Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 92, pp. 1-36.

Title: Stability of stochastic functional differential equations 
       and the W-transform

Authors: Ramazan Kadiev (Dagestan State Univ., Makhachkala, Russia)
         Arcady Ponosov (Inst. for mat. realfag and teknologi, Norway)
 
Abstract: 
  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