Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $\alpha$-stable L\'{e}vy processes

TL;DR

This paper investigates the small noise asymptotics of integrated Ornstein-Uhlenbeck processes driven by alpha-stable Lévy processes, establishing weak convergence to the driving Lévy process and implications for first passage times.

Contribution

It proves weak convergence of integrated Ornstein-Uhlenbeck processes driven by alpha-stable Lévy processes to the Lévy process itself in the M_1 topology, including first passage time convergence.

Findings

Weak convergence of processes in M_1 topology.

Convergence of first passage times.

General approximation results for 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.