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Zbl 0834.60065
Bychkov, A.S.; Khusainov, D.
(Khusajnov, D.)
An exponential decay of solutions of neutral type stochastic equations. (English)
[J] Random Oper. Stoch. Equ. 3, No.3, 245-256 (1995). ISSN 0926-6364; ISSN 1569-397X/e

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.
[C.A.Braumann (Evora)]

MSC 2000:: *60H10 Stochastic ordinary differential equations
60H20 Stochastic integral equations
93E15 Stochastic stability

Keywords: stability; stochastic difference-differential equations; constant delay; exponential decay

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Zentralblatt MATH Berlin [Germany]