SOME STATISTICAL PROPERTIES OF THE SOLUTIONS OF A SYSTEM OF TWO DIMENSIONAL STOCHASTIC FREDHOLM INTEGRAL EQUATIONS CONTAINS GAMMA PROCESSES

Abstract

In this study, we find the two solutions of a system of two dimensional stochastic Fredholm integral equations contains two gamma processes with different values of the two parameters (the first controls the rate of jump arrivals and the second inversely controls the jump size of this arrivals) in two cases and equal in the third that is to introduce two corresponding Probability Density Functions (PDF’s) and to derive some statistical properties (auto-covariance and spectral density functions) depending upon the maximum variance of each p.d.f with respect to the three cases. To indicate which of the three cases gives a highest correlation, the correlation coefficients between any pair of p.d.f’s related to every case are calculated. The solutions of the system of equations are found by the Adomian Decomposition Method (ADM), they are considered as a rapidly converging and geometrical infinite series. It is shown that, the highest correlation coefficient between any pair of p.d.f’s is when the parameters of the two gamma processes are equal to one. This study is interesting as a main goal by combining two fields of mathematics, integral equations and probability theory that is by using the analytical solutions of a system of two dimensional stochastic Fredholm integral equations.