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Lévy process

Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random and independent, and statistically identical over different time intervals of the same length. A Lévy process may thus be viewed as the continuous-time analog of a random walk.

The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths.

Mathematical definition

Astochastic processis said to be a Lévy process if it satisfies the following properties:

  1. almost surely

  2. Independence of increments: For any , are independent

  3. Stationary increments: For any , is equal in distribution to

  4. Continuity in probability: For any and it holds that

Ifis a Lévy process then one may construct a version ofsuch thatisalmost surelyright continuous with left limits.

Properties

Independent increments

A continuous-time stochastic process assigns a random variable X**t to each point t ≥ 0 in time. In effect it is a random function of t. The increments of such a process are the differences X**s − X**t between its values at different times t < s. To call the increments of a process independent means that increments X**s − X**t and X**u − X**v are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent.

Stationary increments

To call the increments stationary means that the probability distribution of any increment X**t − X**s depends only on the length t − s of the time interval; increments on equally long time intervals are identically distributed.

Ifis aWiener process, the probability distribution of Xt − Xsisnormalwithexpected value0 andvariancet − s.
Ifis thePoisson process, the probability distribution of Xt − Xsis aPoisson distributionwith expected value λ(t − s), where λ > 0 is the "intensity" or "rate" of the process.

Infinite divisibility

The distribution of a Lévy process has the property ofinfinite divisibility: given any integer "n", thelawof a Lévy process at time t can be represented as the law of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically distributed by assumptions 2 and 3. Conversely, for each infinitely divisible probability distribution, there is a Lévy processsuch that the law ofis given by.

Moments

In any Lévy process with finitemoments, the nth moment, is apolynomial functionof t;these functions satisfy a binomial identity:

Lévy–Khintchine representation

The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy–Khintchine formula (general for all infinitely divisible distributions):[1]

If is a Lévy process, then its characteristic function is given by

where , , and is a σ-finite measure called the Lévy measure of , satisfying the property

In the above,is theindicator function, and the complements are taken with respect to. Becausecharacteristic functionsuniquely determine their underlying probability distributions, each Lévy process is uniquely determined by the "Lévy–Khintchine triplet". The terms of this triplet suggest that a Lévy process can be seen as having three independent components: a linear drift, aBrownian motion, and a Lévy jump process, as described below. This immediately gives that the only (nondeterministic) continuous Lévy process is a Brownian motion with drift; similarly, every Lévy process is asemimartingale.[2]

Lévy–Itô decomposition

Because the characteristic functions of independent random variables multiply, the Lévy–Khintchine theorem suggests that every Lévy process is the sum of Brownian motion with drift and another independent random variable. The Lévy–Itô decomposition describes the latter as a (stochastic) sum of independent Poisson random variables.

Let— that is, the restriction ofto, renormalized to be a probability measure; similarly, let(but do not rescale). Then
.
The former is the characteristic function of acompound Poisson processwith intensityand child distribution. The latter is a compensated generalized Poisson process (CGPP): a process with countably many jump discontinuities on every intervala.s., but such that those discontinuities are of magnitude less than. If, then the CGPP is apure jump process.[3] [4]

Generalization

A Lévy random field is a multi-dimensional generalization of Lévy process.[5] [6] Still more general are decomposable processes.[7]

See also

References

[1]
Citation Linkopenlibrary.orgZolotarev, Vladimir M. One-dimensional stable distributions. Vol. 65. American Mathematical Soc., 1986.
Sep 29, 2019, 7:46 PM
[2]
Citation Linkopenlibrary.orgProtter P.E. Stochastic Integration and Differential Equations. Springer, 2005.
Sep 29, 2019, 7:46 PM
[3]
Citation Link//doi.org/10.1007%2F978-3-642-37632-0_2Kyprianou, Andreas E. (2014), "The Lévy–Itô Decomposition and Path Structure", Fluctuations of Lévy Processes with Applications, Universitext, Springer Berlin Heidelberg, pp. 35–69, doi:10.1007/978-3-642-37632-0_2, ISBN 9783642376313
Sep 29, 2019, 7:46 PM
[4]
Citation Linkwww.math.uchicago.eduLawler, Gregory (2014). "Stochastic Calculus: An Introduction with Applications" (PDF). Department of Mathematics (The University of Chicago). Archived from the original (PDF) on 29 March 2018. Retrieved 3 October 2018.
Sep 29, 2019, 7:46 PM
[5]
Citation Link//doi.org/10.1007%2F978-1-4612-1732-9_12Wolpert, Robert L.; Ickstadt, Katja (1998), "Simulation of Lévy Random Fields", Practical Nonparametric and Semiparametric Bayesian Statistics, Lecture Notes in Statistics, Springer, New York, doi:10.1007/978-1-4612-1732-9_12, ISBN 978-1-4612-1732-9
Sep 29, 2019, 7:46 PM
[6]
Citation Linkwww2.stat.duke.eduWolpert, Robert L. (2016). "Lévy Random Fields" (PDF). Department of Statistical Science (Duke University).
Sep 29, 2019, 7:46 PM
[7]
Citation Linkportal.issn.orgFeldman, Jacob (1971). "Decomposable processes and continuous products of probability spaces". Journal of Functional Analysis. 8 (1): 1–51. doi:10.1016/0022-1236(71)90017-6. ISSN 0022-1236.
Sep 29, 2019, 7:46 PM
[8]
Citation Linkwww.ams.org"Lévy Processes—From Probability to Finance and Quantum Groups"
Sep 29, 2019, 7:46 PM
[9]
Citation Link//www.worldcat.org/issn/1088-94771088-9477
Sep 29, 2019, 7:46 PM
[10]
Citation Linkdoi.org10.1007/978-3-642-37632-0_2
Sep 29, 2019, 7:46 PM
[11]
Citation Linkweb.archive.org"Stochastic Calculus: An Introduction with Applications"
Sep 29, 2019, 7:46 PM
[12]
Citation Linkwww.math.uchicago.eduthe original
Sep 29, 2019, 7:46 PM
[13]
Citation Linkdoi.org10.1007/978-1-4612-1732-9_12
Sep 29, 2019, 7:46 PM
[14]
Citation Linkwww2.stat.duke.edu"Lévy Random Fields"
Sep 29, 2019, 7:46 PM
[15]
Citation Linkdoi.org10.1016/0022-1236(71)90017-6
Sep 29, 2019, 7:46 PM
[16]
Citation Linkwww.worldcat.org0022-1236
Sep 29, 2019, 7:46 PM
[17]
Citation Linkwww.ams.org"Lévy Processes—From Probability to Finance and Quantum Groups"
Sep 29, 2019, 7:46 PM
[18]
Citation Linkwww.worldcat.org1088-9477
Sep 29, 2019, 7:46 PM
[19]
Citation Linken.wikipedia.orgThe original version of this page is from Wikipedia, you can edit the page right here on Everipedia.Text is available under the Creative Commons Attribution-ShareAlike License.Additional terms may apply.See everipedia.org/everipedia-termsfor further details.Images/media credited individually (click the icon for details).
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