TITLE:
The Distribution of Multiple Shot Noise Process and Its Integral
AUTHORS:
Jiwook Jang
KEYWORDS:
Multiple Shot Noise Process and Its Integral; Multivariate Cox Process; Piecewise Deterministic Markov Process; Martingale Methodology; Multivariate Survival Functions
JOURNAL NAME:
Applied Mathematics,
Vol.5 No.3,
February
12,
2014
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.