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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Received: 22 August 2002 / Revised version: 10 February 2003 Published online: 15 April 2003
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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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DOI: https://doi.org/10.1007/s00440-003-0263-5
Keywords
- Brownian Motion
- High Dimension
- Markov Process
- Poisson Process
- Classical Result