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The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

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Abstract

We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically \(\alpha \)-stable Lévy processes with \(\alpha <1\). Our main result states that if the left tail of the Lévy measure is regularly varying with index \(- \alpha \), and the moving boundary is equal to \(1 - t^{\gamma }\) for some \(\gamma <1/\alpha \), then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary \(1 + t^{\gamma }\) with \(\gamma <1/\alpha \) under the assumption of a regularly varying right tail with index \(-\alpha \).

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Acknowledgments

Frank Aurzada and Tanja Kramm were supported by the DFG Emmy Noether programme. We are grateful to Mikhail Lifshits for valuable discussions on the subject of the article. We would like to thank the two referees for their detailed comments, which improved the exposition of the paper significantly and corrected earlier mistakes.

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  1. Arbeitsgruppe Stochastik, Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Frank Aurzada & Tanja Kramm

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  1. Frank Aurzada
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  2. Tanja Kramm
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Correspondence to Frank Aurzada.

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Aurzada, F., Kramm, T. The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes. J Theor Probab 29, 737–760 (2016). https://doi.org/10.1007/s10959-015-0596-x

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  • DOI: https://doi.org/10.1007/s10959-015-0596-x

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