Approximations of Lévy processes by integrated fast oscillating Ornstein-Uhlenbeck processes

In this paper, we study a smooth approximation of an arbitrary cadlag Levy 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.