# Talk:Lévy process

WikiProject Statistics (Rated Start-class, Mid-importance)

This article is within the scope of the WikiProject Statistics, a collaborative effort to improve the coverage of statistics on Wikipedia. If you would like to participate, please visit the project page or join the discussion.

Start  This article has been rated as Start-Class on the quality scale.
Mid  This article has been rated as Mid-importance on the importance scale.
WikiProject Mathematics (Rated Start-class, Mid-importance)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
 Start Class
 Mid Importance
Field:  Probability and statistics

## Spelling

Corrected spelling of Khintchine. See here http://www.maths.bath.ac.uk/~ak257/LCSB/part1.pdf also Philip Protter's "Stochastic Integration and Differential Equations" —Preceding unsigned comment added by 155.198.190.241 (talk) 18:25, 18 November 2008 (UTC)

## Lévy-Khintchine representation

I didn't get this part. There are many terms that are used without previous definition, and the conclusion so one can see that a purely continuous Lévy process is a Brownian motion with drift is in apparent contradiction with the article Infinite divisibility (probability), where it's implied that we could generate a Lévy process using the Cauchy distribution, that would not generate a tame Brownian motion but a wild Lévy flight. Albmont 17:54, 29 October 2007 (UTC)

Ok, now I get it. See these lectures by Álvaro Cartea. In short, all other Lévy processes are (everywhere) discontinuous. The Cauchy process would be discontinuous everywhere too. Albmont (talk) 20:30, 19 November 2007 (UTC)

## Proper Definition and Levy-Ito Decomposition

I think it would be nice to have a proper mathematical definition of a Levy process instead of a paragraph of woffle. Also I think that the Levy-Ito decomposition should be included as it is one of the fundamental properties of a Levy process. I am willing to write these up unless anyone objects... Hyperbola 15:44, 22 February 2009 (UTC)

## Lévy measure

Article Lévy measure redirects here, but there is no mention/definition. Melcombe (talk) 09:56, 11 May 2010 (UTC).

I added a definition of the Lévy measure (Sept 2012)

## References needed

I think this article desperately needs some good references. The article by Applebaum is not sufficient. In particular it makes no mention at all of the asserted polynomial nature of the moments. Pere Callahan (talk) 00:20, 21 June 2010 (UTC) I have added three more references: graduate textbooks which include introductory sections on Levy processes with detailed proofs of the result given here. (Sept 2012) — Preceding unsigned comment added by 90.61.162.95 (talk) 20:41, 3 September 2012 (UTC)

## Cádlág modification

There should be some clarification of what exactly càdlàg modification is! Following the càdlàg link does not make it clearer. --62.107.83.243 (talk) 21:39, 17 November 2010 (UTC)

## Statistics

The article focuses on probability mathematics. It would be useful to include more on probability models of real phenomena and statistics. For the latter, see Michael Sorensen and co-author on Exponential Familes of Stochastic Processes, etc.: Steffan Iacus (sic) had a Springer monograph on inference for SDEs driven by exponential families, also.  Kiefer.Wolfowitz  (Discussion) 13:18, 13 April 2011 (UTC)

## Constructing a stochastic probability measure

Is this whole section even necessary? It seems excessive and indulgent and not particularly specific to the article at hand. 128.122.20.85 (talk) 00:56, 30 June 2011 (UTC)

I agree: this section is not necessary. It has nothing to do with Levy processes. — Preceding unsigned comment added by 86.195.224.109 (talk) 21:40, 27 August 2012 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Lévy process/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

 You are wrong there about discontinuity. The only Lévy process, which is everywhere continuous is a Brownian motion with a certain covariance matrix and a drift. Every other Lévy process is, with probability one, either discontinous on dense set of points (not everywhere!) or it makes only finite jumps on every compact time intervall.

Last edited at 11:59, 20 June 2008 (UTC). Substituted at 20:05, 1 May 2016 (UTC)

## Characteristic function of a process

In the article is says:

If ${\displaystyle X=(X_{t})_{t\geq 0}}$ is a Lévy process, then its characteristic function ${\displaystyle \phi _{X}(\theta )}$ is given by

${\displaystyle \phi _{X}(\theta ):=\mathbb {E} {\Big [}e^{i\theta X}{\Big ]}=\exp {\Bigg (}ai\theta -{\frac {1}{2}}\sigma ^{2}\theta ^{2}+\int _{\mathbb {R} \backslash \{0\}}{\big (}e^{i\theta x}-1-i\theta x\mathbf {I} _{|x|<1}{\big )}\,\Pi (dx){\Bigg )}}$

What do you mean here by ${\displaystyle \mathbb {E} {\Big [}e^{i\theta X}{\Big ]}}$? I guess it should be ${\displaystyle \mathbb {E} {\Big [}e^{i\theta X_{1}}{\Big ]}}$. The distribution of a Lévy process ${\displaystyle X}$ is uniquely given by the distribution of ${\displaystyle X_{1}}$. I guess you are looking at the characteristic function of ${\displaystyle X_{1}}$ here. 68.174.156.215 (talk) 03:33, 20 July 2016 (UTC)

## Levy-Ito decomposition and Lebesgue decomposition

There is some talk in the section on the Levy-Ito decomposition which suggests a relation with Lebesgue decomposition -- i.e., there is talk of a pure point part and a singular continuous part of a measure W (which so far as I can see is undefined). This relation would be interesting if it is true, but I don't believe that it is. A good discussion of the continuity properties of the one-dimensional distributions of Levy processes (I assume this is what is meant by W in the article) is found in section 27 of Sato's book, and I did not find there such a relation with Levy-Ito. In fact, Proposition 27.16 gives conditions under which the one-dimensional distribution of a Levy process is either absolutely continuous or continuous singular; and yet, under the conditions given, the process ${\displaystyle X^{(2)}}$ in the Levy-Ito decomposition will be non-zero unless the Levy measure integrates the identity function away from zero. Even worse, underneath Remark 27.22 it is stated that the continuity properties of the one-dimensional distributions are time-dependent! So it becomes even less clear what is meant in this section.

Perhaps the person who added these statements (and the corresponding claims in Lebesgue decomposition) could discuss this a bit more here? If not I will remove these parts after some time. 130.88.123.107 (talk) 17:16, 18 January 2017 (UTC)

Sorry about that! Yes, it's just an analogy, with different properties. I'll clean it up next week, clarifying, elaborating, and citing. Thanks!
—Nils von Barth (nbarth) (talk) 16:39, 15 February 2017 (UTC)