Markov additive processes and use Kuznetsov measures to construct the law of transient real self-similar Markov processes issued from the origin. The construction gives a pathwise representation through two-sided Markov additive processes extending the Lamperti- Kiu representation to the origin.

Original languageEnglish
Pages (from-to)1952-2003
Number of pages52
JournalAnnals of Probability
Volume45
Issue number3
Early online date15 May 2017
DOIs
Publication statusPublished - 31 May 2017

Keywords

  • Fluctuation theory
  • Markov additive process
  • Self-similar process

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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