Oscillation and Non-oscillation in Solutions of Nonlinear Stochastic Delay Differential Equations
John A. D. Appleby, Dublin City University
Conall Kelly, Dublin City University, Dublin, Ireland
Abstract
This paper studies the oscillation and nonoscillation 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.
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