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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 definitionEdit

A stochastic process   is 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  

If   is a Lévy process then one may construct a version of   such that   is almost surely right continuous with left limits.

PropertiesEdit

Independent incrementsEdit

A continuous-time stochastic process assigns a random variable Xt 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 XsXt between its values at different times t < s. To call the increments of a process independent means that increments XsXt and XuXv 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 incrementsEdit

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

If   is a Wiener process, the probability distribution of Xt − Xs is normal with expected value 0 and variance t − s.

If   is the Poisson process, the probability distribution of Xt − Xs is a Poisson distribution with expected value λ(t − s), where λ > 0 is the "intensity" or "rate" of the process.

Infinite divisibilityEdit

The distribution of a Lévy process has the property of infinite divisibility: given any integer "n", the law of 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 process   such that the law of   is given by  .

MomentsEdit

In any Lévy process with finite moments, the nth moment  , is a polynomial function of t; these functions satisfy a binomial identity:

 

Lévy–Khintchine representationEdit

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 the indicator function, and the complements are taken with respect to  . Because characteristic functions uniquely 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, a Brownian 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 a semimartingale.[2]

Lévy–Itô decompositionEdit

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 of   to  , renormalized to be a probability measure; similarly, let   (but do not rescale). Then

 .

The former is the characteristic function of a compound Poisson process with intensity  and child distribution  . The latter is a compensated generalized Poisson process (CGPP): a process with countably many jump discontinuities on every interval a.s., but such that those discontinuities are of magnitude less than  . If  , then the CGPP is a pure jump process.[3][4]

GeneralizationEdit

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

See alsoEdit

ReferencesEdit

  1. ^ Zolotarev, Vladimir M. One-dimensional stable distributions. Vol. 65. American Mathematical Soc., 1986.
  2. ^ Protter P.E. Stochastic Integration and Differential Equations. Springer, 2005.
  3. ^ Kyprianou, 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
  4. ^ Lawler, 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.
  5. ^ Wolpert, 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
  6. ^ Wolpert, Robert L. (2016). "Lévy Random Fields" (PDF). Department of Statistical Science (Duke University).
  7. ^ Feldman, 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.