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The purpose of the paper is to show how asymptotic properties, first of all stochastic Lyapunov stability, of linear stochastic functional differential equations can be studied via the property of solvability of the equation in certain pairs of spaces of stochastic processes, the property which we call input-to-state stability with respect to these spaces. Input-to-state stability and hence the desired asymptotic properties can be effectively verified by means of a special regularization, also known as “the W-method” in the literature. How this framework provides verifiable conditions of different kinds of stochastic stability is shown

Input-to-State Stability of Linear Stochastic Functional Differential Equations