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We consider the problem of obtaining effective representations for thesolutions of linear, vector-valued stochastic differential equations (SDEs)driven by non-Gaussian pure-jump L\'evy processes, and we show how suchrepresentations lead to efficient simulation methods. The processes consideredconstitute a broad class of models that find application across the physicaland biological sciences, mathematics, finance and engineering. Motivated byimportant 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 thegeneralised hyperbolic, normal-Gamma, and normal tempered stable cases. Simple,explicit conditions are identified for the convergence of the residual of atruncated shot-noise representation to a Brownian motion in the case of thepure L\'evy process, and to a Brownian-driven SDE in the case of theL\'evy-driven SDE. These results provide Gaussian approximations to the smalljumps of the process under the NVM representation. The resultingrepresentations are of particular importance in state inference and parameterestimation for L\'evy-driven SDE models, since the resulting conditionallyGaussian structures can be readily incorporated into latent variable inferencemethods such as Markov chain Monte Carlo (MCMC), Expectation-Maximisation (EM),and sequential Monte Carlo.

Generalised shot noise representations of stochastic systems driven by non-Gaussian L\'evy processes