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.

翻译：我们研究由非高斯纯跳Lévy过程驱动的线性向量值随机微分方程（SDEs）解的有效表示问题，并展示此类表示如何导出高效模拟方法。所考虑的构成过程涵盖物理与生物科学、数学、金融及工程领域中广泛应用的一类广泛模型。受统计推断中重要相关问题的启发，当驱动Lévy过程存在正态方差-均值（NVM）混合表示时（包括广义双曲、正态-伽马及正态温和稳定情形），我们推导出新型广义散粒噪声模拟方法。针对纯Lévy过程情形，识别出截断散粒噪声表示残差收敛至布朗运动的简洁显式条件；针对Lévy驱动SDEs情形，则识别出收敛至布朗驱动SDE的条件。这些结果为NVM表示下过程的小跳跃提供了高斯近似。所得表示对Lévy驱动SDEs模型的状态推断与参数估计尤为重要，因为由此产生的条件高斯结构可便捷地融入隐变量推断方法，如马尔可夫链蒙特卡洛（MCMC）、期望最大化（EM）及序贯蒙特卡洛。