%0 Generic %1 caballero-lambert-bravo-2008 %A Caballero, Maria-Emilia %A Lambert, Amaury %A Bravo, Geronimo Uribe %D 2008 %K branching_processes lamperti_representation subordinators %T Proof(s) of the Lamperti representation of Continuous-State Branching Processes %U http://arxiv.org/abs/0802.2693v1 %X The representation of continuous-state branching processes (CSBPs) as time-changed Lévy processes with no negative jumps was discovered by John Lamperti in 1967 but was never proved. The goal of this paper is to provide a proof, and we actually provide two. The first one relies on studying the time-change, using martingales and the Lévy-Itô representation of Lévy processes. It gives insight into a stochastic differential equation satisfied by CSBPs and on its relevance to the branching property. The other method studies the time-change in a discrete model, where an analogous Lamperti representation is evident, and provides functional approximations to Lamperti transforms by introducing a new topology on Skorohod space. Some classical arguments used to study CSBPs are reconsidered and simplified.

@misc{caballero-lambert-bravo-2008, abstract = { The representation of continuous-state branching processes (CSBPs) as time-changed L\'evy processes with no negative jumps was discovered by John Lamperti in 1967 but was never proved. The goal of this paper is to provide a proof, and we actually provide two. The first one relies on studying the time-change, using martingales and the L\'evy-It\^o representation of L\'evy processes. It gives insight into a stochastic differential equation satisfied by CSBPs and on its relevance to the branching property. The other method studies the time-change in a discrete model, where an analogous Lamperti representation is evident, and provides functional approximations to Lamperti transforms by introducing a new topology on Skorohod space. Some classical arguments used to study CSBPs are reconsidered and simplified.}, added-at = {2008-09-18T21:03:36.000+0200}, author = {Caballero, Maria-Emilia and Lambert, Amaury and Bravo, Geronimo Uribe}, biburl = {https://www.bibsonomy.org/bibtex/21365b0d00e14a6da4791ce1fb7cf1c3e/peter.ralph}, description = {Proof(s) of the Lamperti representation of Continuous-State Branching Processes}, interhash = {87823eaa41217a132d62da9551ecb535}, intrahash = {1365b0d00e14a6da4791ce1fb7cf1c3e}, keywords = {branching_processes lamperti_representation subordinators}, timestamp = {2008-09-18T21:03:36.000+0200}, title = {Proof(s) of the Lamperti representation of Continuous-State Branching Processes}, url = {\url{http://arxiv.org/abs/0802.2693v1}}, year = 2008 }