R.I.Kadiev

Dagestan State University, Makhachkala 367005, Russia

A.V.Ponosov

Norwegian University of Life Sciences, N-1432 Ås, Norway

The paper contains a systematized presentation of how the so-called "W-transform" can be used to study stability properties 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 the solutions of this equation ("variation-of-constant formula"). The W-method can be used to investigate stability properties of wide classes of linear functional differential equations with driven semimartingales, including also those equations, the study of which by the traditional methods of stability analysis has experienced some serious difficulties.

The W-method, in its present form, was proposed by N.V.Azbelev, but according to his comment it goes back to G.Fubini and F.Tricomi. The method described originally a way to regularize boundary value problems for deterministic differential equations. Later on thЕ method has been developed, generalized and applied in the stability theory.

As in the deterministic case, we can apply the W-transform to the original equation in two different ways: from the left and from the right resulting in what one usually calls "the left" and "the right" W-substitution, respectively. Both cases are thoroughly studied in the paper. We provide also examples illustrating the central general results.

Connections between Lyapunov stability and admissibility of pairs of spaces (the latter is also known as stability under constantly acting perturbations) are crucial in the theory of deterministic functional differential equations, but far less known in the stochastic analysis. We pay therefore much attention to this problem in the paper.

This paper is of expository character. That is why the results presented in it are not new, although many of them are rather recent. The main goal of the paper is to give an impression of how the W-method works in stochastic analysis. On the other hand, we always formulate exact statements, but provide proofs only in exceptional cases. The detailed proofs and other relevant information can be found in the authors' publications: all necessary references are given.