Levy process
Let $$ be a filtered probability space. A Lèvy process on that space is an stochastic process^{} $L:[0,\mathrm{\infty})\times \mathrm{\Omega}\to {\mathrm{\Re}}^{n}$ that has the following properties:

1.
$L$ has increments independent of the past: for any $t\ge 0$ and for all $s\ge 0$, ${L}_{t+s}{L}_{t}$ in independent of ${\mathcal{F}}_{t}$

2.
$L$ has stationary increments: if $t\ge s\ge 0$ then ${L}_{t}{L}_{s}$ and ${L}_{ts}$ have the same distribution^{}. This particulary implies that ${L}_{t+s}{L}_{t}$ and ${L}_{s}$ have the same distribution.

3.
$L$ is continous in probability: for any $t,s\in [0,\mathrm{\infty})$, ${lim}_{t\to s}={X}_{s}$, the limit taken in probability.
Some important properties of any Lèvy processes $L$ are:

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 $t\ge 0$).

2.
${L}_{t}$ is an infinite divisible random variable^{} for all $t\in [0,\mathrm{\infty})$

3.
Lèvy Itô decomposition: $L$ can be written as the sum of a diffusion, a continuous Martingale^{} and a pure jump process; i.e:
$$ where $\alpha \in \mathrm{\Re}$, ${B}_{t}$ is a standard brownian motion^{}. $N$ is defined to be the Poisson random measure of the Lèvy process (the process that counts the jumps): for any Borel $A$ in ${\mathrm{\Re}}^{n}$ such that $0\notin cl(A)$ then $$, where $\mathrm{\Delta}{L}_{s}:={L}_{s}{L}_{s}$; and ${\stackrel{~}{N}}_{t}(\cdot ,A)={N}_{t}(\cdot ,A)tE[{N}_{1}(\cdot ,A)]$ is the compensated jump process, which is a martingale.

4.
Lèvy Khintchine formula: from the previous property it can be shown that for any $t\ge 0$ one has that
$$E[{e}^{iu{L}_{t}}]={e}^{t\psi (u)}$$ where
$$ with $\alpha \in \mathrm{\Re}$, $\sigma \in [0,\mathrm{\infty})$ and $\nu $ is a positive^{}, borel, $\sigma $finite measure called Lèvy measure. (Actually $\nu (\cdot )=E[{N}_{1}(\cdot ,A)]$). The second formula is usually called the Lèvy exponent or Lèvy symbol of the process.

5.
$L$ is a semimartingale (in the classical sense of being a sum of a finite variation process and a local martingale^{}), so it is a good integrator, in the stochastic sense.
Some important examples of Lèvy processes include: the Poisson Process^{}, the Compound Poisson process, Brownian Motion, Stable Processes, Subordinators, etc.
Bibliography

•
Protter, Phillip (1992). Stochastic Integration and Differential Equations^{}. A New Approach. SpringerVerlag, Berlin, Germany.

•
Applebaum David (2004). Lèvy Procesess and Stochastic Calculus. Cambridge University Press, Cambrigde, UK.
Title  Levy process^{} 

Canonical name  LevyProcess1 
Date of creation  20130322 17:58:09 
Last modified on  20130322 17:58:09 
Owner  juansba (18789) 
Last modified by  juansba (18789) 
Numerical id  11 
Author  juansba (18789) 
Entry type  Definition 
Classification  msc 60G20 