Stability of Stochastic Functional Differential Equations and the W-transform

The paper contains a systematic presentation of how the so-called “W-transform” can be used to study stability of stochastic functional differential equations. The W-transform is an integral transform which typically is generated by a simpler differential equation (“reference equation”) via the Cauchy representation of its solutions (“variation-of-constant formula”). This other equation is supposed to have prescribed asymptotic properties (in this paper: Various kinds of stability). Applying the W-transform to the given equation produces an operator equation in a suitable space of stochastic processes, which depends on the asymptotic property we are interested in. In the paper we justify this method, describe some of its general properties, and illustrate the results by a number of examples.