# Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Pierre Lévy is any continuous-time stochastic process that starts at 0, admits càdlàg (right-continuous with left limits) modification and has “stationary independent increments”. The most well-known examples are the Wiener process and the Poisson process.

A continuous-time stochastic process assigns a random variable $X_{t}$ to each point $t\geq 0$ in time. In effect it is a random function of $t$. The increments of such a process are the differences $X_{s}-X_{t}$ between its values at different times $t. To call the increments of a process independent means that increments $X_{s}-X_{t}$ and $X_{u}-X_{v}$ are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) independent. To call the increments stationary means that the probability distribution of any increment $X_{s}-X_{t}$ depends only on the length $s-t$ of the time interval; increments with equally long time intervals are identically distributed.

In the Wiener process, the probability distribution of $X_{s}-X_{t}$ is normal with expected value 0 and variance $s-t$.

In the Poisson process, the probability distribution of $X_{s}-X_{t}$ is a Poisson distribution with expected value $\lambda(s-t)$, where $\lambda>0$ is the intensity or rate of the process.

The probability distributions of the increments of any Lévy process are infinitely divisible. There is a Lévy process for each infinitely divisible probability distribution.

In any Lévy process with finite moments, the $n$th moment $\mu_{n}(t)=E(X_{t}^{n})$ is a polynomial function of $t$; these functions satisfy a binomial identity:

 $\mu_{n}(t+s)=\sum_{k=0}^{n}{n\choose k}\mu_{k}(t)\mu_{n-k}(s).$

It is possible to characterise all Lévy processes by looking at their characteristic function.

This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Levy_processLévy process as of January 25, 2007.

Title Lévy process LevyProcess 2013-03-22 16:37:19 2013-03-22 16:37:19 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Definition msc 62M09 Levy process