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.

Published in at http://dx.doi.org/10.3150/12-BEJ485 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (1)

• On extrema of stable processes

Cited by in corpus (6)

• Codifference as a practical tool to measure interdependence
• Random acceleration process on finite intervals under stochastic restarting
• Inertial Lévy flights in bounded domains
• Non-standard Skorokhod convergence of Levy-driven convolution integrals in Hilbert spaces
• Stochastic kinetics under combined action of two noise sources
• A Wong-Zakai Approximation for Random Slow Manifolds with Application to Parameter Estimation