We present a new approach to positive self-similar Markov processes (pssMps) by reformulating Lamperti's transformation via jump type SDEs. As applications, we give direct constructions of pssMps (re)started continuously at zero if the Lamperti transformed Lévy process is spectrally negative. Our paper can be seen as a continuation of similar studies for continuous state branching processes.
"Jump type SDEs for self-similar processes." Electron. J. Probab. 17 1 - 39, 2012. https://doi.org/10.1214/EJP.v17-2402