Pseudo-poissonian processes with stochastic intensity and a class of processes which generalize the Ornstein Uhlenbeck process

Abstract

The definition of pseudo-poissonian processes is given in the famous monograph of William Feller, vol. II, chapter X. The modern development of the theory of information flows stipulates a new interest to detailed analysis of behavior and characteristics of the pseudo-poissonian processes. Formally, pseudo-poissonian process is a poissonian subordination of the mathematical time of an independent random sequence, the time randomization of a random sequence. We consider the given sequence as independent identically distributed random values with second moments. Although in this case the pseudo-poissonian processes do not have independent increments it is possible to calculate the autocovariance function with the property of the exponentially decreasing. Normalized sums of independent copies of such Pseudo-Poissonian processes tend to the Ornstein Uhlenbeck process. We consider a generalization of the driving poissonian process to the case of a random intensity. We show that under this generalization the autocovariance function of the corresponding Pseudo-Poissonian process is the Laplace transform of the distribution of this random intensity. We shortly discuss stochastic principles to choice of the distribution of the random intensity and illustrate them in two detailed examples. Refs 20. Figs 2.