A stochastic integral equation in Hilbert space with a discrete version and application to stochastic systems

Abstract

A stochastic integral equation of the Volterra type in the form $$x\left( {t;\omega } \right) = h\left( {t;\omega } \right) + \int\limits_0^t k \left( {t,\tau ;\omega } \right)f\left( {\tau ,x\left( {\tau ;\omega } \right)} \right)d\tau , t \geqslant 0,$$ , is formulated in Hilbert space, whereω ∈ Ω, the supporting set of a complete probability measure space (Ω,A, P). A theorem concerning the existence of a random solution of the equation is proven. A discrete version of the above stochastic integral equation is given in the form $$x_n \left( \omega \right) = h_n \left( \omega \right) + \sum\limits_{j = 1}^n {c_{n,j} \left( \omega \right)f_j \left( {x_j \left( \omega \right)} \right), n = 1,2, \cdot \cdot \cdot ,} $$ , and a theorem concerning the existence of a unique random solution is presented. Application to a discrete stochastic control system is given.