A Random Fredholm Integral Equation

Abstract

The aim of this paper is the study of a random or stochastic integral equation of the Fredholm type given by $x(t;\omega ) = h(t;\omega ) + \smallint _0^\infty {{k_0}(t,} \tau ;\omega )e(\tau ,x(\tau ;\omega ))\;d\tau , t \geqq 0$, where $\omega \in \Omega$, the supporting set of the probability measure space $(\Omega ,A,P)$. The existence and uniqueness of a random solution to the above stochastic integral equation is considered. A random solution, $x(t;\omega )$, of such a random equation is defined to be a random function which satisfies the equation almost surely. Several theorems and useful special cases are presented which give conditions such that a random solution exists.