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2014 | OriginalPaper | Buchkapitel

1. Lévy Processes and Applications

verfasst von : Andreas E. Kyprianou

Erschienen in: Fluctuations of Lévy Processes with Applications

Verlag: Springer Berlin Heidelberg

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Abstract

We define and characterise the class of Lévy processes. To illustrate the variety of processes captured within the definition of a Lévy process, we explore briefly the relationship between Lévy processes and infinitely divisible distributions. We also discuss some classical applied probability models, which are built on the strength of well-understood path properties of elementary Lévy processes. We hint at how generalisations of these models may be approached using more sophisticated Lévy processes. At a number of points later on in this text, we handle these generalisations in more detail.

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Nächstes Kapitel The Lévy–Itô Decomposition and Path Structure

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We shall also repeatedly abuse this notation throughout the book as, on occasion, we will need to talk about a Lévy process, X, referenced against a random time horizon, say e, which is independent of X and exponentially distributed. In that case, we shall use \(\mathbb{P}\) (and accordingly \(\mathbb{E}\)) for the product law associated with X and e.

Where we have assumed natural enlargement here, it is commonplace in other literature to assume that the filtration \(\mathbb{F}\) satisfies “les conditions habituelles”. In particular, for each t≥0, \(\mathcal{F}_{t}\) is complete with respect to all null sets of \(\mathbb{P}\). This can create problems, for example, when looking at changes of measure (as indeed we will in this book). The reader is encouraged to read Warning 1.3.39. of Bichteler (2002) for further investigation.

Here and throughout the remainder of the book, we use the convention that, for any n=0,1,2,…, \(\sum_{n+1}^{n}\cdot=0\).

The notation ℜz refers to the real part of z.

We assume that the reader is familiar with the basic notion of a stopping time for a Markov process as well as the strong Markov property. Both will be dealt with in more detail for a general Lévy process in Chap. 3.

Following standard notation, the measure δ ₀ is the Dirac measure, which assigns a unit atom to the point 0.

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Titel: Lévy Processes and Applications
verfasst von: Andreas E. Kyprianou
Verlag: Springer Berlin Heidelberg
Buch: Fluctuations of Lévy Processes with Applications
Print ISBN: 978-3-642-37631-3

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

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-642-37632-0_1