  In this paper a class of Ornstein--Uhlenbeck processes driven by compound
Poisson processes is considered. The jumps arrive with exponential waiting
times and are allowed to be two-sided. The jumps are assumed to form an iid
sequence with distribution a mixture (not necessarily convex) of exponential
distributions, independent of everything else. The fact that downward jumps are
allowed makes passage of a given lower level possible both by continuity and by
a jump. The time of this passage and the possible undershoot (in the jump case)
is considered. By finding partial eigenfunctions for the infinitesimal
generator of the process, an expression for the joint Laplace transform of the
passage time and the undershoot can be found.
  From the Laplace transform the ruin probability of ever crossing the level
can be derived. When the drift is negative this probability is less than one
and its asymptotic behaviour when the initial state of the process tends to
infinity is determined explicitly.
  The situation where the level to cross decreases to minus infinity is more
involved: The level to cross plays a much more fundamental role in the
expression for the joint Laplace transform than the initial state of the
process. The limit of the ruin probability in the positive drift case and the
limit of the distribution of the undershoot in the negative drift case is
derived.
