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February 2014 Small noise asymptotics and first passage times of integrated Ornstein–Uhlenbeck processes driven by $\alpha$-stable Lévy processes
Robert Hintze, Ilya Pavlyukevich
Bernoulli 20(1): 265-281 (February 2014). DOI: 10.3150/12-BEJ485

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.

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Robert Hintze. Ilya Pavlyukevich. "Small noise asymptotics and first passage times of integrated Ornstein–Uhlenbeck processes driven by $\alpha$-stable Lévy processes." Bernoulli 20 (1) 265 - 281, February 2014. https://doi.org/10.3150/12-BEJ485

Information

Published: February 2014
First available in Project Euclid: 22 January 2014

zbMATH: 1309.60059
MathSciNet: MR3160582
Digital Object Identifier: 10.3150/12-BEJ485

Rights: Copyright © 2014 Bernoulli Society for Mathematical Statistics and Probability

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Vol.20 • No. 1 • February 2014
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