SOLUTION MOMENT STABILITY IN STOCHASTIC DIFFERENTIAL-DELAY EQUATIONS

We study the behavior of the first and second solution moments for lin ear stochastic differential delay equations in the presence of additiv e or multiplicative white and colored noise. In the presence of additi ve noise (white or colored), the stability domain of both moments is i dentical to that of the unperturbed system. When these moments lose st ability, there is a Hopf bifurcation and the first moment oscillates w ith a period identical to the solution of the unperturbed equation, wh ile the oscillation period of the second moment is exactly one half th e period of the unperturbed solution and the first moment. When pertur bations are of the parametric (or multiplicative) type and white noise is assumed, under the Its interpretation the first moment of the solu tion preserves properties of the solution of the deterministic equatio n, while the behavior of the second moment depends on the amplitude of the stochastic perturbation. The critical delay value at which the se cond moment loses stability and becomes oscillating is derived, and it is less than the critical delay for the first moment. Under the Strat onovich interpretation, quite different properties were observed for t he moment equations, namely, various critical values of the delay and period of oscillations. For the case of parametric colored noise pertu rbations, sufficient (p-stability) conditions are derived which are in dependent of the value of delay, and it is shown that colored noise ha s a stabilizing effect with respect to white noise.