The Distribution of Multiple Shot Noise Process and Its Integral

Abstract

In this paper, we study multiple shot noise process and its integral. We analyse these two processes systematically for their theoretical distributions, based on the piecewise deterministic Markov process theory developed by Davis [1] and the martingale methodology used by Dassios and Jang [2]. The analytic expressions of the Laplace transforms of these two processes are presented. We also obtain the multivariate probability generating function for the number of jumps, for which we use a multivariate Cox process. To derive these, we assume that the Cox processes jumps, intensity jumps and primary event jumps are independent of each other. Using the Laplace transform of the integral of multiple shot noise process, we obtain the tail of multivariate distributions of the first jump times of the Cox processes, i.e . the multivariate survival functions. Their numerical calculations and other relevant joint distributions ’ numerical values are also presented.