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.
The characteristic functions of infinitely divisible probability measures were completely characterized by Levy and Khintchine in the 1930s. This is a fundamental result.
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.
- 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,
where is the Levy symbol of .
Here are examples of Levy processes.
where 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 form
for some and a Levy measure satisfying