The last part of this lecture will be devoted to compound Poisson processes. This is a very popular model which is essentially based on homogeneous Poisson processes. And according to this model, the process is defined as follows: this is the sum by k from one to some Poisson process N_t, of xi_k. Here random variables xi_1, xi_2 and so on are independent and identically distributed, not necessarily positive, the process N_t is Poisson, I mean homogeneous Poisson, and I will denote its intensity parameter by lambda. And also, we will assume that xi_1, xi_2, and so on and N_t are independent. This model naturally arises in many applications, but a lot of applications can be found in insurance. In fact, X_t can be used for modelling the aggregated claim amounts of one insurance company. According to this interpretation, xi_1, xi_2 and so on are actually individual claim sizes, and N_t expresses the amount of claims till time t. Once more, under this interpretation, X_t is an aggregated claim amount. For analysing of this processes, we need to employ some functionals of this process. Here the main difficulty is the same as in the case of renewal processes. The point is that the distribution of X_t can be written in closed form only in few cases which are rather trivial. In general case it is unknown, and we should use some functionals, and the point is, that a kind of this functional definitely depends on the properties of the random variables xi_1, xi_2 and so on. More precisely, if xi_1, xi_2, and so on have only integer values which are non-negative, then X_t is also an integer non-negative random variable. And in this case we can employ the so-called probability generating function, so, the so-called BGF. This function is defined as follows. Actually, for a random variable xi which has only integer non-negative values this probability generating function is defined as the mathematical expectation of u in the power xi, where this deterministic number u is smaller than one, absolute value is smaller than one. Of course, if xi_1 and xi_2 are independent, then the probability generating function of their sum is in fact a product of the corresponding generating functions. In more general case, when xi_1, xi_2 and so on are non-negative, so they can be non-integer but they are non-negative, X_t is also non-negative. And in this case one can apply the so-called moment generating function, which is very close to the so-called Laplace transform. Let me define it as mathematical expectation of e^(-u * xi). So, once more this definition is correct for any xi taking non-negative values, and the parameter u here is positive. Sometimes you can find in the books here plus without minus, from my point of view, that's a question of taste, but here I defined it in this way just because to be sure that this object exists. In fact, if we look attentively at the probability generating function, we get that this object exists for all xi from this class and for u from this range. The same situation is for moment generating function. Nevertheless, it seems very natural to think that xi can take any values. And in this case we should apply the so-called characteristic function.