Levy process

Let $(\\Omega,\\Psi,P,({\\cal F})_{0\\leq t<\\infty})$ be a filteredprobability space. A Lèvy process on that space is an stochasticprocess $L\\colon[0,\\infty)\\times\\Omega\\rightarrow\\Re^{n}$ that has thefollowing 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 $0\otin cl(A)$ 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
$\\psi(u)=-i\\alpha u+{\\sigma^{2}\\over 2}u^{2}+\\int_{|x|\\geq 1}(1-e^{iux})\\,d\u(%x)+\\int_{|x|<1}(1-e^{iux}+iux)\\,d\u(x)$
with $\\alpha\\in\\Re$ , $\\sigma\\in[0,\\infty)$ and $\u$ is a positive, borel, $\\sigma$ -finite measure called Lèvy measure. (Actually $\u(\\cdot)=E[N_{1}(\\cdot,A)]$ ). 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 PoissonProcess, 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.