We will review definition and some basic properties of Levy process.

The characteristic function of r.v. is the mapping defined by

A random variable is infinitely divisible if its law is infinitely divisible, e.g. , where are i.i.d., for each . Note that, the characteristic function of can be written as in the above.

Definition 1Levy measure is a measure on such that

The characteristic functions of infinitely divisible probability measures were completely characterized by Levy and Khintchine in the 1930s. This is a fundamental result.

Theorem 2 (Levy-Khintchine)A Borel probability measure on is infinitely divisible if there exists a vector , a non-negative symmetric matrix and a Levy measure on such that for all ,

Conversely, any mapping of the form (2) is the characteristic function of an infinitely divisible probability measure on .

The triple is called the *characteristics* of the infinitely divisible random variable , and is called the *Levy symbol* or *Characteristic exponent*.

Definition 3A Levy process is a stochastic process satisfying the following:

- w.p.1
- has independent and stationary increments,
- is stochastically continuous, i.e. for all and for all , .

It follows from the Definition 3 (2) that each Levy process is infinitely divisible. Therefore,

Theorem 4If is a Levy process, then is infinitely divisible for each . Furthermore,

where is the Levy symbol of .

Here are examples of Levy processes.

Example 1 (Brownian motion with drift)Let be a standard Brownian motion in . Then, is a Levy process with its characteristic function given bywhere is Levy symbol of of the form

In fact a Levy process has a continuous sample paths if and only if it is of the form .

Example 2 (The compounded Poisson process)Let be a sequence of i.i.d r.v. in with law . Let be a Poisson process of intensity , which is independent of . The compound Poisson process is defined as follows:Then, has Levy symbol of

If , then is said to be Poisson process. The characteristics of is

Example 3 (Subordinators)A subordinator is a one-dimensional Levy process which is increasing a.s. Such processes can be thought of as a random model of time eveolution. A Levy process be a subordinator if and only if its Levy symbol takes the formfor some and a Levy measure satisfying

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What is the covariance/autocorrelation of Levy processes in general?

Thanks!