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

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 (1980), following some ideas in Chybiryakov (2006) 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.