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