Exponential stability of linear stochastic differential equations with bounded delay and the W-transform

R. I. Kadiev, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, Russia
A. Ponosov, Norwegian University of Life Sciences, As, Norway

E. J. Qualitative Theory of Diff. Equ., No. 23. (2008), pp. 1-14.

Communicated by E. Litsyn.Received on 2007-01-30
Appeared on 2008-07-20

Abstract: We demonstrate how the method of auxiliary ('reference') equations, also known as N. V. Azbelev's W-transform method, can be used to derive efficient conditions for the exponential Lyapunov stability of linear delay equations driven by a vector-valued Wiener process. For the sake of convenience the W-method is briefly outlined in the paper, its justification is however omitted. The paper contains a general stability result, which is specified in the last section in the form of seven corollaries providing sufficient stability conditions for some important classes of It\^{o} equations with delay.

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