 
Summary: Noise Induced Oscillation in Solutions of
Stochastic Delay Di#erential Equations
John A. D. Appleby 1 and Evelyn Buckwar 2
1 School of Mathematical Sciences, Dublin City University,
Dublin 9, Ireland
2 Department of Mathematics, HumboldtUniversit˜at zu Berlin,
Berlin, Germany
ABSTRACT: This paper studies the oscillatory properties of solutions of linear
scalar stochastic delay di#erential equations with multiplicative noise. It is shown
that such noise will induce an oscillation in the solution whenever there is negative
feedback from the delay term. The zeros of the process are a countable set; the
solution is di#erentiable at each zero, and the zeros are simple. The addition of such
noise does not alter the positivity of solutions when there is positive feedback.
AMS (MOS) Subject Classification: 34K50, 34K15, 34F05, 60H10
1. INTRODUCTION
Delay di#erential equations are widely used to model systems in ecology, physics,
and economics. Very often, interest focusses on solutions of such equations which
are oscillatory, as these could plausibly reflect cyclic motion of a system around an
equilibrium. Over the last thirty years, an extensive theory of oscillatory solutions of
deterministic equations has developed. However, the e#ect that random perturbations
