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On some transformations between positive self--similar Markov processes

By Loïc Chaumont and Víctor Rivero

Abstract

A path decomposition at the infimum for positive self-similar Markov processes (pssMp) is obtained. Next, several aspects of the conditioning to hit 0 of a pssMp are studied. Associated to a given a pssMp $X,$ that never hits 0, we construct a pssMp $X^{\downarrow}$ that hits 0 in a finite time. The latter can be viewed as $X$ conditioned to hit 0 in a finite time and we prove that this conditioning is determined by the pre-minimum part of $X.$ Finally, we provide a method for conditioning a pssMp that hits 0 by a jump to do it continuously

Topics: Self-similar Markov processes, Lévy processes, weak convergence, decomposition at the minimum, conditioning, $h$-transforms, MSC: 60G18 (60G17), [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Publisher: HAL CCSD
Year: 2006
OAI identifier: oai:HAL:hal-00016781v1
Provided by: Hal-Diderot
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