Continuous-State Branching Processes and Self-Similarity

In this paper we study the α-stable continuous-state branching processes (for α ∈ (1, 2]) and the α-stable continuous-state branching processes conditioned never to become extinct in the light of positive self-similarity. Understanding the interaction of the Lamperti transformation for continuous-state branching processes and the Lamperti transformation for positive, self-similar Markov processes gives access to a number of explicit results concerning the paths of α-stable continuous-state branching processes and α-stable continuous-state branching processes conditioned never to become extinct.