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Razumikhin-type theorems on exponential stability of stochastic functional differential equations. (English) Zbl 0889.60062
An \(n\)-dimensional stochastic functional differential equation \[ dx(t) = f(t,x_t)dt +g(t,x_t)dw(t),\quad t\geq 0,\quad x_0 =\xi, \] is considered where \(w(t)\) is an \(m\)-dimensional Brownian motion with respect to a filtration \((\mathcal F_t)\),\(\;\xi \in C([-\tau ,0];R^n)\) is bounded and \(\mathcal F_0\)-measurable, \[ f:R_+ \times C([-\tau ,0];R^n)\to R^n,\;\;g:R_+ \times C([-\tau ,0];R^n)\to R^{n\times m} \] and \(x_t = \{x(t+\theta ): -\tau \leq \theta \leq 0\}\). Razumikhin-type theorems on \(p\)th moment exponential stability and almost sure exponential stability are proven and applied to stochastic delay equations and stochastically perturbed equations.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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