We will review some basic properties of Levy process, in particular Levy-Ito decomposition. See more details in [App04b]. This part is continuation of Levy Process-1.
If the filtration satisfies the “usual conditions” of right continuity and completion, then every Levy process has a cadlag modification which is itself a Levy process.
Function is cadlag. Let (it is well defined due to existence of ), and define Poisson Random Measure
Then, 1) , 2) is countable.
Let be bounded below, i.e. . Then is a Poisson process with intensity , where . It follows immediately that whenever is bounded below, hence the measure is -finite.
- For each , , is a counting measure on
- For each bounded below, is a Poisson process with intensity
- The compensator is a martingale-valued measure where
for bounded below, i.e. For fixed bounded below, is a martingale.
Note that each is an -valued r.v. and gives rise to a cadlag stochastic process as we vary t. In the sequel, we will sometimes use to denote the restriction to of the measure .
From Theorem 2, a Poisson integral may fail to have a finite mean if . Thus, for each , we define the compensated Poisson integral by
A straightforward argument about shows that,
Let’s turn to sketch the Levy-Ito decomposition.
First, note that for bounded below, for each is the sum of all the jumps taking values in the set up to the time . Since the paths of are cadlag, this is clearly a finite random sum. In particular, is the sum of all jumps (finite many) of size bigger than one, and hence finite variation. It is a compound Poisson process with finite variation but may have no finite moments. Conversely, it can be shown that is a Levy process having finite moments to all orders. But, it may have unbounded variation. Hence, we can define
then is well defined.
Now, let’s turn to the small jumps. We study compensated integrals, which we know are martingales. Introduce for and bounded below. For each , let . It can be shown that as in , hence is a martingale. Taking the limit in (5), we get
The process is a centered martingale with continuous sample paths. Using Levy’s characterization of Brownian motion we have that is Brownian motion with covariance . Hence we have
The process is the compensated sum of small jumps. The compensation takes care of the analytic complications in the Levy-Khintchine formula in a probabilistically pleasing way, since it is an -martingale.
Levy-Ito decomposition of Theorem 3 is provided with small jumps of and big jumps of . In fact, for any given constant , we can define small jumps and big jumps by and , respectively. The corresponding Levy-Ito decomposition is written by
for some constant defined similar to (6), i.e.
It can be shown is finite and well defined. One can compute in terms of using (3),
- If , then ;
- If , then ;
Example 1 Consider a 1-d Levy process of the form
where is a poisson random measure with Levy measure given by
Show that, does not admit decomposition of (10).
A Levy process has finite variation iff its Levy-Ito takes the form
where . In fact, A necessary and sufficient condition for a Levy process to be of finite variation is that there is no Brownian part (i.e. in the Levy-Khinchine formula), and .
A stochastic process is a semimartingale if it is an adapted process s.t. for each , , where is a local martingale and is an adapted process of finite variation. In particular, every Levy process is a semimartingale. To see this, use the Levy-Ito decomposition to write martingale part as and bounded variation part as
The first three terms on the rhs of (7) have finite moments to all orders, so if a Levy process fails to have a moment, this is due to the large jumps with finite activity part. In fact, for all iff .
An interesting by-product of the Levy-Ito decomposition is the Levy-Khintchine formula for Levy process, which follows easily by independence in the Levy-Ito decomposition and (5).
and the intensity measure of (1) is the Levy measure for .