Electron. J. Diff. Eqns., Vol. 2004(2004), No. 92, pp. 1-36.

Stability of stochastic functional differential equations and the W-transform

Ramazan Kadiev, Arcady Ponosov

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.

Submitted May 12, 2004. Published July 27, 2004.
Math Subject Classifications: 93E15, 60H30, 34K50, 34D20
Key Words: Stability; stochastic differential equations with aftereffect; integral transforms

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Ramazan Kadiev
Department of Mathematics, Dagestan State University
Makhachkala 367005, Russia
email: dgu@dgu.ru
Arcady Ponosov
Institutt for matematiske realfag and teknologi
NLH, Postboks 5003
N-1432 As, Norway
email: arkadi@nlh.no

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