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

Abstract

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 [Formula: see text]-topology will not hold in general. Therefore, we establish an approximation in Skorokhod [Formula: see text]-topology. Note that the convergence is almost surely, which is an extension result of Hintze and Pavlyukevich.