The authors consider the linear stochastic delay differential equation
where is a continuous function, and . The main results of the paper are the following statements:
(a) If , , and is a nondecreasing function, then the equation has an a.s. oscillatory solution on , i.e., a.s., where . Moreover, all points of the zero set are isolated, and is differentiable at all these points.
(b) If, otherwise, and on , then the equation has an a.s. positive solution.