Journal of the Mathematical Society of Japan

Relations between cone-parameter Lévy processes and convolution semigroups

Jan PEDERSEN and Ken-iti SATO

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Abstract

Cone-parameter Lévy processes and convolution semigroups on Rd are defined. Here, cone-parameter Lévy processes have stationary independent increments along increasing sequences on the cone. This property ensures that subordination of a cone-parameter Lévy process by an independent cone-valued cone-parameter Lévy process yields a new cone-parameter Lévy process. It is shown that a cone-parameter Lévy process induces a cone-parameter convolution semigroup. The converse statement, that any convolution semigroup appears in this way, is however not true. In particular we show that there is no Brownian motion with parameter in the set of nonnegative-definite symmetric d×d matrices. The question when a given cone-parameter convolution semigroup is generated by a Lévy process is studied. It is shown that this is the case if one of the following three conditions is satisfied: d=1; the convolution semigroup is purely non-Gaussian; or K is isomorphic to R+N.

Article information

Source
J. Math. Soc. Japan, Volume 56, Number 2 (2004), 541-559.

Dates
First available in Project Euclid: 3 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1191418644

Digital Object Identifier
doi:10.2969/jmsj/1191418644

Mathematical Reviews number (MathSciNet)
MR2048473

Zentralblatt MATH identifier
1056.60010

Subjects
Primary: 60B99: None of the above, but in this section
Secondary: 60G10: Stationary processes 60G51: Processes with independent increments; Lévy processes

Keywords
cone-parameter Lévy processes and convolution semigroups generative and non-generative convolution semigroups

Citation

PEDERSEN, Jan; SATO, Ken-iti. Relations between cone-parameter Lévy processes and convolution semigroups. J. Math. Soc. Japan 56 (2004), no. 2, 541--559. doi:10.2969/jmsj/1191418644. https://projecteuclid.org/euclid.jmsj/1191418644


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