Lévy process

Lévy processes, named after the French mathematician Paul Lévy (1886-1971), are stochastic processes with stationary, independent increments. They describe the temporal evolution of sizes which, although subject to random, but over time ( in distribution) constant and independent effects. Many important processes, such as the Wiener process or the Poisson process are Lévy processes.

Definition

Be a stochastic process over the index set T ( or mostly ). They say Xt have independent increments if the random variables ( the growth of Xt ) are independent for all.

If the distribution of growth over long time intervals equal to the same, that is true

Then we call Xt a process with stationary increments.

As Lévy processes is known precisely those processes that have independent and stationary growth. It is often additionally requires that ( almost certainly ) holds. Is a general Lévy process, then by a Lévy process with defined. In the following it is always assumed.

Discrete-time Lévy processes

Applies specifically, then the class of Lévy processes can be characterized very simple: There is namely for all such processes a representation

Being independent and identically distributed random variables. On the other hand, for every sequence of independent random variables, all having the same any given distribution, and defined by a Lévy process X. In the discrete-time case, a Lévy process is thus basically nothing more than a random walk with arbitrary but constant jump distribution. The simplest example of a discrete time Lévy process is therefore also the simple symmetric random walk, in the symmetric Bernoulli distribution is. Here the process moves X, starting at, at each step with probability ½ up by one, otherwise by one down.

Continuous time Lévy processes

In case the characterization is not so easy: there are, for example, no continuous-time Lévy process in which Bernoulli distributed as above.

However, continuous-time Lévy processes are closely related to the concept of infinite divisibility: If that is a Lévy process, so is infinitely divisible. On the other hand, defines an infinitely divisible random variable already the distribution of total Lévy process clearly established. Each Lévy process thus corresponds to an infinitely divisible distribution function and vice versa.

Important examples of continuous-time Lévy processes (also called Brownian motion ), in which the infinitely divisible distribution of a normal distribution is the Wiener process, or the Poisson process in which the is poisson - distributed. However, many other distributions, for example, the gamma distribution or the Cauchy distribution, can be used for the construction of Lévy processes. In addition to the deterministic process of the Wiener process with constant drift and constant volatility the only continuous Lévy process, that is, from the continuity of a Lévy process already follows the normal distribution of its gains. However, there is for example no Lévy process with uniformly distributed states.

Also important is the concept of finite and infinite activity: Is there an interval infinitely many (and therefore infinitely small ) jumps or not? Information as is also the Lévy measure.

Furthermore, subordinators of importance, which are Lévy processes with almost surely monotonically increasing paths. One example is the gamma process. The difference between the two processes will be referred to as a gamma -gamma process variance.

Further definition

A stochastic process on a probability space is called a Lévy process if

• ,
• Has independent and stationary growth and
• Is stochastic continuously, i.e. for any and is

Lévy- Khinchin formula

For each -valued Lévy process can be its characteristic function written in the form:

With the characteristic exponent

And the characteristic triplet. It is a symmetric positive definite matrix, a vector and a measure on with

The characteristic triplet is uniquely determined by the process.

Named is this representation of the characteristic function of a Lévy process by Paul Lévy and Khinchin Alexandr.

Key Features

• The expected value function of a Lévy process is linear in t, ie

• If true, it is a martingale.