  We consider exit problems for general L\'evy processes, where the first
passage over a threshold is detected either immediately or at an epoch of an
independent homogeneous Poisson process. It is shown that the two corresponding
one-sided problems are related through a surprisingly simple identity.
Moreover, we identify a simple link between two-sided exit problems with one
continuous and one Poisson exit. Finally, Poisson exit of a reflected process
is connected to the continuous exit of a process reflected at Poisson epochs,
and a link between some Parisian type exit problems is established. With the
appropriate perspective, the proofs of all these relations turn out to be quite
elementary. For spectrally one-sided L\'evy processes this approach enables
alternative proofs for a number of previously established identities, providing
additional insight.
