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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 $$ dX(t)=(aX(t)+b(X(t-\tau(t))))\,dt+\sigma X(t)\,dB(t), \quad t\ge 0,$$ where the time lag $\tau$ is a continuous function vanishing at infinity and where $B$ is a standard Brownian motion. Depending on the decay of $\tau(t)$ to zero as $t\to\infty$, 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.

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