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Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations
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  • Published: May 2003

Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations

  • Martin Jacobsen1 &
  • Marc Yor2 

Probability Theory and Related Fields volume 126, pages 1–28 (2003)Cite this article

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  • 7 Citations

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Abstract.

 A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper.

The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift.

Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of Copenhagen, 5 Universitetsparken, 2100 Copenhagen ∅, Denmark. e-mail: martin@math.ku.dk, , , , , , DK

    Martin Jacobsen

  2. Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, Boi te courrier 188, 75252 Paris Cedex 05, France., , , , , , FR

    Marc Yor

Authors
  1. Martin Jacobsen
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  2. Marc Yor
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Additional information

Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003

RID="*"

ID="*" MaPhySto – Centre for Mathematical Physics and Stochastics, funded by a grant from the Danish National Research Foundation

Mathematics Subject Classification (2000): 60G18, 60G51, 60J25, 60J60, 60J75

Key words or phrases: Lévy process – Self-similarity – Time-change – Exponential functional – Brownian motion – Bessel process – Piecewise deterministic Markov process – Moment sequence

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Jacobsen, M., Yor, M. Multi-self-similar Markov processes on ℝ+ n and their Lamperti representations. Probab. Theory Relat. Fields 126, 1–28 (2003). https://doi.org/10.1007/s00440-003-0263-5

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  • Issue Date: May 2003

  • DOI: https://doi.org/10.1007/s00440-003-0263-5

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Keywords

  • Brownian Motion
  • High Dimension
  • Markov Process
  • Poisson Process
  • Classical Result
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