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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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