The Lamperti representation of real-valued self-similar Markov processes

Abstract

In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative invariant processes. Doing so, we complete Kiu's work [Stochastic Process. Appl. 10 (1980) 183-191], following some ideas in Chybiryakov [Stochastic Process. Appl. 116 (2006) 857-872] in order to characterize the underlying processes in this representation. We provide some examples where the characteristics of the underlying processes can be computed explicitly.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ460 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stable Lévy processes, self-similarity and the unit ball

“…It penalises those paths that approach the origin and rewards those that stray away from the origin. In fact, it has been shown in Kyprianou et al (2015) that, for α ∈ (0, 1), in the appropriate sense, the change of measure is equivalent to conditioning the stable process to continuously absorb at the origin, and when α ∈ (1, 2), in Chaumont et al (2013); Kuznetsov et al (2014) it is shown that the change of measure is equivalent to conditioning the stable process to avoid the origin.…”

“…The next theorem generalises its counterpart for positive self-similar Markov processes, namely Theorem 2.2 and is due to Chaumont et al (2013) and Kuznetsov et al (2014). …”

“…With only the sporadic works of e.g. Graversen and Vuolle-Apiala (1986); Vuolle-Apiala and ; Kiu (1980) in the 1980s, only very recently has the bigger picture begun to emerge some 30 or more years later in Xiao (1998); Nane et al (2010); Chaumont et al (2013); Kuznetsov et al (2014); Alili et al (2017); Kyprianou et al (2015); Kyprianou (2016); ).…”

“…This comes about by a change of measure, which corresponds to a Doob h-transform to the semigroup of a two-sided jumping stable process killed on first hitting the origin if α ∈ (1, 2). As it is instructive for future discussion, we give a proof of the following result which is originally from Chaumont et al (2013), for α ∈ (1, 2), and Kyprianou et al (2015), for α ∈ (1, 2). Our proof differs slightly from its original setting.…”

“…We present a dialogue that appeals as much as possible to path decompositions. Most notable in this respect is the Lamperti-Kiu decomposition of self-similar Markov processes given in Kiu (1980); Chaumont et al (2013); Alili et al (2017) and the Riesz-Bogdan-Żak transformation given in Bogdan and Żak (2006).…”