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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.

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