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 $t\ge 0$ 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 . 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:

$${\mu }_n(t+s)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right){\mu }_k(t){\mu }_{n-k}(s).$$

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.