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 timechanged 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 wherethe characteristics of the underlying processes can be computed explicitly.