  Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped
on hitting zero, are generalized by allowing the process governing immigration
to be any L\'evy process without negative jumps. Unlike the CBIs, these newly
introduced processes do not appear to satisfy any natural affine property on
the level of the Laplace transforms of the semigroups. Basic properties are
noted. Explicit formulae (on neighborhoods of infinity) for the Laplace
transforms of the first passage times downwards and of the explosion time are
derived.
