In this paper, we study a smooth approximation of an arbitrary càdlàg Lévy process. Such approximation processes are known as integrated fast oscillating Ornstein–Uhlenbeck (OU) processes. We know that approximating processes are continuous, while the limit of processes may be discontinuous, so convergence in uniform topology or Skorokhod $J_1$-topology will not hold in general. Therefore, we establish an approximation in Skorokhod $M_1$-topology. Note that the convergence is almost surely, which is an extension result of Hintze and Pavlyukevich.

中文翻译：

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通过集成快速振荡 Ornstein-Uhlenbeck 过程近似 Lévy 过程

在本文中，我们研究了任意 càdlàg Lévy 过程的平滑逼近。这种近似过程被称为积分快速振荡奥恩斯坦-乌伦贝克（OU）过程。我们知道，近似过程是连续的，而过程的极限可能是不连续的，因此均匀拓扑或 Skorokhod 中的收敛$J_1$- 拓扑一般不成立。因此，我们在 Skorokhod 中建立一个近似值${\mathrm{中号}}_1$-拓扑。请注意，收敛几乎是肯定的，这是 Hintze 和 Pavlyukevich 的推广结果。