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Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay. (English) Zbl 1064.34068

The authors consider the scalar linear stochastic differential equation

dX(t)=(aX(t)+b(X(t-τ(t))))dt+σX(t)dB(t),t0,

where the time lag τ is a continuous function vanishing at infinity and where B is a standard Brownian motion. Depending on the decay of τ(t) to zero as t, the solution process is proved to be almost surely oscillatory or nonoscillatory. The key ingredients for the proof are a random functional-differential equation solved by X with a geometric Brownian motion as coefficient and related results for deterministic functional-differential equations. In addition, the long-time asymptotics of the solutions are studied in detail.


MSC:
34K50Stochastic functional-differential equations
34K11Oscillation theory of functional-differential equations
60H10Stochastic ordinary differential equations