  Constructing \Levy-driven Ornstein-Uhlenbeck processes is a task closely
related to the notion of self-decomposability. In particular, their transition
laws are linked to the properties of what will be hereafter called the
\emph{a-reminder} of their self-decomposable stationary laws. In the present
study we fully characterize the L\'evy triplet of these a-reminder s and we
provide a general framework to deduce the transition laws of the finite
variation Ornstein-Uhlenbeck processes associated with tempered stable
distributions. We focus finally on the subclass of the exponentially-modulated
tempered stable laws and we derive the algorithms for an exact generation of
the skeleton of Ornstein-Uhlenbeck processes related to such distributions,
with the further advantage of adopting a procedure computationally more
efficient than those already available in the existing literature.
