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We study the pathwise description of a (sub-)critical continuous-statebranching process (CSBP) conditioned to be never extinct, as the solution to astochastic differential equation driven by Brownian motion and Poisson pointmeasures. The interest of our approach, which relies on applying Girsanovtheorem on the SDE that describes the unconditioned CSBP, is that it points outan explicit mechanism to build the immigration term appearing in theconditioned process, by randomly selecting jumps of the original one. Thesetechniques should also be useful to represent more general h-transforms ofdiffusion-jump processes.

On SDE associated with continuous-state branching processes conditioned to never be extinct