# 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.

Title | Lévy process |
---|---|

Canonical name | LevyProcess |

Date of creation | 2013-03-22 16:37:19 |

Last modified on | 2013-03-22 16:37:19 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 62M09 |

Synonym | Levy process^{} |