On the Stochastic Integral Equation of Fredholm Type

Abstract

Given a pair (Z, {ϕn) of a random function Z(t) (t≥≥0) and an orthonormal basis {ϕn) in L2(0,1), we are concerned with the stochastic integral equation of Fredholm type as follows x(t)=f(t)+α∫10: L(t, s, ω) x(s)ds+β ∫10K(t, s, ω)x(s)dϕZ(S), where the term ∫dϕZ stands for the stochastic integral of noncausal type with respect to the pair (Z, {ϕn}) and L, K and f are some random functions. In this article, we will show some results on the question of existence and uniqueness of solutions and apply them to the boundary value problems of stochastic differential equations containing (d/dt)Z(t) as a coefficient. Furthermore, we will discuss the approximation of the solution of the integral equation by those of ordinary random differential equations. © 1986, Kinokuniya Company Ltd. All rights reserved.