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 L\'{e}vy processes of small amplitude. We prove that the integrated Ornstein-Uhlenbeck process converges weakly to the underlying $\alpha$-stable L\'{e}vy 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 L\'{e}vy process by continuous integrated Ornstein-Uhlenbeck processes in the $M_1$-topology.