Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Pierre Lévy is any continuous-time stochastic process that starts at 0, admits càdlàg (right-continuous with left limits) modification and has “stationary independent increments”. The most well-known examples are the Wiener processMathworldPlanetmath and the Poisson process.

A continuous-time stochastic process assigns a random variableMathworldPlanetmath Xt to each point t0 in time. In effect it is a random function of t. The increments of such a process are the differencesPlanetmathPlanetmath Xs-Xt between its values at different times t<s. To call the increments of a process independentPlanetmathPlanetmath means that increments Xs-Xt and Xu-Xv 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. To call the increments stationary means that the probability distribution of any increment Xs-Xt depends only on the length s-t of the time interval; increments with equally long time intervals are identically distributed.

In the Wiener process, the probability distribution of Xs-Xt is normal with expected valueMathworldPlanetmath 0 and varianceMathworldPlanetmath s-t.

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

The probability distributions of the increments of any Lévy process are infinitely divisible. There is a Lévy process for each infinitely divisible probability distribution.

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


It is possible to characterise all Lévy processes by looking at their characteristic functionMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Levy_processLévy process as of January 25, 2007.

Title Lévy process
Canonical name LevyProcess
Date of creation 2013-03-22 16:37:19
Last modified on 2013-03-22 16:37:19
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 62M09
Synonym Levy processPlanetmathPlanetmath