On a stochastic integrodifferential evolution equation of Volterra type. (English) Zbl 0583.60060

We initiate a study of the stochastic integrodifferential evolution equation \[ \dot x(t,\omega)=Ax(t,\omega)+\int^{t}_{0}B(t- s)x(s,\omega)d\beta (s,\omega)+f(t) \] of Volterra type in a Hilbert space set up, where \(\beta\) (t) is a Hilbert-Schmidt operator valued Brownian motion. We first present some existence-uniqueness theorems. Next, we consider approximation theorems of the following types with mean-square convergence uniformly on compacta:
(1) Approximate unbounded operators \(A\) and \(B\) by Yosida type approximations with bounded operators; (2) Approximate the generator A by generators \(A_ n\), and \(B\) by a sequence \(B_ n\); and (3) Zeroth order approximation of the stochastic equation by a deterministic evolution equation \(\dot y(\)t)\(=Ay(t)+f(t).\)
We also prove the weak convergence of measures induced by solutions of approximate equations to the measure induced by the solution of the given equation.


60H25 Random operators and equations (aspects of stochastic analysis)
60B10 Convergence of probability measures