Search

Let $X=(X_t, t\geq 0)$ be a self-similar Markov process taking values in $\mathbb{R}$ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of $X$ that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (2006) and Rivero (2005, 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable L\'evy process which to the best of our knowledge has not been studied before.

The purpose of this paper is to construct the law of a L\'evy process conditioned to avoid zero, under mild technicals conditions, two of them being that the point zero is regular for itself and the L\'evy process is not a compound Poisson process. Two constructions are proposed, the first lies on the method of $h$-transformation, which requires a deep study of the associated excessive function; while in the second it is obtained by conditioning the underlying L\'evy process to avoid zero up to an independent exponential time whose parameter tends to $0.$ The former approach generalizes some of the results obtained by Yano \cite{Yano10} in the symmetric case and recovers some of main results in Yano's work \cite{Yano13}, while the latter is reminiscent of \cite{Chaumont-Doney05}. We give some properties of the resulting process and we describe in some detail two examples: alpha stable and spectrally negative L\'evy processes., Comment: 35 pages

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)