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Relations between stability and admissibility for stochastic linear functional differential equations. (English) Zbl 1093.34046
The authors consider the stochastic functional (or delay) differential equation \[ dx(t)=(Vx)(t)+f(t))dZ(t), \] where \(Z\) is a semimartingale and \(V\) is a linear Volterra operator. The idea employed is to consider a simpler reference equation of a similar form. Then, it is shown by Azbelev’s W-method that stability properties of the reference equation are inherited by the equation under consideration if a certain operator is invertible. Other sufficient conditions to guarantee this property are formulated and the theory is illustrated with an example of a delay differential equation driven by Brownian motion.

34K50 Stochastic functional-differential equations