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Oscillation and non-oscillation in solutions of nonlinear stochastic delay differential equations. (English) Zbl 1060.60059
Summary: This paper studies the oscillation and non-oscillation of solutions of a nonlinear stochastic delay differential equation, where the noise perturbation depends on the current state, and the drift depends on a delayed argument. When the restoring force towards equilibrium is relatively strong, all solutions oscillate, almost surely. However, if the restoring force is superlinear, positive solutions exist with positive probability, and for suitably chosen initial conditions, the probability of positive solutions can be made arbitrarily close to unity.
##### MSC:
 60H10 Stochastic ordinary differential equations 34K11 Oscillation theory of functional-differential equations 34K50 Stochastic functional-differential equations