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：本文研究由非高斯纯跳跃 Lévy 过程驱动的线性向量值随机微分方程（SDE）有效解的表示问题，并展示此类表示如何导出高效模拟方法。所考虑的过程构成了一类广泛应用于物理、生物科学、数学、金融和工程领域的模型。受统计推断中重要相关问题的驱动，我们推导了当驱动 Lévy 过程存在正态方差-均值（NVM）混合表示（包括广义双曲线、正态伽马和正态时变稳定情形）时的新型广义散粒噪声模拟方法。针对纯 Lévy 过程情形，我们识别出了截断散粒噪声表示余项收敛到布朗运动的简单显式条件；针对 Lévy 驱动 SDE 情形，则识别出了余项收敛到布朗驱动 SDE 的相应条件。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 SDE 模型的状态推断和参数估计具有特殊意义，因为由此产生的条件高斯结构可便捷地纳入隐变量推断方法（如马尔可夫链蒙特卡洛（MCMC）、期望最大化（EM）和序贯蒙特卡洛）中。