  In this paper we introduce a new class of state space models based on
shot-noise simulation representations of non-Gaussian L\'evy-driven linear
systems, represented as stochastic differential equations. In particular a
conditionally Gaussian version of the models is proposed that is able to
capture heavy-tailed non-Gaussianity while retaining tractability for inference
procedures. We focus on a canonical class of such processes, the
$\alpha$-stable L\'evy processes, which retain important properties such as
self-similarity and heavy-tails, while emphasizing that broader classes of
non-Gaussian L\'evy processes may be handled by similar methodology. An
important feature is that we are able to marginalise both the skewness and the
scale parameters of these challenging models from posterior probability
distributions. The models are posed in continuous time and so are able to deal
with irregular data arrival times. Example modelling and inference procedures
are provided using Rao-Blackwellised sequential Monte Carlo applied to a
two-dimensional Langevin model, and this is tested on real exchange rate data.
