# Lévy process

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In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below. The most well-known examples are the Wiener process and the Poisson process.

## Properties

### Independent increments

A continuous-time stochastic process assigns a random variable Xt to each point t ≥ 0 in time. In effect it is a random function of t. The increments of such a process are the differences XsXt between its values at different times t < s. To call the increments of a process independent means that increments XsXt and XuXv 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.

### Stationary increments

To call the increments stationary means that the probability distribution of any increment XsXt depends only on the length st of the time interval; increments with equally long time intervals are identically distributed.

In the Wiener process, the probability distribution of Xs − Xt is normal with expected value 0 and variance s − t.

In the Poisson process, the probability distribution of Xs − Xt is a Poisson distribution with expected value λ(s − t), where λ > 0 is the "intensity" or "rate" of the process.

### Divisibility

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

### Moments

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

${\displaystyle \mu _{n}(t+s)=\sum _{k=0}^{n}{n \choose k}\mu _{k}(t)\mu _{n-k}(s).}$

## Lévy-Khinchin representation

It is possible to characterise all Lévy processes by looking at their characteristic function. This leads to the Lévy-Khinchin representation. If ${\displaystyle X_{t}}$ is a Lévy process, then its characteristic function satisfies the following relation:

${\displaystyle \mathbb {E} {\Big [}e^{i\theta X_{t}}{\Big ]}=\exp {\Bigg (}ait\theta -{\frac {1}{2}}\sigma ^{2}t\theta ^{2}+t\int _{\mathbb {R} \backslash \{0\}}{\big (}e^{i\theta x}-1-i\theta x\mathbf {I} _{|x|<1}{\big )}\,W(dx){\Bigg )}}$

where ${\displaystyle a\in \mathbb {R} }$, ${\displaystyle \sigma \geq 0}$ and ${\displaystyle \mathbf {I} }$ is the indicator function. The Lévy measure ${\displaystyle W}$ must be such that

${\displaystyle \int _{\mathbb {R} \backslash \{0\}}\min\{x^{2},1\}W(dx)<\infty .}$

A Lévy process can be seen as comprising of three components: a drift, a diffusion component and a jump component. These three components, and thus the Lévy-Khinchin representation of the process, are fully determined by the Lévy-Khinchin triplet ${\displaystyle (a,\sigma ^{2},W)}$. So one can see that a purely continuous Lévy process is a Brownian motion with drift.