If I will formulate result which gives us the formula for the characteristic function of any Levy process, I would like to introduce the so-called Levy measure. Levy measure is a characteristic of jumps of a Levy process and its formal definition is the following one. Levy measure of a process X t is a measure nu such that nu of B for any subset of a real line without zero is equal to the mathematical expectation of the amount of jumps between zero and one with psis which belongs to B. By delta X t, we mean a size of a jump. So, you know that a typical trajectory of Levy process looks as follows, so this continues and at some points it has discontinuous type and the size of this jump- so the difference between this value and this one is exactly the size of a jump. So, delta X t is equal to X t minus the value at the left limit, so minus X t minus. And what is really here is that we should count the amount of jumps which occur between zero and one and take mathematical expectation of the amount of jumps. This is exactly the Levy measure. Let me provide a couple of examples. The first example is a Brownian motion, you know that for a Brownian motion, the Levy measure is equal to zero because Brownian motion doesn't have any jumps. So, for Brownian motion nu is equal to zero. Okay, the second example is a Poisson process. Here, situation is a bit more difficult because you know that Poisson process, the size of any jump of Poisson process is equal to one. Well, if you plot the trajectory of a Poisson process, this process is equal to zero till the first jump occur. Then the jump is at one, then some more time goes and afterwards the jump is also at one and so on. So all jumps have the size one. And therefore in this case, nu of B is equal to zero if one does not belong to B. And if one belongs to B, should be equal to the amount of jumps between zero and one, so here is zero, here is one and if you look at this picture, you can realize that the amount of jumps is equal exactly to the value of the process at time one. That is it is equal to mathematical expectation of N 1 if one belongs to B. Mathematical expectation of N 1 is of course equal to lambda. And finally, we conclude that nu of B is equal to lambda multiplied with the indicator that one belongs to B. Okay, just keep in mind these outcomes for Poisson process, the Levy measure is equal to lambda multiplied by this indicator. Third example, similarly one can consider the compound Poisson process. Sum K from one to N t, psi K where N t is a Poisson process [inaudible] lambda and psi one, psi two and so on is a sequence of independent identically distributed random variables. And obviously through a Poisson process, one can show that nu of B is equal to lambda multiplied by the probability that one summand, psi one belongs to B. And if psi one has a density which I will denote by by P psi, then it turns out that for nu of beta, there exists also a density. In the sense that nu of beta is equal to the integral of B of lambda, P psi of X d x for any Borel subset B. Okay. In this case when nu of beta is equal to the integral B psi function to x, we will say that this function, lambda P psi in this example is equal to S of X, and this S of X is called a Levy density. In most situations, nu of beta has a density and this helps a lot to understand the Levy measure. In the next subsection, I will present the Levy-Khintchine formula, which connects the measure nu with a characteristic function of the process x t. And one of the parts of this formula of the corresponding theorem is that for any Levy measure nu, the following facts hold. So actually the integral of X squared nu d x, for all x smaller than one is finite. If nu has a density like in this example, then nu d x is equal to s of x dx. And second fact which also follows from the theorem which will be later, that the integral nu the x for any x larger than one is also finite. These are two basic properties of the Levy measure. And it turns out that any Levy measure possesses this property or in another side any measure which possesses these two properties is a Levy measure for some Levy process. This is basically all what you should know about Levy measure at the moment. And let me now proceed to the Levy-Khintchine formula, the most important fact in the theory of the Levy processes.