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Centre for Mathematical Physics and Stochastics
Department of Mathematical Sciences, University of Aarhus

Funded by The Danish National Research Foundation

MPS-LN 2000-7
May 2000

Density Transformation in Lévy Processes


Ken-iti Sato


From the preface:

These notes are outgrowth of the Concentrated Advanced Course on Léevy Processes, MaPhySto, University of Aarhus, January 24-28, 2000. The course started with the definition of Lévy processes and discussed their elementary properties and their transformations. It was based on the book K. Sato, Lévy Processes and Infinitely Divisible Distributions, 1999, Cambridge University Press. One of the subjects in the course was the density transformation of Lévy processes. This is also discussed in Chapter 6 of the book, but I treated it in a different way in the course, fully utilizing the power of the Hellinger-Kakutani inner product and distance of order $\alpha$. This method was adopted by C.M. Newman in 1972-73 but it is not widely known. In pursuing this method, I found that the Lebesgue decomposition of path space measures of Lévy processes could be obtained easily. Together with the description of the Radon-Nikodym densities of the absolutely continuous parts, this clarifies the relationship of the path space measures on a finite time interval of two given Lévy processes on Rd. The main part of these notes concentrates on this subject and gives the results with complete proofs.

The other parts of the lectures of the course are attached here as Appendix A.

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