  We study the speed of extinction of continuous state branching processes in a
L\'evy environment, where the associated L\'evy process oscillates. Assuming
that the L\'evy process satisfies the Spitzer's condition and the existence of
some exponential moments, we extend recent results where the associated
branching mechanism was stable. Our study relies on the path analysis of the
process together with its environment, when this latter is conditioned to have
a non negative running infimum. This approach is inspired from the discrete
setting with i.i.d. environment studied in (Afanasyev et al. 2005).
