Top

Published in:

2014 | OriginalPaper | Chapter

3. More Distributional and Path-Related Properties

Author : Andreas E. Kyprianou

Published in: Fluctuations of Lévy Processes with Applications

Publisher: Springer Berlin Heidelberg

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We consider some more distributional and path-related properties of general Lévy processes. Specifically, we examine the strong Markov property, duality, moments and exponential change of measure.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter The Lévy–Itô Decomposition and Path Structure

next chapter General Storage Models and Paths of Bounded Variation

Available only for authorised users

As we shall see later, this is a phenomenon which is not exclusive to compound Poisson processes with strictly negative drift. The same behaviour is experienced by, for example, Lévy processes of bounded variation with strictly negative drift.

It is worth reminding oneself, for the sake of clarity, that \(X_{t_{n}}\rightarrow X_{t}\) \(\mathbb{P}\)-a.s. as n↑∞ means that, for all ε>0, there exists an almost surely finite N>0 such that \(|X_{t_{n}} - X_{t}|<\varepsilon\) for all n>N. This does not contradict the fact that there might be an infinite number of discontinuities in the path of X in an arbitrary small neighbourhood of t.

In the case that β<0, simply consider the forthcoming argument for −X. For β=0 the statement of the theorem is trivial.

As noted just after Definition 1.1, we are making an abuse of notation in the use of the measure \(\mathbb{P}\) here. Strictly speaking, we should work with the measure \(\mathbb{P}\times \mathcal{P}\), where \(\mathcal{P}\) is the probability measure on the space in which the random variable e _q is defined. This abuse of notation will be repeated at various points throughout this text.

Bingham, N.H. (1975) Fluctuation theory in continuous time, Adv. Appl. Probab. 7, 705–766. CrossRefMATHMathSciNet

Blumenthal, R.M. and Getoor, R.K. (1968) Markov Processes and Potential Theory. Academic, New York. MATH

Dellacherie, C. and Meyer, P.A. (1975–1993) Probabilités et Potentiel. Hermann, Paris. Chaps. I–VI, 1975; Chaps. V–VIII, 1980; Chaps. IX–XI, 1983; Chaps. XII–XVI, 1987; Chaps. XVII–XXIV, with Maisonneuve, B., 1992.

Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Vol II. 2nd Edition. Wiley, New York. MATH

Title: More Distributional and Path-Related Properties
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_3