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2014 | OriginalPaper | Chapter

7. Lévy Processes at First Passage

Author : Andreas E. Kyprianou

Published in: Fluctuations of Lévy Processes with Applications

Publisher: Springer Berlin Heidelberg

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Abstract

This chapter is devoted to studying how the Wiener–Hopf factorisation can be used to characterise the behaviour of any Lévy process at first passage over a fixed level. The case of a subordinator will be excluded throughout this chapter, as this has been dealt with in Chap. 5. Nonetheless, the analysis of how subordinators make first passage will play a crucial role in understanding the case of a general Lévy process.

To some extent, the results we present on the first-passage problem suffer from a lack of analytical explicitness. This is due to the same symptoms present in our understanding of the Wiener–Hopf factorisation. Nonetheless there is sufficient mathematical structure to establish qualitative statements concerning the characterisation of the first-passage problem. This becomes more apparent when looking at asymptotic properties of the established characterisations.

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previous chapter The Wiener–Hopf Factorisation

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Note that \(\int_{1}^{\infty}t^{-1}\mathbb{P}(X_{t}\geq 0){\mathrm{d}}t + \int_{1}^{\infty}t^{-1} \mathbb{P}(X_{t}\leq0){\mathrm {d}}t \geq \int_{1}^{\infty}t^{-1}{\mathrm{d}}t =\infty\) and hence at least one of the integral tests in (i) or (ii) fails.

Theorem 7.4 is built on the foundational, but weaker, result of Chung and Fuchs (1951). See also Kingman (1964).

Recall the definition in (5.2).

Specifically we use the independence of the pairs \((\underline{G}_{\mathbf{e}_{q}}, \underline{X}_{\mathbf{e}_{q}})\) and \((\mathbf{e}_{q} - \underline{G}_{\mathbf{e}_{q}}, X_{\mathbf{e}_{q}} - \underline{X}_{\mathbf{e}_{q}})\).

Recall from the discussion at the end of Sect. 5.3 that, formally speaking, a compound Poisson subordinator cannot creep, despite the fact that a given point may lie in its range with positive probability.

Formally speaking, any distribution F on [0,∞) is subexponential if, when X ₁ and X ₂ are independent random variables with distribution F, \(\mathbb{P}(X_{1} + X_{n} >x)\sim2\mathbb{P}(X_{1}>x)\), as x↑∞.

The case that X or −X is spectrally negative is dealt with later in Exercise 8.11.

There is a typographic error in Lemma 3 of Rogozin (1972) for the two-sided exit formula. In the notation of that paper, the roles of q and (1−q) should be exchanged and the upper delimiter in the integral should be x and not ∞.

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Title: Lévy Processes at First Passage
Author: Andreas E. Kyprianou
Publisher: Springer Berlin Heidelberg
Book: Fluctuations of Lévy Processes with Applications
Print ISBN: 978-3-642-37631-3

Electronic ISBN: 978-3-642-37632-0

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-642-37632-0_7