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Exponential stability in mean square of neutral stochastic differential difference equations. (English) Zbl 0956.60069

The authors discuss the exponential stability in mean square for a neutral stochastic differential difference equation of the form \[ d(x(t)- G(x(t- \tau)))= f(t, x(t), x(t-\tau))dt+ \sigma(t, x(t),x(t- \tau)) dw(t). \] In the case when \((t,x,y)= 0\) this neutral stochastic differential difference equation becomes a deterministic neutral differential difference equation \[ {d\over dt} (x(t)- G(x(t- \tau)))= f(t,x(t), x(t-\tau)). \] So as corollaries the authors obtain a number of useful criteria for this deterministic neutral differential difference equation to be exponentially stable. Several interesting examples are also given for illustration.

MSC:

60H20 Stochastic integral equations
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
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