  In this paper, we study the speed of extinction of continuous state branching
processes in subcritical L\'evy environments. More precisely, when the
associated L\'evy process to the environment drifts to $-\infty$ and, under a
suitable exponential martingale change of measure (Esscher transform), the
environment either drifts to $-\infty$ or oscillates. We extend recent results
of Palau et al. (2016) and Li and Xu (2018), where the branching term is
associated to a spectrally positive stable L\'evy process and complement the
recent article of Bansaye et al. (2021) where the critical case was studied.
Our methodology combines a path analysis of the branching process together with
its L\'evy environment, fluctuation theory for L\'evy processes and the
asymptotic behaviour of exponential functionals of L\'evy processes. As an
application of the aforementioned results, we characterise the process
conditioned to survival also known as the $Q$-process.
