Asymptotic and oscillatory properties of linear stochastic delay differential equations with vanishing delay.

*(English)*Zbl 1064.34068The authors consider the scalar linear stochastic differential equation

$$dX\left(t\right)=\left(aX\right(t)+b(X(t-\tau (t\left)\right)\left)\right)\phantom{\rule{0.166667em}{0ex}}dt+\sigma X\left(t\right)\phantom{\rule{0.166667em}{0ex}}dB\left(t\right),\phantom{\rule{1.em}{0ex}}t\ge 0,$$

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 $\tau \left(t\right)$ 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.

Reviewer: Markus Reiß (Berlin)