  We study the pathwise description of a (sub-)critical continuous-state
branching process (CSBP) conditioned to be never extinct, as the solution to a
stochastic differential equation driven by Brownian motion and Poisson point
measures. The interest of our approach, which relies on applying Girsanov
theorem on the SDE that describes the unconditioned CSBP, is that it points out
an explicit mechanism to build the immigration term appearing in the
conditioned process, by randomly selecting jumps of the original one. These
techniques should also be useful to represent more general h-transforms of
diffusion-jump processes.
