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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

dx ( t ) - D x ( t - τ )=A 0 x (t) + A 1 x (t-τ)+B 0 x (t) + B 1 x (t-τ)dw(t),

with A 0 , A 1 , B 0 , B 1 , D constant n×n-matrices with |D|<1, τ>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 |·| is the Euclidean norm.

60H10Stochastic ordinary differential equations
60H20Stochastic integral equations
93E15Stochastic stability