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Transactions of the American Mathematical Society

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A jump-type SDE approach to real-valued self-similar Markov processes

Author: Leif Döring
Journal: Trans. Amer. Math. Soc. 367 (2015), 7797-7836
MSC (2010): Primary 60G18; Secondary 60G55
Published electronically: February 18, 2015
MathSciNet review: 3391900
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Abstract: In his 1972 paper, John Lamperti characterized all positive self-similar Markov processes as time-changes of exponentials of Lévy processes. In the past decade the problem of representing all non-negative self-similar Markov processes that do not necessarily have zero as a trap has been solved gradually via connections to ladder height processes and excursion theory.

Motivated by a recent article of Chaumont, Panti, and Rivero, we represent via jump-type SDEs the symmetric real-valued self-similar Markov processes that only decrease the absolute value by jumps and leave zero continuously.

Our construction of these self-similar processes involves a pseudo excursion construction and singular stochastic calculus arguments ensuring that solutions to the SDEs spend zero time at zero to avoid problems caused by a ``bang-bang'' drift.

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Additional Information

Leif Döring
Affiliation: Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI, 4, Place Jussieu, 75005 Paris, France

Received by editor(s): October 27, 2012
Received by editor(s) in revised form: August 9, 2013
Published electronically: February 18, 2015
Additional Notes: The author was supported by the Fondation Science Matématiques de Paris
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.