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An exponential decay of solutions of neutral type stochastic equations. (English) Zbl 0834.60065

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.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
93E15 Stochastic stability in control theory
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