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 dier- ential equations. The W-transform is an integral transform which typically is generated by a simpler dierential 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 pro- cesses, 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.