Study of a Mild Solution of a Stochastic Integrodifferential Equation with Non-lipschitzian Coefficients in a Complex Hilbert Space

In this work we consider a system of stochastic integrodifferential equations in a complex Hilbert space. We first establish the existence and uniqueness of mild solutions for our equation (1) under non-Lipschitz conditions. Then we show under certain assumptions that the mild solution found is asymptotically stable in mean order n. We obtain our existence and uniqueness results by using the Lipschitz global and growth conditions and applying the properties of the analytic semigroup with those of stochastic calculus. The application of the fixed point theorem together with the properties of the stochastic integral gives us the asymptotic stability result.