On the existence of a random solution to a nonlinear perturbed stochastic integral equation

Abstract

There are two basic classes of random or stochastic integral equations cur ren t ly under s tudy by probabilists and mathemat ical statisticians. Those integral equations involving Ito or Ito-Doob type stochastic integrals and those which can be considered as probabilistic analogues of classical determinist ic integral equations whose formulation involves only the Lebesque integral. I t has been shown [1]-[4], [6]-[11], [13], [14], [16], [17], among others tha t the theory of random integral equations of the l a t t e r ca tegory are ex t remely important in stochastically character izing many physical situations in life sciences and engineering. In this paper we shall be concerned with a stochastic integral equation of the Ito-Doob type. We shall be concerned with a nonlinear stochastic integral equation of the form given by