# Lévy process

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 *X*_{t} 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 *X*_{s} − *X*_{t} between its values at different times *t* < *s*. 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.

### Stationary increments

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 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 *n*th moment is a polynomial function of *t*; these functions satisfy a binomial identity:

## 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 is a Lévy process, then its characteristic function satisfies the following relation:

where , and is the indicator function. The Lévy measure must be such that

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 . So one can see that a purely continuous Lévy process is a Brownian motion with drift.

## External links

- Applebaum, David (December 2004), "Lévy Processes—From Probability to Finance and Quantum Groups" (PDF),
*Notices of the American Mathematical Society*, Providence, RI: American Mathematical Society,**51**(11): 1336–1347, ISSN 1088-9477

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