On Asymptotic Stability of Nonlinear Stochastic Systems with Delay

Abstract

We consider the system of stochastic differential equations with delay and with non-autonomous nonlinear main part

Here h ≥ 0, [X]_t^{t - h} (s) = X(s), when s ⋲ [t - h, t], t > h, [X]_t^{t - h} (s) = ðœ™(s), when s ⋲ [-∞, 0], ðœ™(s) is a given initial process, X= (x₁, x₂,..., x_n)^T, u_i > 1 are rational numbers with odd numerators and denominators, w_t is a Wiener process. For different types of delays in coefficients f_i (t, [X]_t^{t - h}) and ðœŽ_i (t, [X]_t^{t - h}) we prove almost sure asymptotic stability of a trivial solution to the system (1) when ðœ™(s) ≡ 0.