Bychkov, A. S.; Khusainov, D. An exponential decay of solutions of neutral type stochastic equations. (English) Zbl 0834.60065 Random Oper. Stoch. Equ. 3, No. 3, 245-256 (1995). The paper considers stochastic difference-differential equations with constant delay \[ d \bigl( x(t) - Dx(t - \tau) \bigr) = \bigl( A_0 x(t) + A_1x(t - \tau) \bigr) + \bigl( B_0 x(t) + B_1 x(t - \tau) \bigr) dw (t), \] with \(A_0\), \(A_1\), \(B_0\), \(B_1\), \(D\) constant \(n \times n\)-matrices with \(|D |< 1\), \(\tau > 0\), \(w(t)\) a scalar standard Wiener process, and \(x(t)\) an \(n\)-vector. The paper gives conditions for exponential decay in mean square of the solutions and obtains rates of convergence. It also shows exponential decay of \(dE |x(t) |^2/dt\), where \(|\cdot |\) is the Euclidean norm. Reviewer: C.A.Braumann (Evora) Cited in 2 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations 93E15 Stochastic stability in control theory Keywords:stability; stochastic difference-differential equations; constant delay; exponential decay PDFBibTeX XMLCite \textit{A. S. Bychkov} and \textit{D. Khusainov}, Random Oper. Stoch. Equ. 3, No. 3, 245--256 (1995; Zbl 0834.60065) Full Text: DOI