Levy process

Let $(\Omega,\Psi,P,({\cal F})_{0\leq t<\infty})$ be a filtered probability space. A Lèvy process on that space is an stochastic process $L\colon[0,\infty)\times\Omega\rightarrow\Re^{n}$ that has the following properties:

1.

$L$ has increments independent of the past: for any $t\geq 0$ and for all $s\geq 0$ , $L_{t+s}-L_{t}$ in independent of ${\cal F}_{t}$
2.

$L$ has stationary increments: if $t\geq s\geq 0$ then $L_{t}-L_{s}$ and $L_{t-s}$ 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 $t,s\in[0,\infty)$ , $\lim_{t\rightarrow s}=X_{s}$ , 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 $t\geq 0$ ).
2.

$L_{t}$ is an infinite divisible random variable for all $t\in[0,\infty)$
3.

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

$L_{t}=\alpha t+\sigma B_{t}+\int_{|x|<1}x\,d\tilde{N}_{t}(\cdot,dx)+\int_{|x|% \geq 1}x\,dN_{t}(\cdot,dx)\quad\hbox{for all $t\geq 0$}$

where $\alpha\in\Re$ , $B_{t}$ 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 $\Re^{n}$ such that $0\notin cl(A)$ then $N_{t}(\cdot,A)\colon=\sum_{0<s\leq t}1_{A}(\Delta L_{s})$ , where $\Delta L_{s}\colon=L_{s}-L_{s-}$ ; and $\tilde{N}_{t}(\cdot,A)=N_{t}(\cdot,A)-tE[N_{1}(\cdot,A)]$ is the compensated jump process, which is a martingale.
4.

Lèvy -Khintchine formula: from the previous property it can be shown that for any $t\geq 0$ one has that

$E[e^{iuL_{t}}]=e^{-t\psi(u)}$

where

$\psi(u)=-i\alpha u+{\sigma^{2}\over 2}u^{2}+\int_{|x|\geq 1}(1-e^{iux})\,d\nu(% x)+\int_{|x|<1}(1-e^{iux}+iux)\,d\nu(x)$

with $\alpha\in\Re$ , $\sigma\in[0,\infty)$ 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. Springer-Verlag, Berlin, Germany.
•

Applebaum David (2004). Lèvy Procesess and Stochastic Calculus. Cambridge University Press, Cambrigde, UK.

Title	Levy process
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