  We consider the problem of obtaining effective representations for the
solutions of linear, vector-valued stochastic differential equations (SDEs)
driven by non-Gaussian pure-jump L\'evy processes, and we show how such
representations lead to efficient simulation methods. The processes considered
constitute a broad class of models that find application across the physical
and biological sciences, mathematics, finance and engineering. Motivated by
important relevant problems in statistical inference, we derive new,
generalised shot-noise simulation methods whenever a normal variance-mean (NVM)
mixture representation exists for the driving L\'evy process, including the
generalised hyperbolic, normal-Gamma, and normal tempered stable cases. Simple,
explicit conditions are identified for the convergence of the residual of a
truncated shot-noise representation to a Brownian motion in the case of the
pure L\'evy process, and to a Brownian-driven SDE in the case of the
L\'evy-driven SDE. These results provide Gaussian approximations to the small
jumps of the process under the NVM representation. The resulting
representations are of particular importance in state inference and parameter
estimation for L\'evy-driven SDE models, since the resulting conditionally
Gaussian structures can be readily incorporated into latent variable inference
methods such as Markov chain Monte Carlo (MCMC), Expectation-Maximisation (EM),
and sequential Monte Carlo.
