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On the Relation Between The Lévy Measure And The Jump Function Of A Lévy Process

Title:

On the Relation Between The Lévy Measure And The Jump Function Of A Lévy Process

Mozumder, Md. Sharif and Garrido, José (2007) On the Relation Between The Lévy Measure And The Jump Function Of A Lévy Process. Technical Report. Concordia University. Department of Mathematics & Statistics, Montreal, Quebec.

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Abstract

The Lévy and jump measures are two key characteristics of Lévy processes. This paper fills what seems to be a simple gap in the literature, by giving an explicit relation between the jump measure, which is a Poisson random measure, and the L´evy measure. This relation paves the way to a simple proof of the classical result on path continuity of Lévy processes in Section 2.
The jump function in Paul Lévy’s version of the Lévy-Khinchine formula and the Lévy measure in more recent characterizations essentially play the same role, but with different drift and Gaussian components. This point is shown in detail in Section 3, together with an explicit relation between the jump function and the Lévy measure.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Monograph (Technical Report)
Authors:Mozumder, Md. Sharif and Garrido, José
Series Name:Department of Mathematics & Statistics. Technical Report No. 2/07
Corporate Authors:Concordia University. Department of Mathematics & Statistics
Institution:Concordia University
Date:October 2007
ID Code:6682
Deposited By:DIANE MICHAUD
Deposited On:03 Jun 2010 16:07
Last Modified:08 Dec 2010 18:21
References:
Bertoin, Jean (1996). L´evy Processes.Cambridge University Press: Cambridge, UK.

Cont, Rama and Tankov, Peter (2004). Financial Modelling With Jump Processes. Chapman and Hall/CRC Financial Mathematics Series: London.

Kyprianou, Andreas, E. (2007). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Springer.

Mozumder, Sharif, Md. (2007). Estimation of the Lévy Measure for the Aggregate Claims Process in Risk Theory, MSc. Thesis, Department of Mathematics and Statistics, Concordia University: Montreal, Canada.

Sato, Ken–Iti (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press: Cambridge, UK.
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