Levy process

Let (Ω,Ψ,P,()0t<) be a filtered probability space. A Lèvy process on that space is an stochastic processMathworldPlanetmath L:[0,)×Ωn that has the following properties:

  1. 1.

    L has increments independent of the past: for any t0 and for all s0, Lt+s-Lt in independent of t

  2. 2.

    L has stationary increments: if ts0 then Lt-Ls and Lt-s have the same distributionPlanetmathPlanetmathPlanetmath. This particulary implies that Lt+s-Lt and Ls have the same distribution.

  3. 3.

    L is continous in probability: for any t,s[0,), limts=Xs, the limit taken in probability.

Some important properties of any Lèvy processes L are:

  1. 1.

    There exist a modification of L that has càdlàg paths a.s. (càdlàg paths means that the paths are continuous from the right and that the left limits exist for any t0).

  2. 2.

    Lt is an infinite divisible random variableMathworldPlanetmath for all t[0,)

  3. 3.

    Lèvy -Itô decomposition: L can be written as the sum of a diffusion, a continuous MartingaleMathworldPlanetmath and a pure jump process; i.e:

    Lt=αt+σBt+|x|<1x𝑑N~t(,dx)+|x|1x𝑑Nt(,dx)for all t0

    where α, Bt is a standard brownian motionMathworldPlanetmath. N is defined to be the Poisson random measure of the Lèvy process (the process that counts the jumps): for any Borel A in n such that 0cl(A) then Nt(,A):=0<st1A(ΔLs), where ΔLs:=Ls-Ls-; and N~t(,A)=Nt(,A)-tE[N1(,A)] is the compensated jump process, which is a martingale.

  4. 4.

    Lèvy -Khintchine formula: from the previous property it can be shown that for any t0 one has that




    with α, σ[0,) and ν is a positivePlanetmathPlanetmath, borel, σ-finite measure called Lèvy measure. (Actually ν()=E[N1(,A)]). The second formula is usually called the Lèvy exponent or Lèvy symbol of the process.

  5. 5.

    L is a semimartingale (in the classical sense of being a sum of a finite variation process and a local martingalePlanetmathPlanetmath), so it is a good integrator, in the stochastic sense.

Some important examples of Lèvy processes include: the Poisson ProcessMathworldPlanetmath, the Compound Poisson process, Brownian Motion, Stable Processes, Subordinators, etc.


Title Levy processMathworldPlanetmath
Canonical name LevyProcess1
Date of creation 2013-03-22 17:58:09
Last modified on 2013-03-22 17:58:09
Owner juansba (18789)
Last modified by juansba (18789)
Numerical id 11
Author juansba (18789)
Entry type Definition
Classification msc 60G20