Now, let me define the characteristic exponent of a levy process. This exponent is the same as cumulant. This notion is defined as a following proposition. For any levy process, L_t, there exists a function psi from R to the set of complex number, numbers which is called a characteristic exponent, such that characteristic function of the process L_t, is a point u, that is mathematical expectation exponent iuL_t is equal to exponent in the power t Psi(u). At the first glance, this proposition is very simple. We just claims as there is sum function of Psi. Nevertheless, the existence of this function yields a lot of interesting effects. I will formulate the effects right after a little bit later, but now let me provide a couple of examples. Okay. So first example, if L_t is a trivial process this is just a constant Mu multiplied by t. Then the characteristic function of L_t is equal to exponent in the power i, u, Mu, t. And therefore, Psi(u), is equal to iuMu. Secondly, if L_t is a Brownian motion. Brownian motion let me multiply this Brownian motion by a positive constant sigma, at least a positive constant sigma. Okay in this case, the function of Psi is equal to minus one half sigma squared u squared. And the such example if L_t is a compound Poisson process, so it is equal to the sum K from one to a Poisson process and T with sum of that to lambda psi_k; where psi_1, psi_2, and so on is a sequence of independent identical distributor random variables then the function of Psi is equal to lambda multiplied by the characteristic function of one sum, psi_1 of u minus 1, and we can actually write this formula in the following form. This is the integral exponent to the power iux minus one, multiply it by lambda F psi_1(dx). So it is difficult for you to work with the stieltjes integral. You can just imagine that random variable psi_1 has a density by Psi that in this case is F psi to x is exactly equal to pi psi(x)dx. Okay. And finally, let me provide one more example. If L_t is a sum of Mu_t, the sigma W_t and the compound Poisson process, then Psi(u) is a sum of the corresponding size, that is equal to iuMu minus one half of square sigma squared and plus integral exponent as a power iux minus one lambda F psi(dx). This result will be rather important and let me now box it. Just keep in mind that for some levy processes namely for levy process which Kimbro represents this form the characteristic function can be computed explicitly. Okay. These are examples, four examples, of characteristic exponent and now let me show that the notion of characteristic exponent helps a lot to understand the theory of levy processes. Let me formulate a couple of important qualities from this proposition. The first quality is that the distribution of L_t is determined by the distribution of L_t in one moment, time moment. For instance, it is determined by L_1. Why this quality holds? This is just because if you know the distributions one time moment, say at the moment t to one. So you know phi L_1(u). Then you know that this phi L_1(u) is equal to the exponents of power of psi(u). And therefore, if you know phi L_1, then from this phi L_1, you can simply calculate psi. And if you know psi, you know the distribution to any other time point because this psi completely determines the characteristic function of L_t for any t, and therefore determines the distribution. Okay. In this respect, levy processes are associated with linear functions because you know any linear function which is zero at zero, can be completely determined by the value in only one point. And this situation is very similar. You know that a levy process is zero at zero. And to determine the distribution at any time point, it's enough to determine at one time point. For instance, it's the point t equal to one. Okay this is the first quality. And the second quality is that, the basic elements, the basic characteristics of the levy process can be also determined by the characteristics in one time point. And namely that means the following. With mathematical expectation of L_t is equal to t multiplied by mathematical expectation at the moment by time moment one. The variance of L_t is equal to t multiply it by the variance of L_1. And third property which is also rather important is a covariance function of any levy processes is equal to the minimum between t and s multiplied by the variance of L_1. It is a very interesting property which actually says, that the levy process is typically not stationary. In fact, you know that the minimal function doesn't poses a properties that minimum of t, between t and s is equal to sum function of the argument t minus s. Therefore, this process is not stationary. And you should use some other methods in compression with a situation where the process is stationary in a broad sense. Okay this is very important thing, the characteristic exponent and the question which placed essential role in the theory of levy processes is what is a form of the function of psi. You are provided one example how is this function can look. And in the next subsection, I would like to formalize a general result which actually reveals the clause four of psi for any levy process.
