[MUSIC] As I said at the beginning of this lecture, Levy processes are widely used for modeling job type dynamics. And in this subsection, I would like to discuss more precisely how Levy processes can help describe jobs. You know that the Levy measure, which is included in the Levy-Khintchine formula, is a characterization of jumps, of jump structure. And from the Levy-Khintchine formula, or more precisely, from the Levy-Khintchine theorem, it follows that the following property of the measure hold. Integral x squared nu dx for all x with absolute value smaller than 1 is finite. And also, the integral nu dx, for all x's with absolute value larger than 1 is also finite. And from these two properties, it follows, for instance, that the integral in the Levy-Khintchine formula is finite for any level measure. Why is this so? Basically, you can rewrite this integral as a sum of two integrals. First integral is integral for absolute values of x smaller than 1, (exponent iux- 1- iux) nu dx. Plus integral for all x's with absolute value larger than 1 (exponent iux- 1) nu dx. And we claim that both integrals are finite for any Levy measure. In fact, the first integral is finite because it's a function, which is in the brackets, is O of x squared multiplied by u squared. And therefore the first integral is upper bounded by u squared multiplied by the integral of all x's smaller than 1, x squared nu dx, multiplied by a constant. And you know that since this integral is finite, the first integral is also finite. Similar situation with the second integral, because this thing's absolute value is smaller or equal than 2. And this means that this integral can be upper bounded by two integrals or nu(dx) over absolute value of x larger than 1. You know this integral is finite, and therefore this integral is also finite. And finally, we conclude that the sum of these two integrals is also finite. So from this fact, it follows that the Levy-Khintchine formula is at least correct. And as a corollary from these properties of the Levy measure is this so-called Blumenthal-Getoor Index. So you see that here we have the integral of x squared nu dx. And the question which will be interesting, is it possible to make this power smaller? That is, is it possible to determine some r such that integral x the power r nu dx is finite? Well, the integral is for all x with absolute value smaller than 1. You know that if there is some r such that this integral is finite, then it is also finite for any r larger than this one. This is just because this integral is smaller while r grows. Okay, and the question which will be natural is to find the minimal r such that this integral is finite. And this infimal over all r such that this integral is finite is known as the Blumenthal-Getoor Index of the Levy measure nu. So once more, this is called Blumenthal, Getoor index. The role of this index can be illustrated by the so-called stable processes. Stable processes are Levy processes which have stable distribution at any time moment. It can be also defined from other sources. For instance, one can say that St is a Levy process if, for any a larger than 0, there exists some b, such that the distribution of Sat. This process is equal, To the distribution of the process a in the power 1 divided by alpha St + b. But this b should actually depend on t. B is a function from R plus to R. This index alpha is very important, this index is between 0 and 2. And in order to emphasize which alpha is used for the given stable process, one can call this process alpha stable Okay, for instance, Brownian motion is a stable process because you know that Brownian motion is a moment at is equal to a to the power one-half St, and therefore is a stable distribution with alpha equal to 2. And then turns out that for stable processes, the Blumenthal-Getoor index is exactly equal to alpha. And if we take a stable process with alpha close to 0, then the trajectory of such process will be very close to the trajectories of the compound Poisson process. If you now take, so this is when alpha is close to 0. Well, when alpha is close to 2, you will get something similar to the Brownian motion, so alpha is close to 2. And if alpha is between 0 and 2, say alpha is close to 1. Then the picture will be exactly the mixture of the left and the right pictures. So it will be something like this. And therefore, this parameter alpha determines the structural jumps, it can be used as a characteristic of the jump activity. So we have actually two characteristics of jumps. The first characteristic is a Levy measure, this is a universal tool which can completely describe the jump structure of a given Levy process. And the second characteristic is a Blumenthal-Getoor index, which can also give us an impression how these jumps are organized. But nevertheless, it's very pleasant to have such characteristic, because it's much more pleasant to work with numbers than with measures. Well, in the next subsection, I would like to show how one can estimate the Levy measure from the data. And therefore, how one can describe the activity of jumps in practice.
