Small noise asymptotics and first passage times of integrated Ornstein–Uhlenbeck processes driven by $\alpha$-stable Lévy processes

Abstract

In this paper, we study the asymptotic behaviour of one-dimensional integrated Ornstein–Uhlenbeck processes driven by $\alpha$-stable Lévy processes of small amplitude. We prove that the integrated Ornstein–Uhlenbeck process converges weakly to the underlying $\alpha$-stable Lévy process in the Skorokhod $M_{1}$-topology which secures the weak convergence of first passage times. This result follows from a more general result about approximations of an arbitrary Lévy process by continuous integrated Ornstein–Uhlenbeck processes in the $M_{1}$-topology.