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

Robert Hintze; Ilya Pavlyukevich

doi:10.3150/12-bej485

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arXiv · 1205.6116

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

Robert Hintze·Ilya Pavlyukevich

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Abstract

In this paper, we study the asymptotic behaviour of one-dimensional integrated Ornstein-Uhlenbeck processes driven by $α$-stable Lévy processes of small amplitude. We prove that the integrated Ornstein-Uhlenbeck process converges weakly to the underlying $α$-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. 2014-02-05. Small noise asymptotics and first passage times of integrated Ornstein-Uhlenbeck processes driven by $α$-stable Lévy processes. https://doi.org/10.3150/12-bej485

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