On a Stochastic Integro-Differential Equation of Volterra Type

Abstract

A nonlinear stochastic integro-differential equation of the form\[x'( {t;\omega } ) = h( {t,x( {t;\omega } )} ) + \int_0^t {k( {t,\tau ;\omega } )f( {x( {\tau ;\omega } )} )d\tau ,} \]where $t\geqq 0( ' = ( d/dt ) )$, and $\omega \in \Omega $, the supporting set of a complete probability measure space $( {\Omega ,A,P} )$, is studied with respect to the existence of a unique random solution. Results are also given concerning the statistical behavior of the random solution as $t \to \infty $, and an application to differential systems with random parameters is presented.