Levy process

Levy process captures the idea of being a noise process, namly, it has independent increments. Wiener process and Poisson process are good examples of such. Wiener process has long been assumed as the natural noise process largely because of Central Limit Theorem. However, this is quite limiting because CLT requires finite variance of each summand. As nature is often `heavy tailed', CLT no longer holds in these scenarios.

Def. Levy process in law:
1. Xt has independent increments.
2. Xt has stationary increments.
3. Xt is stochasticlly continuous.

Some facts about Levy process:
1. Every Levy process in law has an unique modification which is cadlag. The Levy process is the Levy process in law which is also cadlag.
2. The natural filtration of Levy process is right continuous
3. If a Levy process has bounded jumps, then all of its moments are bounded.

Def. Jump of the Levy process


Def. Jump counting process

Def. Levy measure

The jump counting process is a Poisson process with arrival rate
Note that the jump counting process is a random counting measure

Thm. Let f be Borel and finite on . is a Levy process.

Def. Jump process


Thm. is a Levy process

Thm. f belongs to L1,
and the variance of the process is

Thm. Levy process can be decomposed as
where Y_t is a martingale with bounded jumps and Z_t is a finite variation process.
Simply let W(t)=X(t)-J(t), where J(t) is the jump process associated with X(t) which has jumps larger than 1. Y(t) = W(t)-E[W(t)] is the martingale part. And Z(t) = E[W(t)]+J. We can further decompose Y(t) as in the next theorem.

Thm. Let Y(t) be the Levy process with bounded jumps a and has mean removed (i.e E[Y(t)]=0). Then
where Y^c_t is the continuous martingale and
Y^c_t and Y^d_t are independent Levy processes.

Thm. The Levy Decomposition Theorem

Levy process can be decomposed into a Brownian motion, a drift, a Poisson process with jumps greater than 1 and the compensated Poisson process with jumps bounded by 1.

Thm. The Levy-Khintchine formula




Remark. We can say more about the term

in Levy-Khintchine formula. Note that the characterisitc function of the sum of independent Poisson process is

where each X_t^{(k)} has arrival rate \lambda_k and jump size x_k. If there are infinitly many such Poisson process, then we can integrate over all jump size:

which is of the same form as in Levy-Khintchine formula.