Levy process

Let be a filteredprobability space. A Lèvy process on that space is an stochasticprocess $L:[0,\mathrm{\infty })\times \mathrm{\Omega }\to {\mathrm{\Re }}^n$ that has thefollowing 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 ${ℱ}_t$

2. $L$ has stationary increments: if $t\ge s\ge 0$ 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 $t,s\in [0,\mathrm{\infty })$, ${lim}_{t\to s}=X_s$, 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 ${\tilde{N}}_t(\cdot ,A)=N_t(\cdot ,A)-tE[N_1(\cdot ,A)]$ is the compensated jump process, which is a martingale.

4. Lèvy -Khintchine formula: from the previous property it can be shown that for any $t\ge 0$ 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 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.