On the existence, uniqueness, and stability behavior of a random solution to a nonlinear perturbed stochastic integro-differential equation

The purpose of the present paper is to study a perturbed nonlinear stochastic integro-differential equation of the form x′(t;ω)=h(t,x(t;ω))−∫0tk1(t,τ;ω)f1(τ,x(τ;ω))dτ+∫0tk2(t,τ;ω)f2x(τ;ω))dβ(τ;ω),t⩾0,, where ′ denotes the sample path derivitive; w, the sample point of a complete probability measure space (ω, A, P) and β(t; ω), a martingale adapted to an increasing family of sub-σ-algebras At in A. Conditions which guarantee the existence and uniqueness of a random solution are obtained using functional analytic techniques. Also, the stability and boundedness of the second moments of the random solution are discussed, along with an application to a stochastic feedback system